Two-dimensional quantum antiferromagnets
Gr\'egoire {\sc Misguich}
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\begin{document}
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%\title{Two-dimensional quantum antiferromagnets}

\chapter*{TWO-DIMENSIONAL QUANTUM ANTIFERROMAGNETS}
\markboth{G. Misguich and C. Lhuillier}{Two-dimensional
quantum antiferromagnets}


\author{Gr\'egoire {\sc Misguich}}
\address{Service de Physique Th\'eorique, CEA/DSM/SPhT\\
Unit\'e de recherche associ\'ee au CNRS\\
CEA/Saclay 91191 Gif-sur-Yvette C\'edex, France\\
E-mail: gmisguich@cea.fr}

\author{Claire {\sc Lhuillier}}
\address{Laboratoire de Physique Th\'eorique des Liquides\\
Universit\'e P. et M. Curie and UMR 7600 of CNRS\\
Case 121, 4 Place Jussieu, 75252 Paris C\'edex, France\\
E-mail: claire.lhuillier@lptl.jussieu.fr}

\date{October 17$^{\rm th}$, 2003}
\setcounter{tocdepth}{4} % Pour avoir les subsubsections dans la TOC
\tableofcontents


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\section{Introduction}

In this review  we present some theoretical advances  in the field  of
quantum     magnetism   in    two-dimensional     (2D)   systems.    The
spin-$\frac{1}{2}$   next  neighbor 2-dimensional  Heisenberg model on
Bravais  lattices (square, triangular) is N\'eel ordered\footnote{This
generic kind of order,  with a macroscopic sublattice magnetization is
called in  the following {\it magnetic  } long-ranged order  (LRO), in
contrast  to  other ordered  phases   where  the  long-ranged  ordered
correlations concern    $S=0$    scalar   observables   (on    dimers,
quadrumers...)}     at $T=0$.\cite{manousakis91,bllp94}   Frustration,
small coordination  number, competition  between interactions can lead
to specific quantum phases without magnetic  long-ranged order.  Since
a  decade  this subject is  a highly  debated  issue  in  the field of
quantum magnetism.   It  was revived by  the   discovery of high-$T_c$
superconductivity in  the  doped   cuprates and   fueled   by numerous
experimental studies of 2D antiferromagnetic insulators.\cite{HFM2000}

Section~\ref{sec:j1j2}  is  devoted to   the  first academic  model of
quantum  frustration:  the spin-$\frac{1}{2}$ Heisenberg model  on the
square     lattice   with   first-   and second-neighbor  interactions
($J_1$--$J_2$ model).  This  model is one of the  most studied in  the
field and this  section  is a short guide   to the literature, with  a
special emphasis on the various methods used for this problem.

Section~\ref{sec:VBC} deals  with general properties  of  valence-bond
crystals (VBC) and   related  states, the  simplest  phase which  is 
commonly realized in frustrated spin systems without magnetic LRO.

In  section~\ref{sec:largeN}  we present  large-$N$ generalizations of
the Heisenberg model.  This approach was extensively developed by Read
and Sachdev from the   early 90's and has  been  the first to give  an
insight into  the  alternative between  VBC  and related phases, which
have long ranged  order in local  singlet patterns (whence the name of
crystals), and  resonating valence-bond  (RVB) spin-liquids (SL) which
are translationally invariant phases where the  quantum coherence is a
central issue.

Section~\ref{sec:QDM}  presents some   results  of  quantum dimer  models
(QDM).  These models are effective approaches to the quantum phases of
antiferromagnets which are  dominated by short-range valence-bonds (or
singlets).  They have received recently  some special attention and provide
useful insights onto the phenomenology of VBC and RVB SL.

In Section~\ref{sec:MSE} we review  some  results concerning models  with
multiple-spin    exchange (MSE)    (also    called    ring   exchange)
interactions. These interactions are  now recognized to be  present in
several physical systems  and appear to play  an important role in the
stabilization of RVB liquid ground-states.

The  last section  is devoted to  the Heisenberg  model  on the kagome
lattice  (and  related models).  Despite of  an  important activity on
this subject,  the   understanding of the  low-energy   physics of the
spin-$\frac{1}{2}$  kagome   antiferromagnet remains   a   challenging
problem and we discuss some of the important results and questions.

We should warn the readers that this review is quite ``inhomogeneous''
and   cannot,   of    course,     replace    textbooks.\cite{fradkinbook,auerbachbook,tsvelikbook,sachdevbook}
While some parts deal  with some rather recent works  (QDM or MSE  for
instance), some others are devoted to older results which we think are
still  of importance for   current research (beginning  of the  section
$J_1$--$J_2$,  large-$N$).  The final part  devoted to kagome reflects
our own views and some unpublished material
 on still unsettled issues. Some parts are intended to be
more pedagogical and concrete (QDM and  beginning of large-$N$ section)
while some others contain more qualitative discussions of the physical
issues (end of the section $J_1$--$J_2$, VBC, kagome).


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\section{$J_1$--$J_2$ model on the square lattice}
\label{sec:j1j2}

We consider the following Heisenberg model on the square lattice:
\begin{equation}
	{\cal H} =  2 J_1 \sum_{\langle ij\rangle}  {\vec S}_i\cdot{\vec S}_j
	+  2 J_2 \sum_{\langle\langle ij \rangle\rangle}  {\vec S}_i\cdot{\vec S}_j
\label{J1J2}
\end{equation}
where $\langle ij\rangle$ and $\langle\langle ij \rangle\rangle$ denote
pairs of nearest and next-nearest neighbors respectively.
Although  quite simple in appearance,  this spin model realizes several
interesting   phenomena which are    relevant   to a  large class   of
2D  frustrated    quantum magnets: classical degeneracy,
order by disorder, destruction of some long-ranged order by quantum
fluctuations, break down of the spin-wave expansion, opening of a spin
gap and (possibly ?)   spontaneous translation symmetry breaking, etc.
For this reason  we start  with  a general overview of  some important
results  concerning this  system.  We  will   focus on  the properties
related  to {\em frustration}.  A  review  on the non-frustrated model
($J_2=0$) can be found in Ref.\cite{manousakis91}

\subsection{Classical ground-state and spin-wave analysis}
It  is  easy  to find    {\em    some} classical ground-state  of    a
translation   invariant  Heisenberg  model  on   a Bravais lattice
because the energy can always be minimized by a planar helix
\begin{equation}
	\vec{S}_{\bf    r}=
		\vec{e}_1 \cos ({\bf q} \cdot  {\bf r})
	+   \vec{e}_2 \sin ({\bf q}\cdot {\bf r})
\end{equation}
provided  that the pitch  ${\bf q}$  minimizes  the Fourier  transform
$J({\bf q})$ of the coupling.\cite{lt47}  In the case of the $J_1$--$J_2$
model one has
\begin{equation}
	J({\bf q})=2J_1\left(\cos(q_x)+\cos(q_y)\right)
	+2J_2\left(\cos(q_x+q_y)+\cos(q_x-q_y)\right)
\end{equation}

\begin{itemize}

\item $J_2<0.5J_1$:
$J({\bf q})$ has a single minimum at $(\pi,\pi)$. It corresponds to the
``usual'' N\'eel state.

\item $J_2>0.5J_1$:
$J({\bf q})$ has two isolated minima at $(0,\pi)$ and $(\pi,0)$.  They
correspond to ferromagnetic   lines  (resp.  columns)  arranged  in an
antiferromagnetic   way.  These   states are    sometimes  called {\em
collinear}  (in {\em real} space).  From these planar helix states one
can build many other ground-states  by rotating globally all the spins
of one sublattice with  respect to the other.   Although this costs no
energy  for classical  spins at  zero  temperature, it is known (order
from disorder, see  below) that configurations  where both sublattices
have  their staggered magnetization {\em collinear  in spin space} are
selected by thermal or quantum fluctuations.

\item $J_2=0.5J_1$:
$J({\bf q})$  has  lines of minima  around  the edges of the  Brillouin
zone. At this point the classical ground-state  is highly degenerate :
We can write $\mathcal{H}={\rm cst}+J_2 \sum
\left(S_1+S_2+S_3+S_4\right)^2$ where the   sum runs over   all square
plaquettes and any  state where each plaquette  has a vanishing  total
spin minimizes the classical energy.
\end{itemize}

Even   at the  lowest    order   in $1/S$, zero-temperature    quantum
corrections to the  sublattice magnetization (order parameter) diverge
around $J_2=0.5J_1$  (Chandra    and  Dou{\c  c}ot\cite{cd88}).   Such
large-$S$ approximation usually tends to overestimate the stability of
magnetic phases, therefore this breakdown around $J_2\sim 0.5J_1$ is a
strong  evidence for the existence  of quantum  disordered phase(s) in
this region of parameter space.

\subsection{Order by disorder ($J_2>J_1/2$)}

The concept of ``order by disorder'' was introduced in 1980 by Villain
and  co-workers\cite{vbcc80} in the study  of a frustrated Ising model
on the square lattice. In this model the next neighbor couplings along
the rows are ferromagnetic as well as those on  the odd columns (named
$A$ in the following).  The couplings  on the even columns (named $B$)
are  antiferromagnetic.  At  $T=0$   the ground-state has   no average
magnetization   and  is   disordered.    This  changes when    thermal
fluctuations are introduced:    a $B$-chain sandwiched  between  two A
chains  with parallel  spins  has {\em lower  excitations}  than a $B$
chain between two $A$-chains with  anti-parallel spins.  This gives  a
larger Boltzmann weight  to ferrimagnetically ordered states.  Villain
{\it  et al.}    have  exactly  shown    that  the system is    indeed
ferrimagnetic at  low temperature.  They were  also able  to show that
site dilution (non-magnetic  sites) selects the  same ordered pattern,
whence the name  of ``order by disorder''.

A somewhat less drastic     phenomenon has been observed  in   quantum
systems.  It is the  selection    of particular long-ranged    ordered
quantum states among a larger  family of  ordered solutions which  are
classically degenerate at $T=0$. 
\footnote{In Villain's model
the  system is truly   disordered at $T=0$  and  an ordered solution is
entropically   selected   at finite   temperature.     In  the quantum
$J_1$--$J_2$ model above,  the  classical solutions can  adopt various
ordered patterns:  quantum fluctuations select  among these  patterns
the most ordered one, that is the situation  with the highest symmetry
and the smallest  degeneracy.   The ultimate  effect of these  quantum
fluctuations can be the destruction of the  N\'eel order in favor of a
fully quantum ground-state with $\mathcal{O}(1)$ degeneracy.}
Consider a  spin system in which the
molecular field  created by the  spins of one  sublattice on the other
cancels, which is  the case when $J_2>0.5J_1$. Shender\cite{shender82}
showed that  if  fluctuations are included,   the system  will  select
states in  which all spins are  collinear to each other.  This follows
from the fact that (moderate)  fluctuations at one site are orthogonal
to the mean value of the magnetization at that site and the system can
gain    some magnetic  exchange energy   by   making such fluctuations
coplanar on neighboring sites, that is to  making the spins collinear.
Such a selection of order by quantum  fluctuations (and dilution)
was discussed by Henley\cite{h89} and appears also quite straightly
in a spin-wave expansion.\cite{mdjr90}

This  selection of the   $(\pi, 0)$ or  $(0,\pi)$ order spontaneously
breaks a four-fold lattice symmetry.  An Ising order parameter is thus
generated.  It  takes two values  depending whether  the ferromagnetic
correlations are locally arranged  horizontally or vertically. Chandra
and co-workers\cite{ccl90a} have  studied this mechanism and predicted
the  existence of   a   finite  temperature   Ising  phase  transition
independent  of   the    subsequent  development  of     a  sublattice
magnetization.  This result has been questioned recently\cite{szosh03}
and  the  transition    has    not been observed   so      far in  the
spin-$\frac{1}{2}$ model.\cite{mbp03,szosh03}  It    has however  been
confirmed by some   recent    Monte Carlo simulations of     the  {\em
classical}   Heisenberg model.\cite{wm03}  Very  similar phenomena are
present in the   $J_1$--$J_2$  quantum Heisenberg  model  on  the {\em
triangular} lattice.\cite{jdgb90,cj92,k93,lblp95}

Melzi   {\it   el   a.}\cite{melzi00,melzi01}  have  studied a   quasi
2D spin-$\frac{1}{2}$ system  which  is believed to be  a
$J_1$--$J_2$ square  lattice Heisenberg antiferromagnet.  They found some
evidence (splitting   of NMR  lines)  for a   collinear ($(\pi,0)$  or
$(0,\pi)$)              magnetic        ordering.              Several
estimates\cite{melzi01,rosner02,rosner03,mbp03} indeed  point to $J_2>J_1$    in
this compound.

\subsection{Non-magnetic region ($J_2\simeq J_1/2$)}

Consider the two classical ``Ising  states'' corresponding to the wave
vectors $(\pi,\pi)$ and $(\pi,0)$.    They  can be taken  as   (crude)
variational states for the Hamiltonian Eq.~\ref{J1J2}.  Their energies
(per  site) are  $E_{\pi,\pi}=-J_1+J_2$   and  $E_{\pi,0}=-J_2$.    As
discussed above, these states cross at $J_2=\frac{1}{2}J_1$.  However,
one  can also consider  any  first-neighbor singlet (or  valence-bond)
covering of the lattice  as   another variational  state.  In such   a
completely dimerized state the  expectation  value of the  energy  per
site is $E_{dimer}=-\frac {3}{4}J_1$,   which is below the  two  Ising
states around $J_2\simeq J_1/2$.  Of course this very simple argument
does not  prove anything  since  ``dressing'' these classical  states
with quantum fluctuations (spin   flips in the N\'eel-like  states  or
valence-bond motions in the  dimerized wave-functions) will  lower the
energies of all these trial  states and it  is absolutely not clear
which  one may eventually win.   Nevertheless,  this shows in a simple
way why non-magnetic states  ({\it i.e} rotationally invariant or spin
singlet) such as dimerized states can be a route to minimize the energy in
a   frustrated   magnet.\footnote{Klein\cite{klein82}  introduced   a
general procedure to generate local and $SU(2)$ symmetric Hamiltonians
for which any first-neighbor dimerized state is an exact ground-state.
These Hamiltonians   are simply defined  as  sums of  projectors which
annihilate all dimer  coverings.   The Majumdar-Gosh\cite{mg69} chain
is the simplest example of a ``Klein model''.}


\subsubsection{Series expansions}

High-order  series  expansions  can   be   a powerful  technique   to
investigate frustrated quantum magnets. The general method to generate
zero-temperature perturbation expansions  in quantum many-body systems
was described  by Singh {\it et  al.}\cite{sgh88} and Gelfand {\it et
al.}\cite{gsh90}   For instance,  one   can  consider  the following
anisotropic model:
\begin{eqnarray}
	{\cal H}(\lambda) =  &&2 J_1 \sum_{<ij>}
	\left[
	S^z_i S^z_j
	+ \lambda \left(S^x_i S^x_j + S^y_i S^y_j \right)
	\right] \nonumber \\
	&& +  2 J_2 \sum_{<<ij>>}
	\left[
	S^z_i S^z_j
	+ \lambda \left(S^x_i S^x_j + S^y_i S^y_j \right)
	\right]
\end{eqnarray}
${\cal H}(\lambda=0)$ is a classical Ising model which ground-state is
known. The series expansion about  the Ising limit amounts to  compute
expectation values in the ground-state $\left|\lambda\right>$ of ${\cal
H}(\lambda)$ in powers of $\lambda$:
\begin{equation}
	\frac{
	\left<\lambda\right| {\hat O} \left|\lambda\right>
	}{
	\left<\lambda | \lambda\right>}=\sum_n a_n \lambda^n,
\end{equation}
(energy  gaps, dispersion  relations and susceptibilities  can also be
computed in the same approach).  The calculation of $a_n$ requires the
enumeration  and evaluation  of  the {\em connected  clusters} of size
$\sim n$, whose number grows exponentially with $n$.  Depending on the
quantity ${\hat  O}$  and  on the model,    orders from  7 to   20 can
typically be  obtained   on present computers.   The   series  is then
extrapolated to  $\lambda=1$  by  standard  Pad\'e, Dlog   Pad\'e   or
integrated differential approximations.  Such a series expansion about
the  Ising limit was done  by Weihong~{\it et al.}\cite{woh91} for the
first  neighbor   square-lattice   antiferromagnet.     Oitmaa     and
Weihong\cite{ow96}  extended  the  series  to the $J_1$--$J_2$  model,
where  each $a_n$ is   now  a polynomial   in  $J_1$ and  $J_2$.   The
disappearance of N\'eel order in the Heisenberg model manifests itself
by   a  vanishing sublattice magnetization as    well as some singular
behavior of  the series for  $\lambda_c<1$.   The results indicate the
absence of N\'eel long-ranged  order in the strongly frustrated region
$0.4\le  J_2/J_1\le  0.6$.   Such  an   expansion can  locate  with  a
reasonable accuracy  the phase boundary  of the N\'eel ordered regions
but provides no direct information  on the nature of the  non-magnetic
phase.

To study the model around $J_2\simeq  J_1/2$, several other expansions
have been carried  out.  A dimer  expansion about an exactly dimerized
model was       done       by  Gelfand~{\it    et    al.},\cite{gsh89}
Gelfand,\cite{gelfand90} Singh {\it   et   al.}\cite{swho99} and Kotov
{\it et al.}.\cite{kosw00}  In this approach $J_1$  and $J_2$ are set
to zero everywhere except on isolated bonds arranged in a columnar way
and all  the other  couplings are  treated perturbatively.   At zeroth
order  the  ground-state is simply  a  product of singlets.   In these
calculations the dimerized   phase remains stable  in the intermediate
region.  Singh  {\it et al.}\cite{swho99}  also performed a different
kind of zero-temperature series expansion.  They  started from a model
of isolated  4-spin plaquettes in order to  check a prediction made by
Zhitomirsky  and    Ueda\cite{zu96}   that   such   plaquettes  could
spontaneously  form around $J_2\simeq  J_1/2$ to produce a state which
is   invariant    under $\pi/4$  lattice   rotations.     Although the
ground-state energy they  obtained is very  close to the  one obtained
from   the  dimerized limit  (within   error bars of the extrapolation
procedure) they observed an instability in the plaquette scenario (the
triplet gap vanishes   before  reaching the isotropic   square-lattice
model)   which suggests  that plaquette  order  is not  the issue (the
analysis of  the exact numerical  spectra for 36  sites confirmed this
result\cite{cbps01}).


Sushkov {\it et al.}\cite{sow02}  (improved numerical results compared
to Ref.\cite{sow01})  computed the susceptibility  $\chi_D$ associated
with the appearance of columnar dimer  order in the $(\pi,\pi)$ N\'eel
phase  by   a series   expansion   about  the   Ising limit.   Such  a
susceptibility  seems    to       diverge  at    $J_2/J_1=g_{c1}\simeq
0.405\pm0.04$.   On the other hand  the  disappearance of the magnetic
LRO is   observed (through the N\'eel  order  parameter or through the
anisotropy   in   spin space    of   the  spin-spin  correlations)  at
$J_2/J_1=g_{c2}\simeq 0.39\pm0.02$.  This point could a priori be {\em
different} from $g_{c1}$.  In such a case the system would first break
the $\pi/4$ lattice rotation  symmetry at $g_{c1}$, while magnetic LRO
remains (gapless  spin waves).   Only  at $g_{c2}>g_{c1}$  the $SU(2)$
rotation  symmetry is restored and the  magnetic excitations acquire a
gap.  From  field  theoretical  arguments based  on  effective actions
valid close to  the critical points,  Sushkov {\it et al.}\cite{sow02}
argue   that  the proximity  (or  possible  equality)  of $g_{c1}$ and
$g_{c2}$ is a general  feature  in frustrated magnets which  originate
from the coupling of triplet and singlet excitations.

Sushkov  {\it   et al.}\cite{sow01}  computed  susceptibility $\chi_P$
associated  to plaquette  order by an   expansion around the dimerized
limit, assuming that  the system  has  columnar dimer  LRO. The result
shows a  divergence of  $\chi_P$ when  $J_2/J_1\to g_{c3}=0.5\pm0.02$.
From these results Sushkov {\it et al.} suggested that the translation
symmetry along the columns is broken  down at $g_{c3}$ (giving rise to
an   eight-fold  degenerate ground-state in  the  thermodynamic limit)
before  the $(\pi,0)-(0,\pi)$   magnetically ordered phase  appears at
$g_{c4}\simeq0.6$. This picture  is qualitatively consistent  with the
spin-spin    correlations computed in  a  $10\times10$   system with a
density matrix renormalization group (DMRG) algorithm.\cite{cls00}

Due to the relatively short series (typically of order 7) involved and
the uncertainties in the extrapolation procedures, such results should
be confirmed  by  other methods  but this succession  of quantum phase
transitions  represents a very interesting scenario.   We note that if
the model has a fully symmetric  liquid ground-state in some parameter
range, it should be difficult to  capture from series expansions about
limits where some lattice symmetries are explicitly broken.

\subsubsection{Exact diagonalizations}
Exact diagonalizations have a priori no bias, and were used very early
in      this   field.\cite{dm89,fkksrr90,pgbd91} Large-size  computations and
sophisticated finite size scaling  analysis are nevertheless needed to
extract   significant      information.        Schulz      {\it     et
al.}\cite{schulz} performed extensive exact  diagonalizations
of  the $J_1$--$J_2$ model  for system sizes up  to  $36$ sites.  They
analyzed the behavior of  several quantities such as structure factors
(N\'eel order  parameter), ground-state  energy, spin-wave  velocities
(obtained  from the   finite  size  corrections   to the  ground-state
energy), spin  stiffness and  uniform susceptibility.  Their analysis,
including  quantitative  comparisons   with  non-linear  sigma   model
predictions,\cite{nlsm}  concluded to the absence
of N\'eel long-ranged order in  the strongly frustrated region $0.4\le
J_2/J_1\le0.6$.   There, they   show  enhanced   columnar  dimer-dimer
correlations  as  well as  chiral ones but  the size  effects were not
clear  enough to discriminate between short  or  long-ranged order for
these order parameters.



\subsubsection{Quantum Monte Carlo}

Quantum Monte Carlo (QMC) methods have been extensively applied to the
$J_1$--$J_2$  model  in the low  frustration   regime giving an highly
accurate description of the  N\'eel phase (Sandvik\cite{s97} and Refs.
therein).  In the non-magnetic  and highly frustrated regime a  simple
QMC approach  is ineffective  due  to the  so-called sign problem. The
fixed node  approach is the  first answer  to this  problem: the exact
imaginary time propagator   $e^{-\tau\mathcal{H}}$ used to  filter out
the ground-state from a variational guess $|\psi_g\rangle$ is replaced
by  an    approximate  propagator,  which   has  the    same  nodes as
$|\psi_g\rangle$.  The quality of the result depends on the quality of
the nodal regions of $|\psi_g\rangle$.  Various schemes have been used
to    try to  go  beyond  this  limitation: stochastic reconfiguration
(Sorella\cite{s98}), eventually   associated  to  a   few    Lanczos
iterations.\cite{s01,cbps01} An alternative method has been devised by
du Croo de Jongh {\it et  al.},\cite{cls00} where the guiding function
is replaced  by the result of  a  DMRG calculation.\cite{w92,w93} Both
methods have their  own bias.  Using  the first of them, Capriotti and
Sorella\cite{cbps01}    concluded  that  for   $J_2/J_1\sim  0.45$   a
Gutzwiller-projected   BCS  wave-function $|p\;BCS\rangle$  was an
excellent guiding wave-function:
\begin{eqnarray}
	|p\;BCS\rangle&=&\hat\Pi\left|BCS\right>  \\
	|BCS\rangle&=&\exp{\left(
		\sum_{i,j}f_{i,j}
		c^\dag_{i\uparrow}c\dag_{j\downarrow}
	\right)}|0\rangle
\end{eqnarray}
where      $\left|0\right>$    is     the        fermion        vacuum,
$c^\dag_{i\uparrow}c\dag_{j\downarrow}$ creates  a     valence-bond on
sites  ($i,j$)  and   $\hat\Pi$  projects   out  states   with  double
occupancy. The pairing  amplitude $f_{i,j}$ (often called gap function
$\Delta_k$)\footnote{After the  Gutzwiller projection $\Delta_k$ is no
longer the observable gap.\cite{gl91}} is optimized with a Monte Carlo
algorithm in order to minimize the energy.  Capriotti and Sorella gave
convincing indications that their wave-function is quite accurate. The
best variational energies are obtained in the frustrated region with a
pairing amplitude    which     mixes   $d_{x^2-y^2}$ and      $d_{xy}$
symmetries. In particular it reproduces the correct nodal structure of
the ground-state in the frustrated region at least for moderate system
sizes where the  variational result can be  checked against the  exact
result.   This is a   subtle and non-trivial  information for  systems
which do not  obey the Marshall's sign rule  as this frustrated model.
They  concluded from these results   that  the system probably had  no
long-ranged order neither in dimer-dimer correlations nor in four-spin
plaquette correlations.  On  the other hand,  du Croo de Jongh {\it et
al.} argued in favor of columnar dimerized phase  which also break the
translation  symmetry along  the columns (plaquette-like  correlations
similar to those found by series expansions\cite{sow01}).

The comparison of the results of these different approaches shows that
this  problem  remains a  very challenging  one.    The  model in  the
frustrated regime  is probably never very  far from a quantum critical
point and in these conditions none of the available methods seems able
to discriminate between  a VBC with tiny  gaps both in the singlet and
triplet  sectors, a critical phase  with a quasi  order  in dimers and
gapless singlet  excitations, or a true SL  with gaps in any sector of
spin but no long ranged order in any observable. As we will explain in
the following sections some other frustrated models are happily deeper
in the strong  coupling regime  and  exhibit quantum phases which  are
easier to characterize.

%______________________________________________________________________________

\section{Valence-bond crystals}\label{sec:VBC}
\setcounter{footnote}{0}
%______________________________________________________________________________
\subsection{Definitions}
Among the different quantum  solutions to overcome frustration the VBC
is  the simplest scenario.   In this phase,  neighboring spins arrange
themselves in  a regular pattern  of singlets: dimers,\footnote{whence
the  name   Spin Peierls  phase sometimes  given   to the  VBC phase.} 
quadrumers or 2$n$-mers  $S=0$ plaquettes.  The  stability of this phase
comes from the extreme stability of  small $S=0$ clusters (recall that
the  energy  of a  singlet  of two spins  $\frac{1}{2}$  is -3/4 to be
compared to the energy of two classical (or Ising) spins which is only
-1/4), and eventually from the  fact that frustrated bonds between two
different singlets do not contribute to the total energy.

In a  VBC phase there is no  $SU(2)$ symmetry breaking, no long-ranged
order in spin-spin correlations,  but long-ranged order in dimer-dimer
or  larger singlet units.  Except  at  a quantum  critical point,  all
excitations  of a VBC are gapped.   Depending on the lattice geometry,
such a wave  function can  spontaneously  break some  lattice symmetry
({\em  spontaneous VBC}) or can  remain fully symmetric ({\em explicit
VBC}).   In a  strict sense,  the name  VBC  should be  reserved  for
systems with a spontaneous lattice  symmetry breaking. However,  since
these two kinds  of systems share   many similarities we  will discuss
both in this section. 

When the Hamiltonian has some  inequivalent bonds and an integer  spin
in the unit cell (even number  of spin-$\frac{1}{2}$ for instance) the
system can take  full advantage of the  strong bonds and  minimize the
effects of the frustrating ones.  In that case  the {\em explicit} VBC
is  the ``natural'' strong coupling solution.   One can build a simple
Hamiltonian in which the bonds  which are not  occupied by the singlet
objects  are   turned off.  The   resulting  model is a   set of small
decoupled clusters (dimers  or larger plaquettes) and the ground-state
is a trivial product  of singlets.  Importantly, this strong  coupling
limit has  the same lattice symmetry  as the original one.  Going back
to   the  original Hamiltonian {\em  no   quantum  phase transition is
encountered when  going from  the trivial  singlet  product up to real
interacting ground-state}. Models with an half-odd-integer spin in the
unit cell cannot realize a VBC unless they {\em spontaneously} enlarge
their unit  cell.   In these  situations  there is no   unique elected
position for the 2$n$-mers and a symmetry  breaking must take place in
order to form a VBC.  Examples of these two kinds of VBC will be given
below.


%______________________________________________________________________________

\subsection{
One-dimensional and quasi one-dimensional examples (spin-$\frac{1}{2}$
systems)}


One of the  simplest example of  (spontaneous) VBC is observed in  the
$J_1$--$J_2$ model  on  the chain for   $J_2/J_1 >  (J_2/J_1  )_c \sim
0.24$.\cite{mg69,ss81,h82,a89,ys97}  For $J_2/J_1=0.5$   the    doubly
degenerate   ground-states        are  exact    products            of
dimers:\cite{mg69,auerbachbook}
\begin{equation}
|MG_{\pm}> =\prod_{n=1}^{N/2} |(2n, 2n \pm 1)>.
\label{mg} 
\end{equation}
Here   and in the  following we  call ``dimer''  a  pair of spins in a
singlet state, and note it:
\begin{equation}
|(i,j)> =\frac{1}{\sqrt2} \left[ |i, +>|j, ->  - |i, -> |j, +>\right].
\label{dimere} 
\end{equation}

For  all $J_2/J_1 >  (J_2/J_1 )_c $,
the ground-states are products
of dimers, dressed by fluctuations of valence bonds, dimer long-ranged
order persists in all the range of  parameters.  This model has gapful
excitations  which  can be   described     as pairs   of    scattering
spin-$\frac{1}{2}$       solitons      separating    the     two exact
ground-states\cite{ss81} (these fractionalized      excitations    are
specific of the 1D chain).

The  Heisenberg chain  with alternating  strong and  weak  bonds (Spin
Peierls  instability),  has indeed  a  unique  ground-state where  the
dimers  are mainly  located on  the strong  bonds. In  that  case, the
spin-$\frac{1}{2}$   excitations  are   confined  by   the  underlying
potential    and   the   true    excitations   are    gapful   integer
magnons~(\cite{nt97a,asrp99} and refs. therein). It is an explicit VBC.

A    two-leg ladder with   AF rung   exchange has   also a  unique VBC
ground-state  and  gapped magnons as  excitations.\cite{dr96}  On  the
other hand   Nersesyan  and Tsvelik\cite{nt97,km98} have   proposed an
example  of frustrated  ladder,  with  a   spontaneously dimerized
ground-state, and gapful  excitations.  Excitations of this last model
are identified as pairs of singlet and triplet domain walls connecting
the two ground-states, they form a continuum.

As  can be seen  from this  rapid and  non exhaustive enumeration, VBC
ground-states are relatively  frequent in  frustrated  one-dimensional
spin-$\frac{1}{2}$  models.  All   these systems are   gapful  but the
excitations could be of different nature emerging as modes (associated
to integer spin excitations) or continuum of pairs of excitations that
could be  fractionalized   (it  is then specific   of  one dimensional
systems) or not.
%______________________________________________________________________________

\subsection{Valence Bond Solids}
\label{ssec:VBS}
The  VBS  wave-function was introduced  by  Affleck, Kennedy, Lieb and
Tasaki (AKLT).\cite{aklt87,aklt88} It  can be constructed whenever the
spin $S$ on a site  is a multiple  of one half the coordination number
$z$: $2S=0$ mod $z$.   Let us consider the  simplest case $2S=z$.   In
that case the local spin $S$ can  be seen as the symmetric combination
of $2S$ (fictitious) spin-$\frac{1}{2}$.    Now on  each bond  of  the
lattice    one can    make    a  singlet    between   two   fictitious
spins-$\frac{1}{2}$.  Such a  product  of singlets does not  belong to
the  physical Hilbert space  of the  original spin-$S$  model but to a
much larger space.  The VBS wave-function is defined as the projection
of the singlet-product state onto the physical space.  This projection
amounts  to symmetrize (for all  lattice sites) the wave-function with
respect to the fictitious spins to force them into a physical spin-$S$
state. A VBS can be viewed as  an {\em explicit  VBC of the fictitious
spins}.  Simple  Hamiltonians with  short ranged and $SU(2)$-symmetric
interactions for   which  the VBS is  an  exact   ground-state can  be
constructed (sum of projectors\cite{aklt87,aklt88}).  By  construction
the VBS wave-function  is a  spin  singlet and {\em  breaks no lattice
symmetry}.  By extension  we may say that a   system is in  a VBS {\em
phase} if  its ground-state can be  adiabatically transformed into the
VBS wave-function without  crossing a phase  transition.  As the  VBC,
models in  the VBS phase have a  gap  to all excitations\footnote{This
may however not always  be  true when  the coordination number  of the
lattice is large.\cite{aklt87} In such cases  the VBS wave-function is
still a spin  singlet but has  long-ranged spin-spin correlations.  We
do  not consider such  cases  here.}   but  their wave-functions   are
slightly more   complex and their order parameter   is non-local.  The
order of VBS is associated to long-ranged singlet-singlet correlations
in the {\em fictitious spins}.  Expressing such observable in terms of
the physical  spins leads to a  non-local quantity  called {\em string
order parameter}.\cite{nr89,kt92} Contrary to  explicit VBC,  VBS have
fractionalized degrees of freedom at  the edges of the system with
open boundary conditions.  These are simply associated to the unpaired
fictitious  spins.  To our  knowledge  these properties  have not been
explored in  quantum 2D systems.

The spin-1 Heisenberg chain is the prototype of VBS in 1D.\footnote{In
1D, some authors call ``Haldane systems''  all the spin-gapped phases,
whatever  their true nature: VBC  or VBS.}  Such a  state has a unique
ground-state, a  gap in the   excitations and exponentially decreasing
spin-spin and dimer-dimer\cite{j02}  correlations. See  the chapter by
P.~Lecheminant in this volume for more details  about the VBS phase of
the spin-1 chain.

A spin-$\frac{3}{2}$ specific $SU(2)$-invariant model on the honeycomb
lattice\cite{aklt87,aklt88} is another  example of 2D VBS.  The spin-1
Heisenberg  model on  the kagome  lattice   was proposed  to realize a
VBS-like  ground-state\cite{h00}  in  which  singlets  form   on every
hexagon without any spontaneous symmetry breaking
(hexagonal singlet solid).\footnote{Each kagome   site belongs to  two
hexagons.      Each   physical spin-1 can     be     split  into   two
spin-$\frac{1}{2}$, each of them being  involved in the formation of a
singlet on one neighboring  hexagon.}  A similar approach  was carried
out for  the spin-1  pyrochlore  Heisenberg model.\cite{yk00}  In that
case a  lattice distortion was invoked  to lift the degeneracy between
the two singlet states of each tetrahedron.


%______________________________________________________________________________

\subsection{Two-dimensional examples of VBC}
\subsubsection{Without spontaneous lattice symmetry breaking}


Two spin-$\frac{1}{2}$ experimental examples of 2D (explicit) VBC have
recently                   attracted                        attention:
CaV$_4$O$_9$\cite{CaV4O9,uksl96,am96b,tku96,sr96,szksu96,wgsoh97}
and
SrCu$_2$(BO$_3$)$_2$.\cite{k99,nkoum99,mu99,k00,kk00,msku00,tmu01,lws02,kthb02,mu03}
Both  of  them have a   lattice derived from  the   square lattice: in
CaV$_4$O$_9$ the  V$^{4+}$ ions are on a  1/5  depleted square  lattice and
SrCu$_2$(BO$_3$)$_2$ is   an illustration  of  the  Shastry Sutherland
model\cite{ss81a} . A lattice embedding the main couplings of these two physical problems
is drawn in  Fig.~\ref{qss}.   Interactions   are of  the
Heisenberg type.\footnote{Small Dzyaloshinsky-Moriya interactions have
been identified in SrCu$_2$(BO$_3$)$_2$.\cite{cz02}}

\begin{figure} 
\begin{center}
\includegraphics[height=4cm]{QSS}
\end{center}
\caption[99]{
The $\frac{1}{5}$-depleted lattice and the
Shastry  Sutherland  lattice.  The
strong  bonds  of the  Shastry Sutherland  model  are the   bonds $J'$
(dotted dashed lines): they can  accommodate orthogonal dimers  ($J=K$
acts as  a perturbation  in  the  real SrCu$_2$(BO$_3$)$_2$).    The
lattice formed by the  strongest exchanges in CaV$_4$O$_9$ is obtained
with $J=0$.    The phase diagram  of   this model\cite{lws02} contains
collinear N\'eel phases, dimer and 4-spin plaquette VBC.}
\label{qss}
\end{figure}

VBC are obvious  ground-states in the strong  coupling limits of each
problem.

The  exact ground-state of the Shastry  Sutherland model is built from
singlets   on   the  $J'$  bonds.\cite{ss81a,mu99}   For  $J'/J\gtrsim
1.55\pm0.05 $ the  model has a gap in  the spectrum of excitations and
is in a dimerized VBC phase.
\footnote{
Consider a decomposition  of the  Shastry-Sutherland lattice as  edge-
and  corner-sharing  triangles.    For  $J'=2J$  the Hamiltonian    is
accordingly written  as a  sum of $J(\vec{S}_1+\vec{S}_2+\vec{S}_3)^2$
terms for each  triangle ($J'$-bonds are  shared by two triangles) and
each such  term is   minimized by the   dimerized state.   This  shows
rigorously that  the dimerized  state is  the ground-state as  soon as
$J'/J\gtrsim 2$.}  For $J'/J\lesssim 1.15\pm0.05$ the system is in the
$(\pi,\pi)$    N\'eel  state  of   the   square   lattice~(results  of
zero-temperature series expansion\cite{zoh02}).  The possibility of an
intermediate phase,  possibly with helical  short-ranged correlations,
has  been actively discussed in the literature.\cite{am96a,kk00,cms01}
S.~Miyahara and  K.~Ueda have recently written  a review of the theory
of    the    orthogonal     dimer    Heisenberg   spin  model      for
SrCu$_2$(BO$_3$)$_2$.\cite{mu03}

The $\frac{1}{5}$-depleted Heisenberg square lattice model ($J=0$) has
been studied  as   a  function of   the  ratio  of the  two  different
couplings: bonds within   a plaquette ($K$)   and dimer bonds   ($J'$)
between   plaquettes.     At isotropic   coupling   ($J'=K$) collinear
long-ranged N\'eel order survives  the depletion, (the decrease in the
order parameter is roughly $50\%$\cite{tku96}).   A small unbalance in
couplings drives the system either  in a 4-spin plaquette VBC ($K>J'$)
or in a dimer  VBC ($K<J'$). Both  (explicit) VBC  phases have a  spin
gap.  A  recent generalization  of these  models  by L\"auchli {\em et al}
encompasses  both the
$\frac{1}{5}$-depleted Heisenberg square lattice model and the Shastry
Sutherland model\cite{lws02} (see  Fig.~\ref{qss}).   Its phase diagram
exhibits collinear   N\'eel   phases ($(\pi,  \pi)$   or  $(0,  \pi)$)
separated from the  VBC   phases by second  order  phase  transitions.
Transition between the two  VBC phases which have different symmetries
occurs via  a  first order phase transition.\footnote{A  recent $Sp(N)$
study of the Shastry Sutherland model\cite{cms01} suggests that a spin
liquid phase with deconfined spinons might appear in  such a model. No
evidence of such a phase has emerged from the $SU(2)$ studies.}

Excitations in these models come from the  promotion of local singlets
to triplet excitations.  In 2D the ordered dimer background provides a
confining   force  for  the   spin-$\frac{1}{2}$ excitations.  Indeed,
separating two  unpaired spins  (that  is two {\em  spinons})  creates
disruption in the ordered  pattern all the  way from the first  to the
second.  The  energy cost is  thus  proportional to the length  of the
string of defaults and  both spin-$\frac{1}{2}$ excitations remains in
fact confined.   Only integer spin  excitations are expected.   On the
other hand in   these strongly coupled models  single-triplet hoppings
can be difficult and correlated motions might be important, leading to
a  large   zoology of  excited modes   (see Ref.\cite{tmu01} and Refs.
therein).  This potential frustration of the triplet motion favors the
appearance   of     magnetization      plateaus     in             VBC
models.\cite{fo99,k99,mt00,mt00b,mu00,mjg01}  This  aspect was briefly
discussed in the lecture notes published by the authors.\cite{lm02}

%______________________________________________________________________________

\subsubsection{With spontaneous lattice symmetry breaking}

In  the previous models  the (explicit)  VBC phases  do not  break any
lattice  symmetry.  They can be directly  related  to the geometry and
relative strength of the couplings.  In more symmetric situations with
frustration, spontaneously  symmetry breaking VBC  can appear as a way
to overcome  this frustration by  taking full advantage of the quantum
fluctuations.  This is probably the  case in the $J_1$--$J_3$ model on
the square lattice for intermediate  $J_3/J_1 \sim 0.6$,\cite{ll96} in
the  $J_1$--$J_2$ model   on  the hexagonal  lattice  for intermediate
$J_2/J_1\sim  0.4$,\cite{fsl01}  and in the  Heisenberg  model  on the
checker-board    lattice.\cite{canals02,fsl03,sfl02,bh02,baa03,tsma03} In the
two   first cases  the  ground-states are   dressed  columnar VBC   of
dimers. Translation and $C_4$  (resp.   $C_3$ only) symmetries of  the
lattice are spontaneously  broken.  The ground-state is   4 (resp.  3)
times  degenerate.  Spin-spin correlations decrease exponentially with
the  system size.  All  excitations   are  gapped.  Contrary to    the
$J_1$--$J_2$    model     on     the     square     lattice,     exact
diagonalizations\cite{ll96,fsl01,fsl03} give a rather  straightforward
information on  these systems where the  correlation lengths are small enough
(far enough from  the critical points which limit  the boundary of the
VBC phases).

The spin-$\frac{1}{2}$ Heisenberg model  on the checker board lattice,
which  can  also  be  seen  as  a planar  lattice  of  corner sharing
tetrahedrons (see Fig.~\ref{ckb}),  has received the largest attention
for different reasons.\cite{mts01,pc02,canals02,bh02,sfl02,fsl03,baa03,tsma03}
\begin{figure}
\begin{center}
\includegraphics[height=4cm]{checkerboard2}
\hspace*{0.5cm}
\includegraphics[height=4cm]{checkerboard}
\end{center}
\caption[99]{Left:  the checkerboard  lattice.  The spins  sit at  the
vertices shown  by bullets, all couplings are  identical, ${\bf u}_1$,
${\bf u}_2$  are the unit vectors  of the Bravais  lattice. Right: the
ground-states of the Heisenberg model on the checker board lattice are
translational  symmetry  breaking  VBC  of 4-spin  plaquettes  on  the
uncrossed squares.}
\label{ckb}
\end{figure}
The problem  has   classically  a  continuous  local degeneracy:   the
Hamiltonian can  be rewritten as the sum  of the squares of  the total
spin of each tetrahedron, and every  configuration with a zero spin on
each tetrahedron is a  ground-state.  Classically this problem  shares
this  property      with   the Heisenberg   model     on   the kagome,
pyrochlore\cite{mc98,mc98a},    garnet\cite{pp01}   and     pyrochlore
slab\cite{ka02}  lattices (these lattices    made  of corner   sharing
``simplexes'' with  2, 3  or 4  spins  each were  dubbed ``bisimplex''
lattices by Henley\cite{henley2000}).

The quantum spin-$\frac{1}{2}$
antiferromagnet  on the  kagome  lattice has been  found  to  be quite
specific with a  small gap (if any) toward  triplet excitations and an
anomalous density  of  gapless  low-lying singlets   excitations  (see
\S\ref{sec:kagome}).

The quantum  scenario on the  checkerboard lattice is quite different.
The ground-state is a (dressed) product  of 4-spin $S=0$ plaquettes on
uncrossed squares: this  state  breaks translational symmetry  but not
$C_4$  (the  point group being  defined  at  the  center of  an  empty
plaquette).  It has a two fold degeneracy  in the thermodynamic limit:
this is easily seen in the  symmetries and finite  size scaling of the
low  lying   levels of  exact spectra.\cite{fsl03}  The  choice of the
4-spin $S=0$  states on the  uncrossed squares corresponds to the most
stable local configuration.   The product of  $S=0$ quadrumers is only
weakly renormalized in the exact ground-state.\cite{fsl03}


Excitations of this model  have been studied in different  approaches,
exact diagonalizations\cite{f03}, series  expansions,\cite{bh02}  real
space renormalization transformation.\cite{baa03} All the  excitations
(singlet and  triplet) are gapped.   The triplet excitations originate
from the triplet excitation of an uncrossed plaquette, they are weakly
dispersive with  a  large  gap.   The singlet   excitations  cannot be
described so simply: from exact diagonalizations  data one can suspect
that some  of these  excitations come  from the  reorganization of two
adjacent triplet  on crossed squares.   The Contractor Renormalization
(CORE)  method  of Berg  {\it et  al.}\cite{baa03} on   the other hand
suggests that  these  excitations are  domain  walls between   the two
translated plaquette-VBC ground-states.

%______________________________________________________________________________

\subsection{Methods}

Spin    waves\cite{uksl96} and  Schwinger bosons\cite{am96a,am96b} are
simple  approximations  to   study  the  phase  diagram  of  a quantum
frustrated  magnet.  But the  first  method only gives an  approximate
knowledge  on the range of  existence of the N\'eel  phases, and it is
rather difficult  to  include the effect  of  fluctuations  beyond the
mean-field  approximation within  the Schwinger-Boson  formulation. As
discussed   in   the   section     devoted to  large-$N$    approaches
(\S\ref{sec:largeN}), spontaneous  lattice symmetry  breaking are very
likely  to be due to  topological effects (Berry  phase of instantons)
which cannot be   captured  by the mean-field  state.    This probably
explains why no spontaneous VBC has ever been found (to our knowledge)
in  Schwinger-Boson   calculations.\footnote{It  seems   however  that
spontaneous VBC naturally arise in large-$N$ approaches with fermionic
representation of  $SU(N)$ when $1/N$  corrections are considered (see
section
\ref{ssec:spin2QDM}).}  For  unfrustrated problems,  as the Heisenberg
model on    the  $\frac{1}{5}$-depleted   square  lattice,\cite{tku96}
QMC  is considered  as  the   method which  can  give
benchmarks for other approaches.

Although  VBC  are   naturally  described     with   the help       of
spin-$\frac{1}{2}$ valence-bonds, the appearance of such states in the
low-spin  limit can  sometimes   be anticipated  from  an  appropriate
large-$S$ approach.   An example is given in  the work of Tchernyshyov
{\it   et     al.}\cite{tsma03} (see  also   a    previous    paper by
Henley\cite{henley2000})  on the  checkerboard  Heisenberg  model.  In
this  model, when both   couplings (square lattice bonds and  diagonal
bonds)  are    equal,     ground-state  has    a   continuous    local
degeneracy. However  leading $1/S$ corrections select collinear states
out  of  this    huge  manifold.\cite{henley2000} There   remains   an
exponential number of such  states and they  do not have  any magnetic
order.  However    they exhibit long-ranged  {\em   bond}  order and a
spontaneous symmetry  breaking\cite{tsma03} which is  analogous to the
one observed numerically in the spin-$\frac{1}{2}$ case.

For frustrated problems, exact diagonalizations can be useful tools in
situations where the system  is not too  close from a critical  point,
that is when  the correlations length is  not  too large.   Successful
applications of   exact  diagonalizations  methods  to   2D Heisenberg
magnets realizing a VBC include studies of  the $J_1$--$J_3$ model on the
square  lattice,\cite{ll96}    $J_1$--$J_2$  model   on    the  hexagonal
lattice,\cite{fsl01}     Heisenberg model       on    the checkerboard
lattice.\cite{fsl03}    In such  situations    analysis of  the  quasi
degeneracy of the  low-lying levels    of  the spectra and  of   their
finite-size scaling gives an unbiased  and direct informations on  the
symmetry breakings in the thermodynamic limit.\cite{fsl03} However the
boundaries of  the phases  and  the quantum critical  points cannot be
 accurately determined  with  this method.   The  series expansions
described in the previous section appears to be a powerful approach to
determine  phase  boundaries.\cite{gwsoh96,wgsoh97} If  the  spin-spin
correlation length is   not very  short, as it   is  the case in   the
$J_1$--$J_2$    model       on        the   square       lattice      for
$J_2/J_1\sim0.5$,\cite{cbps01}  it  is very  difficult  to decide from
exact diagonalizations between a VBC, a critical  phase or an RVB SL.

Concerning excitations, exact  diagonalizations give the gaps in  each
sector and provide a crude approximation of the dispersion laws of the
first  excitations. The large-scale nature  of the excitations (as for
examples  domain  walls  excitations) can  escape   this method.   The
semi-analytical  approaches which  can be  used  for the study  of the
excitations of the    VBC, all use  as  a  basic departure point   the
excitations of a local cluster  of spins conveniently renormalized  by
perturbation\cite{szksu96,bh02}       or    effective  renormalization
technique.\cite{baa03}    Contrary  to   exact  diagonalizations these
methods are not limited by finite-size effects  but the results can be
biased by the departure point.\cite{bh02,baa03}

%______________________________________________________________________________

\subsection{Summary of the properties of VBC phases}
The   generic  features of  VBC   (whatever the dimensionality  of the
lattice) are:

\begin{itemize}
	\item  A    spin gap,   no   $SU(2)$  symmetry   breaking  and
	short-ranged spin-spin correlations,
	
	\item Long-ranged order   in dimer-dimer  and/or larger  $S=0$
	plaquettes.  The coupling of this order to lattice distortions
	is probable in experimental realizations of spontaneous VBC.

	\item In {\em spontaneous}   VBC  phases the  ground-state  is
	degenerate.  From  the theoretical point  of view the discrete
	symmetry of the order parameter of the VBC which spontaneously
	breaks  a   lattice symmetry  may   give  birth   to a  finite
	temperature Ising-like  transition.\cite{ccl90a}  Simultaneity
	between this transition  and a  possible structural transition
	is  likely when  the  couplings of  the  spins  to the lattice
	degrees of freedom (phonons) is considered.\cite{bm02}

	\item VBC have gapped excitations, in the $S=0$ sector as well
	as  in other $S$  sectors.  A wide zoology  of  modes is to be
	expected  as well  as  continua  associated to  multi-particle
	excitations or  scattering of domain walls (in   the case of a
	spontaneous symmetry  breaking of  the ground-state).   In two
	dimensions  all   these excitations  have  integer  spins (the
	ordered   back-ground   inducing     a  confinement   of   the
	spin-$\frac{1}{2}$ excitations)
\end{itemize}

 Frustration on the square  lattice  or more  generally  on bipartite
  lattices is often  overcome by
  VBC phases.  No  theorem  forbids   the appearance  of VBC   in
triangular  geometries but  there is up  to  now  no examples of  such
phases   in pure  spin-$\frac{1}{2}$  models  (in   Sec.~\ref{sec:QDM}
examples will be given within the framework of quantum dimer models).

It     has  been advocated     in    the  large-$N$    approaches (see
section~\ref{sec:largeN}) that, at least  in two dimensions, collinear
spin-spin correlations  generically lead  to    VBC or VBS and    only
non-collinear spin-spin correlations  can give birth   to RVB SL  with
unconfined spin-$\frac{1}{2}$  excitations.  The present  knowledge of
$SU(2)$ phase diagrams supports this prediction.  The VBC found so far
numerically in $SU(2)$   spin  models appear   to  be in  regions   of
parameter space where the  spin-spin correlations are characterized by
some   short-ranged   collinear order in     the large-$S$ limit.  The
$J_1$--$J_2$   model on  the  honeycomb    lattice   has a   classical
incommensurate phase  in the regime of  high frustration and there are
some evidences that  in the quantum phase  diagram the collinear phase
is separated from the columnar VBC phase  by a RVB SL.\cite{fsl01} The
multiple-spin exchange  (MSE) model on  the triangular lattice is also
believed to be  a RVB SL\cite{mlbw99}  and the corresponding classical
ground-states   generically  have  non-coplanar spin   configurations.
Capriotti {\it et al.}\cite{cbps01} argued that the spin-$\frac{1}{2}$
square   lattice $J_1$-$J_2$ model could be   a RVB SL.  If confirmed,
this would be the first counter-example to  the general rule explained
above (The  Heisenberg  model on  the pyrochlore  lattice  might be an
other counter-example\cite{baa03}).

%_____________________________________________________________________

\section{Large-$N$ methods}\label{sec:largeN}
%_____________________________________________________________________

Introduced  by   Affleck,\cite{a85}  Affleck  and  Marston,\cite{am88}
Arovas       and      Auerbach\cite{aa88}      and       Read      and
Sachdev\cite{rs89,rs90,rs91} in the  context of spin models, large-$N$
approaches    are   powerful    methods    to   investigate    quantum
antiferromagnets.  When $N$ is taken  to infinity many of these models
can be solved  by saddle point methods and  finite-$N$ corrections can
be, at least in principle, explored in a controlled way.  A success of
these  approaches is  that they  can describe  the phenomenology  of a
large variety of phases encountered  in quantum magnets : Magnetic LRO
(possibly  with  order  by  disorder  selection)  as  well  as  phases
dominated  by short-ranged  valence-bonds: VBC,  VBS and  RVB liquids.
One       crucial       result       (due      to       Read       and
Sachdev\cite{rs89,rs90,rs91,sr91,s93})  concerning  these three  later
phases  is  that  the  analysis  of  finite-$N$  corrections  to  some
large-$N$ formulations  ($Sp(N)$ for  instance, see below)  provides a
general criterion to  decide which of these three  phase should appear
in a given model.

This criterion is the following in 2D: if the (large-$N$ equivalent of
the) ``spin'' $S$ at each site matches the lattice coordination number
$z$ by $2S=0$~mod~$z$ a VBS is to be expected.  If  it is not the case
(as for a spin-$\frac{1}{2}$  model on the  square lattice) one should
look at the local spin-spin correlations.  If  they are reminiscent of
a   collinear   order, a VBC   with  spontaneous  translation symmetry
breaking is  expected whereas  non-collinear short-ranged  correlation
generically  give rise to a RVB  phase without any broken symmetry and
deconfined spinon excitations.  These results are of course based on a
large-$N$  generalization of the original  spin model and  there is no
guaranty at all that these rules should always  apply to $SU(2)$ models.
To our knowledge they have however not  been manifestly found in error
up  to now.\footnote{The   spin-$\frac{1}{2}$  kagome  antiferromagnet
might however be such an  example.  See section~\ref{sec:kagome}.}  In
the following we will  present some of  the important reasoning  steps
leading to this result.


\subsection{Bond variables}
The $SU(2)$  algebra of a spin $S$ at one site
can be  represented by $2$ species of  particles $a^\dag_\sigma$ (with
$\sigma=\uparrow,\downarrow$),  provided  that  the  total  number  of
particles on one site is constrained to be $a^\dag_\uparrow a_\uparrow
+  a^\dag_\downarrow  a_\downarrow=2S$.   The raising  operator  $S^+$
(resp.  $S^-$) is simply represented by $a^\dag_\uparrow a_\downarrow$
(resp.  $a^\dag_\downarrow a_\uparrow$). These particles can be chosen
to   be   fermions   (Abrikosov   fermions)   or   bosons   (Schwinger
bosons). These particles  carry a magnetization $\pm\frac{1}{2}$ since
$S^z=\frac{1}{2}(a^\dag_\uparrow           a_\uparrow-a^\dag_\downarrow
a_\downarrow)$.  For this reason  they are often called {\em spinons}.
The  Heisenberg  interaction  is   a  quartic  interaction  for  these
particles:
\begin{equation}
	{\vec S}_i\cdot{\vec S}_j =
	S^2 - \frac{1}{2} A_{ij}^\dagger A_{ij}
	\label{eq:SSAij}
\end{equation}
with the bond operator $A_{ij}$ defined by:
\begin{equation}
	A_{ij} = a_{j\downarrow} a_{i\uparrow}
	- a_{j\uparrow} a_{i\downarrow}
	\label{eq:ASU2}
\end{equation}
Acting on the  vacuum, $A_{ij}^\dagger$ creates a spin  singlet on the
bond $(ij)$. Physically $A_{ij}^\dagger A_{ij}$ measures the number of
singlets  on   that  bond   and  Eq.~\ref{eq:SSAij}  shows   that  the
antiferromagnetic Heisenberg  interaction just tries  to maximize that
number.

The idea of large-$N$ methods is to generalize the $SU(2)$ symmetry of
the spin$-S$ algebra to a larger group $SU(N)$ (or $Sp(N)$) by letting
the index  $\sigma$ go from  $1$ to  $N$  (or $2N$).  The $SU(N)$  (or
$Sp(N)$) generalization of the Heisenberg  model is solved by a saddle
point calculation of the action.  The $N=\infty$ limit is very similar
to a mean-field decoupling    of  the four-body interaction   of   the
physical    $SU(2)$       model:       $A_{ij}^\dagger    A_{ij}\simeq
A_{ij}^\dagger\left<A_{ij}\right>+\left<A_{ij}^\dagger\right>A_{ij}-\left|\left<A_{ij}\right>\right|^2$.

\subsection{$SU(N)$}
The generalization of the  Heisenberg model to  such a  symmetry group
depends  only  on the choice  of   the  irreducible representation  of
$SU(N)$  according to which the  ``spin'' operators transform (and not
on    the  choice    of fermions   or      bosons  to implement    the
representation). For $SU(2)$ this amounts to specify the magnitude $S$
of  the spin. For $SU(N)$  irreducible  representations are labeled by
Young tableaux.  The case of a general  rectangular tableau with $n_c$
columns and $m$ rows was discussed by Read and Sachdev\cite{rs89b} and
$n_c$ appears  to  continue  to play a   role similar  to $2S$  in the
large-$N$ phase  diagrams.\footnote{Taking the limit $N\to\infty$ with
$m$ fixed  of order 1 and $n_c\sim  N$ is most conveniently  done with
bosons $b^\dagger_{\alpha p}$  where $\alpha=1\cdots N$ is the $SU(N)$
index, while $p=1\cdots m$ label the  different ``colors''.  There are
therefore $N m$ kinds of bosons. On the other hand it is convenient to
use $n_c$ ``colors'' of fermions (still with an $SU(N)$ index) to deal
with   the case $n_c\sim\mathcal{O}(1)$   and    $m\sim N$.    Bosonic
representations   with   $n_c\sim  N$  are   appropriate  to  describe
magnetically ordered  phases\cite{aa88}    but cannot  be  used    for
frustrated models (such  a representation is  not self-conjugate).  On
the other  hand fermionic  representations,  such as the  $m=N/2$  and
$n_c=1$ one,\cite{am88,rokhsar90} can be used on  any lattice but they
do not display magnetically ordered phases and tend to favor dimerized
states.\cite{am88,rs89b,rokhsar90}} In this  review we will focus on a
slightly different large-$N$ generalization of the $SU(2)$ model which
is both able to deal with frustration and magnetic states.

\subsection{$Sp(N)$}
To perform a large-$N$ extension of {\em frustrated} Heisenberg models
one has  to use either fermions\cite{am88} or  bosons with an $Sp(N)$
symmetry.  The  latter  seems to  produce phase diagrams  that closely
resemble the $SU(2)$ problems and we will focus on this representation
which was introduced by  Read and Sachdev.\cite{rs91} The presentation
below           is       largely      inspired        from       their
papers.\cite{rs89,rs90,rs91,sr91,s93}

We now have  $2N$ flavors of bosons  at each site: $b_{i,\sigma}$ with
$\sigma=1..2N$ and we define an $Sp(N)$-invariant bond operator:
\begin{equation}
	A_{ij} = \sum_{\sigma,\sigma'=1..2N}
	\mathcal{J}_{\sigma,\sigma'} \;b_i^\sigma \;b_j^{\sigma'}
\end{equation}
where the  $2N\times2N$ antisymmetric   tensor $\mathcal{J}$ is  block
diagonal
\begin{equation}
	\mathcal{J}=\left[
	\begin{array}{ccc}
	\begin{array}{cc}
	0&1\\-1&0
	\end{array}&	&\\
		&\ddots	&\\
		&	&\begin{array}{cc}0&1\\-1&0\end{array}
	\end{array}
	\right]	
\end{equation}
and   generalizes the  $SU(2)$   antisymmetric tensor  $\epsilon_{ij}$
($SU(2)$ is identical to $Sp(1)$).    Up to some constant the  $Sp(N)$
Hamiltonian is
\begin{equation}
	\mathcal{H}=-\frac{1}{N} \sum_{ij} J_{ij} A_{ij}^\dagger A_{ij}
\end{equation}
with the constraints
\begin{equation}
	\forall i\;\;
	\sum_{\sigma=1}^{2N} b^{\dag}_{i\sigma}\;b_{i\sigma}=n_c
	\label{eq:constraint}
\end{equation}
$n_c=2S$ in the $SU(2)$ case and  $n_c/N=\kappa$ will be kept constant
when  taking  the  large-$N$  limit.  The  partition function  can be
represented by an imaginary time functional integral:
\begin{equation}	
	Z = \int \mathcal{D}[\lambda_i,b_i^\sigma,b_i^{\dag\sigma}]
	\exp{\left(-\int_0^\beta (L_0+\mathcal{H})d\tau\right)}
	\label{eq:Z}
\end{equation}
\begin{eqnarray}
	L_0&=&\sum_{i \sigma}
		b^{\dag\sigma}_i ( \partial_\tau+i\lambda_i) b_{i\sigma}
		-iN\kappa\sum_i \lambda_i
	\label{eq:lL0}
\end{eqnarray}
The $\lambda_i(\tau)$   are  Lagrange multipliers  that   enforce  the
constraint  Eq.~\ref{eq:constraint} at  every  site.   Bond degrees of
freedom  $Q_{ij}$  are introduced   in order  to  decouple the  bosons
(Hubbard-Stratonovitch).  The partition function is now
\begin{equation}	
	Z = \int \mathcal{D}[Q_{ij},\bar{Q}_{ij},\lambda_i,b_i^\sigma,b_i^{\dag\sigma}]
	\exp{\left(-\int_0^\beta ( L_0 + L_1) d\tau\right)}
	\label{eq:ZQ}
\end{equation}
with
\begin{equation}	
	L_1 = \sum_{ij} \left[
		N \frac{ | Q_{ij} |^2}{J_{ij}}
		- (A_{ij}^\dagger Q_{ij} + {\rm h.c})
		\right]\;,
\end{equation}
so that   a   Gaussian  integration  of   the    $Q_{ij}$  gives  back
Eq.~\ref{eq:Z}. The bond variable  are $Sp(N)$ invariant and they  can
take  non-zero  expectation  values  at   a mean-field level   without
breaking the $Sp(N)$ symmetry.  As we explain  below, they are however
not {\em gauge-invariant}.

\subsubsection{Gauge invariance}
An   important property   of  Eq.~\ref{eq:ZQ}   is  the $U(1)$   gauge
invariance associated to the following transformations:
\begin{eqnarray}
	b_{i\sigma}&\to&b_{i\sigma}e^{i\phi_i} \label{eq:phi}\\
	Q_{ij} &\to& Q_{ij} e^{i(\phi_i+\phi_j)} \label{eq:Qtheta}\\
	\lambda_i &\to& \lambda_i -\partial_\tau \phi_i \label{eq:lambdatheta}
\end{eqnarray}
where  $\phi_i(\tau)$ are arbitrary site- and imaginary-time-dependent
angles.  This gauge invariance comes from the  the conservation of the
local boson number and reflects the fact the magnitude of the spin
is constant at each site. If we focus on the  phase degrees of freedom
of  the bond variables, the    Eq.~\ref{eq:ZQ} describes a system   of
charge-1    bosons     coupled    to    a    $U(1)$     lattice  gauge
theory.\cite{wilson74,bdi74}  These gauge  degrees  of freedom play a
crucial  role in  the analysis  of  the fluctuations  about mean-field
solutions.

{\em Effective action for the bond variables} --- The boson degrees of
freedom can be integrated out to give an effective action for the bond
variables:
\begin{equation}	
	Z = \int \mathcal{D}[Q_{ij},\bar{Q}_{ij},\lambda_i]
	\exp{\left(-S^{\rm eff} \right)}
\end{equation}
\begin{equation}	
	S^{\rm eff}=N \int_0^\beta \left[
		\sum_{ij} \frac{ | Q_{ij} |^2}{J_{ij}}
		-i\kappa \sum_i \lambda_i
	\right]
	- N  {\rm Tr} \log G
	\label{eq:Seff}
\end{equation}
where $G^{-1}$  is  the  quadratic form which  couples   the bosons in
Eq.~\ref{eq:ZQ} (propagator).  It depends on the bond variables and on
$\lambda_i$.             We     may          write            formally
$G^{-1}=\partial_\tau-i\lambda-Q$. The term $N {\rm Tr} \log G$ is the
free energy    of the  bosons  in  presence  of   the bond  fields. By
construction the action $S^{\rm  eff}$ is gauge-invariant with respect
to  the transformations of  Eqs.~\ref{eq:phi}--\ref{eq:lambdatheta}.
So far  this   is an exact   formulation of  the  original  model  for
arbitrary $N$.


\subsubsection{Mean-field ($N=\infty$  limit)  } Since $N$  factorizes  (no
flavor index  is left in Eq.~\ref{eq:Seff}),  $Z$  is dominated by the
saddle  point of $S^{\rm eff}$ when  $N$ is  large.  For simple models
such as the  first-neighbor antiferromagnet on  the cubic lattice (any
space dimension), the saddle point can be determined analytically. The
$N=\infty$  limit    is almost   equivalent   to   the Schwinger-boson
mean-field theory.\cite{schwingerboson,ceccatto} This can otherwise be
done numerically.\footnote{To our   knowledge, all  the saddle  points
considered  so far are static  (expectation values of the $Q_{ij}$ are
time-independent) and  the  corresponding $\left<Q_{ij}\right>$  could
all be made  real with an  appropriate gauge transformation.  There is
no  chiral order  and  the  time-reversal  symmetry is  unbroken.  The
(oriented) sum of  the complex phases of the  bond variables  around a
plaquette defines a  $U(1)$ flux.  This flux is  related to  the solid
angle formed by the spins  and it vanishes  in collinear as well as in
coplanar states.  In such cases the phases  can be therefore be gauged
away and the $\left<Q_{ij}\right>$ can be  made real.  For this reason
complex bond variables are  usually not observed.\cite{ot03} } In this
large-$N$  limit, two kinds of mean-field   solutions can appear.  For
large  enough  $\kappa$ the  bosons condense  at some wave-vector, the
spectrum of the  mean-field Hamiltonian is gapless.  This  corresponds
physically to N\'eel long-range order.  On the other hand, for smaller
$\kappa$  (smaller ``spin'') the  mean field Hamiltonian is gapped and
the ground-state preserves  the $Sp(N)$ symmetry.  Fluctuations around
the saddle  point are not   expected to change drastically  the N\'eel
ordered phases  but  they play  an important  role in   the physics of
$Sp(N)$ symmetric phases.  The following  is a brief discussion of the
effects of fluctuations in these non-magnetic phases.


\subsubsection{Fluctuations about the mean-field solution} At the mean-field
level described  above some  $Q_{ij}$ acquire a  (static in  all known
cases) non-zero expectation value: $\left<Q_{ij}\right>=\bar{Q}_{ij}$.
For  this reason  such a  state spontaneously  breaks the  local gauge
invariance        of       Eqs.~\ref{eq:phi},\ref{eq:Qtheta}       and
\ref{eq:lambdatheta}.   However  this does  not  mean  that the  gauge
degrees are  all gapped and  do not play  any role at low  energy.  In
fact, as remarked by Read and Sachdev, depending on the {\em geometry}
of  the  lattice  defined   by  the  bonds  where  $Q_{ij}\ne0$,  some
long-wavelength gapless  gauge excitations survive  and the associated
fluctuations  must  be  taken   into  account.   More  precisely,  the
fluctuations  of  the  bond  variables  about  the  saddle  point  are
decomposed into an amplitude and a phase
\begin{eqnarray}
	Q_{ij}=\left(\bar{Q}_{ij} + q_{ij}\right)\exp(i\theta_{ij})
	\label{eq:qtheta}
\end{eqnarray}
and we  expand $S_{\rm eff}$ with these  new variables. Two cases must
then be considered:
\begin{itemize}
\item[i)]  The  lattice  made  of  the  sites  connected  by  non-zero
$\bar{Q}_{ij}$ bonds is bipartite.  This is  automatically the case if
the original lattice defined  by bonds where the exchange $J_{ij}\ne0$
is  bipartite.  This can   also be true  if the  original lattice is a
non-bipartite lattice but some bonds have $\bar{Q}_{ij}=0$ so that the
remaining lattice  is bipartite.  This  is the case,  for instance, in
the  $J_1$--$J_2$  model on  the   square lattice,\cite{rs91}  in some
regions    of   the     phase  diagram   of   the   Shastry-Sutherland
model\cite{cms01}    and  in  on      the   checkerboard    Heisenberg
model.\cite{mts01} Such configurations of the bond variables give {\em
collinear}   spin  structures: spin-spin  correlations  can  either be
long-ranged (large $\kappa$, N\'eel phase) or short-ranged but in both
cases the magnetic structure factor is  peaked at a simple wave-vector
${\bf k}_0$   such that $2{\bf k}_0$ is    a reciprocal lattice vector
($k_0=(\pi,0)$,   $k_0=(0,\pi)$    or     $k_0=(\pi,\pi)$ in    square
geometries).
\item[ii)]
The  lattice  made of the sites   connected by non-zero $\bar{Q}_{ij}$
bonds  is   not  bipartite.  This    happens  in some   phases  of the
$J_1$--$J_2$--$J_3$  model  on the square  lattice,\cite{rs91}  on the
triangular  or         kagome    lattices\cite{sachdev92},   in    the
Shastry-Sutherland model      for  some    values of     the  exchange
parameters\cite{cms01}     and    on    an    anisotropic   triangular
lattice.\cite{cmm01}  Such mean-field  states  generically have planar
but non-collinear spin-spin correlations.
\end{itemize}
It is simple to check that case i) preserve a {\em global} continuous
symmetry while such a symmetry is absent in ii). Consider the
following {\em global} gauge transformation in case i) :
\begin{eqnarray}
	b_{i\sigma}\to b_{i\sigma}e^{i\phi} \;&&\;
	b_{j\sigma}\to b_{j\sigma}e^{-i\phi} \label{eq:phiij} \\
	Q_{ii'} \to Q_{ii'} e^{2i\phi}  \;&&\;
	Q_{jj'} \to Q_{jj'} e^{-2i\phi}\label{eq:Q2}\\
	Q_{ij} &\to& Q_{ij} \label{eq:Qij}
\end{eqnarray}
where  $i,i'$  belongs  to   sublattice  $A$  and  $j,j'$  belongs  to
sublattice $B$.   This transformation  does not change  the mean-field
parameters   $\bar{Q}_{ij}$.   The  only   fields  affected   by  this
transformation   are  those   connecting   two  sites   on  the   same
sublattice.\footnote{In  a gauge theory  language the  fields $Q_{ii'}$
(resp.  $Q_{jj'}$) of Eq.~\ref{eq:Q2} transform like a charge-2 scalar
for  the $U(1)$  gauge field.   Instead, from  Eq.~\ref{eq:phiij}, the
bosons (spinons)  carry a  charge 1.}  They  have a  zero expectation
values in  case i) (or  do not even  exist if the physical  lattice is
itself bipartite).  For this reason  it is possible to make low-energy
(and long-wavelength) gapless gauge excitations about the saddle-point
by   replacing   the   global   staggered  phase   shift   $\phi$   of
Eq.~\ref{eq:phiij}  by  a  slowly  varying  (staggered)  $\theta_{ij}$
(Eq.~\ref{eq:qtheta}).  A  gradient expansion of  the effective action
performed  at the  appropriate points  in the  Brillouin zone  for the
phase   fluctuations  only  involves   gradients  of   $\theta$.   The
corresponding action is that of $U(1)$ lattice gauge theory coupled to
charge-1 boson (spinons).


\subsubsection{Topological effects - instantons and spontaneous dimerization}
So far only small fluctuations around the saddle point were considered
and   the  contribution   of  topologically   non-trivial  gauge-field
configurations  were  ignored.   Consequently,  the magnitude  of  the
``spin'' $n_c/2$  was a continuous parameter and  {\em the information
about the integer  or half-integer (for instance) character  of $S$ as
disappeared}.   From  Haldane's  work   on  quantum  spin  chains  and
non-linear sigma  models\cite{haldane83} it  is well known  that Berry
phases  in  spin  systems  give  rise  to  topological  terms  in  the
low-energy effective action which can play a crucial role depending on
the parity of $2S$.

In non-linear sigma models in  2+1 dimensions the Berry phase vanishes
for  configurations which  are  smooth  on the  scale  of the  lattice
spacing\cite{haldane88,noHopfTerm2D88}  (unlike  the 1+1  dimensional
case).  However  ``hedgehog'' space-time singularities\cite{haldane88}
give  non-trivial  Berry phases.   Read  and  Sachdev  found that  the
closely  related  instantons  of  the effective  $U(1)$  gauge  theory
described  before\footnote{In  that  gauge  theory associated  to  the
phases of the  link variables an instanton corresponds  to a tunneling
event during  which the total  magnetic field piercing the  lattice is
changed by $\pm2\pi$.}  also play a crucial role in the physics of the
$Sp(N)$ (as well as $SU(N)$) spin models.

The Berry phase associated to such a singular configuration depends on
details   of  the  lattice  geometry.    In  the   short-range ordered
$(\pi,\pi)$ phase of the   square lattice antiferromagnet  this  Berry
phase is  a multiple of $in_c\pi/2$.   Although dealing  with a gas of
interacting ($1/r$ Coulomb-like  potential) instantons is a  difficult
problem (see  Ref.~\cite{rs90}  and references therein),  we can guess
that  the  physics will   depend on  $n_c\;{\rm mod}\;4$.  A  detailed
analysis\cite{rs90} shows  that   when   $n_c\ne0\;{\rm mod}\;4$   the
instantons  condense and   {\em   spontaneously  break  the    lattice
translation symmetry}.  This generates a static electric field for the
$U(1)$   gauge field.   Since  the  electric field  is  coupled to the
difference  of amplitudes of the  bond  variables, such state acquires
spatially inhomogeneous expectation values of the bond variables, {\em
it  is a  VBC  and spinons are confined   in pairs}.  In the $J_1$--$J_2$
model around $J_2/J_1\simeq0.5$ the   mean-field state is  short-range
ordered with correlations peaked at $(\pi,\pi)$.  A columnar dimerized
state is  predicted by Read  and Sachdev   from this analysis  of  the
fluctuations, in agreement  with  a number  of numerical works  on the
$SU(2)$ $J_1$--$J_2$ spin-$\frac{1}{2}$ model.

In a recent work by Harada, Kawashima and Troyer\cite{hkt03} the phase
diagram of the (first neighbor - unfrustrated) $SU(N)$ antiferromagnet
on  the  square  lattice  with $n_c=1$ was  found   to be in  complete
agreement with Read and Sachdev's predictions.  They showed by quantum
Monte Carlo  simulations   that for $N\leq4$   the  systems  is N\'eel
ordered whereas  it  is a columnar   VBC for $N>5$.   This provides an
additional support to the field theory  arguments described above.  It
also underlines that   the mechanism of spontaneous  symmetry breaking
and formation of a VBC  may  come from  quantum fluctuations only  and
that frustration is not  always required (although it clearly enhances
quantum fluctuations).

On the other hand when $n_c=0\;{\rm mod}\;4$ the  analysis of Read and
Sachdev shows that fluctuations   should not bring any broken  lattice
symmetry.  Spinons are also  confined and this state closely resembles
the     valence-bond   solid (VBS)  proposed     by   Affleck {\it  et
al.}\cite{aklt87} as a possible ground-state when the spin $S$ matches
the  coordination  number  $z$   according  to $2S=0$   mod  $z$  (see
\S\ref{ssec:VBS}).


\subsubsection{Deconfined phases}
Now  we suppose    that,  starting from   a mean-field  solution  with
collinear correlations (case i),   a  parameter of the  original  spin
model is varied  so that the mean field  solution is changed  and some
bonds $Q_{ii'}$ ($i$ and $i'$ belong to the same sublattice) acquire a
non zero expectation   value (case ii).  In  the  framework of  square
lattice  antiferromagnets,  a finite  third-neighbor coupling  ($J_3$)
would  be needed.\cite{rs91}  From   the   point   of  view  of    the
long-wavelength gauge fluctuations (related  to the continuum limit of
the  phases  $\theta_{ij}$)   discussed   above,  the appearance    of
$\bar{Q}_{ii'}\ne0$ is  equivalent  to the condensation   of a (Higgs)
boson of charge 2.   This is a spontaneous  break  down of  the global
$U(1)$ staggered symmetry of Eqs.~\ref{eq:phiij}--\ref{eq:Qij} down to
a    $\Z$ one   since the   field   $Q_{ii}$ is   not invariant  under
Eq.~\ref{eq:Q2}  except if  $\phi=0$  or $\pi$.   Based on  results of
Fradkin  and Shenker\cite{fs79}   concerning  confinement  in  compact
lattice gauge theories coupled to matter, Read and Sachdev argued that
this Higgs mechanism  suppresses the low-energy gauge fluctuations and
liberate the spinons.  This  confinement transition is described by  a
$\Z$ gauge theory.  The suppression  of the $U(1)$ gauge  fluctuations
also forbids  the condensation of instantons   discussed above and the
ground-state remains unique\footnote{Except for a discrete topological
degeneracy.} and  {\em   bond  variables    have  uniform  expectation
values.  It is a genuine SL  without any broken symmetry and
deconfined spinons.}

%_____________________________________________________________________

\section{Quantum Dimer Models}\label{sec:QDM}
\setcounter{footnote}{0}
%_____________________________________________________________________

In  a previous  section   (\S\ref{sec:VBC})  we  showed  that  pairing
spins-$\frac{1}{2}$   into singlets at  short   distances is a  rather
natural    way     to     overcome   frustration     in     Heisenberg
antiferromagnets.  QDM   are    defined  in the     Hilbert   space of
nearest-neighbor valence-bond (or dimer) coverings of the lattice.  By
construction these models focus  on the dynamics  in the singlet space
and     ignore magnetic   (gapped      magnons   or gapped    spinons)
excitations. For  this reason they  are (a priori)  not appropriate to
describe  the physics of spin  systems  where magnetic excitations are
gapless.

The Hamiltonian of a QDM usually contains kinetic as well as potential
energy terms for these dimers.  Such Hamiltonians can often be simpler
than   their   spin parents  and  are   amenable  to  several analytic
treatments  because of their    close  relations to  classical   dimer
problems,\cite{k61,f61,k63}    Ising     models   and     $\Z$   gauge
theory.\cite{msf02,msp02,ms02}    These   models  can   offer   simple
descriptions    of       VBC\cite{rk88}   as      well      as     RVB
liquids.\cite{ms01,msp02} It is  in particular possible to write  down
some QDM    that  have a   simple   and exact VBC    ground-state with
spontaneous broken symmetries (such as Rokhsar and Kivelson's model on
the square lattice\cite{rk88}  with attractive potential energy only -
in which case the exact ground-state is very simple).  Simple solvable
QDM    which    have  a   dimer-liquid   ground-state   can   also  be
constructed.\cite{msp02}

\subsection{Hamiltonian}

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The first QDM was introduced by Rokhsar and Kivelson.\cite{rk88} It is
defined   by  an   Hamiltonian  acting   in  the   Hilbert   space  of
first-neighbor dimer  (valence-bonds) coverings of  the square lattice
and reads:
\begin{equation}
\mathcal{H}=\sum_{\rm Plaquette}
\left[
-J\left(\left|\vv\right>\left<\hh\right| +{\rm H.c.}\right)
+V\left(\left|\vv\right>\left<\vv\right|+\left|\hh\right>\left<\hh\right|\right)
\right]
\label{eq:sQDM}
\end{equation}

Flipping two parallel dimers around a square plaquette is the simplest
dimer move on the square lattice and the $J$ terms precisely represent
such dynamics.  The $V$   terms are diagonal in   the dimer basis  and
account for an    attraction or  repulsion between    nearest-neighbor
dimers.   These are the two  {\em  most local}  terms (respecting  all
lattice  symmetries) which   can  be   considered.\footnote{They  were
originally  derived\cite{rk88} as the lowest  order  terms of a formal
overlap     expansion  (see \S\ref{ssec:spin2QDM}     below)   of  the
Heisenberg. In that  calculation $J\sim x^4$  and $V\sim x^8$.  Notice
that  a  three-dimer kinetic term     (extending over two  neighboring
plaquettes)  is generated  at order    $x^6$ and  is not included   in
Eq.~\ref{eq:sQDM}.}


\subsection{Relation with spin-$\frac{1}{2}$ models}
\label{ssec:spin2QDM}

There exists  different interesting  mappings between frustrated  Ising
models  and QDM.\cite{msc00,ms02} Here,     however, we focus   on the
relations between  QDM and $SU(N)$ (or $Sp(N)$)  spin models  in which
dimers are related to singlet valence-bonds.

{\em Overlap expansion}. --- A valence-bond state (product of two-spin
singlets -  belongs  to the spin-$\frac{1}{2}$  Hilbert space)  can be
associated  to any dimer   covering.\footnote{There is however  a sign
ambiguity (a valence-bond is antisymmetric under  the exchange of both
spins)  that can be fixed by  choosing  an orientation on every bond.} 
Two such valence-bond states $\left|a\right>$ and $\left|b\right>$ are
not orthogonal but,  as first discussed by  Sutherland\cite{s88} their
overlap decays exponentially with the length $L$ of the loops of their
transition    graphs     (defined    in     \S\ref{sssec:trgr})     as
$|\left<a|b\right>|=2^{1-L/2}$.   Rokhsar   and    Kivelson\cite{rk88}
introduced  a    formal    expansion  parameter   $x$    and  replaced
$|\left<a|b\right>|$  by   $2x^{L}$.   Their  idea  is that   although
$x=\frac{1}{\sqrt{2}}$ for physical $SU(2)$ spins, the physics of some
models may     be captured  by  the   first  orders  of  a  small  $x$
expansion. Truncating this expansion to order $x^n$ gives an effective
Hamiltonian which contains {\em  local}  terms involving at most   $n$
dimers.\footnote{The {\em signs}  of the non-diagonal  (kinetic) terms
of the effective QDM  obtained by such small  $x$ expansion depends on
the   sign  convention which  was  chosen   to  map   valence-bonds to
dimers. An important  question  is to know  whether,  at least at  the
lowest non-trivial  order, a sign convention  giving the same sign for
all kinetic terms exists (as in  Eq.~\ref{eq:sQDM}).  This is the case
on the square\cite{rk88}  and triangular lattices.\cite{ms01}} In this
approach the dimer states of the QDM  are in one-to-one correspondence
with    {\em  orthogonalized}   valence-bonds   in   the spin  Hilbert
space.\footnote{This implicitly assumes  that the  valence-bond states
are  linearly  independent.  This  can be  demonstrated on  the square
lattice and appears   to be the   case on  the triangular  and  kagome
lattices for large enough sizes. The operator which orthogonalizes the
valence-bond basis  into  the  dimer  basis is   $\Omega^{-1/2}$ where
$\Omega_{a,b}=\left<a|b\right>$ is the overlap matrix.}
	 
{\em Fluctuations about large-$N$ saddle points}.--- From the argument
above it could   seem that the connection between   spin-$\frac{1}{2}$
models and QDM relies on a variational approximation: the spin Hilbert
space  is   restricted     to   the   nearest-neighbor    valence-bond
subspace. This connection is in fact probably deeper, as some theories
describing  fluctuations  about  some  large-$N$ saddle points  ($1/N$
corrections) are equivalent to  (generalized)  QDM.  This mapping  was
discussed by  Read   and Sachdev\cite{rs89b} for  representation  with
$m=1$ (number  of   rows  in  the  Young    tableau of  the    $SU(N)$
representation), $N\to\infty$ and $n_c\sim\mathcal{O}(1)$, it leads to
a {\em generalized} QDM where $n_c$ dimers emanate  from each site.  A
QDM  also    describes $1/N$  corrections      in  the case  of    the
fermionic\footnote{Young  tableau with    $n_c=1$ column  and  $m=N/2$
rows.}   $SU(N)$ generalization\cite{am88,rokhsar90} of the Heisenberg
model:
\begin{eqnarray}
  \mathcal{H}&=&\frac{1}{N}\sum_{ij} J_{ij}:B_{ij}^\dagger B_{ij}: \nonumber \\
  &=& -\frac{1}{N}\sum_{ij} J_{ij}\; B_{ij}^\dagger B_{ij} +{\rm cst}
  \label{eq:HSUNf} \\
	{\rm where}\;\;B_{ij}&=& \sum_{\sigma=1}^N c_{j\sigma}^\dagger c_{i\sigma}
\end{eqnarray}
and where  the $c_{i\sigma}$ are  $N$ flavors of  fermions satisfying
a  constraint similar to Eq.~\ref{eq:constraint} : 
\begin{equation}
  \sum_{\sigma=1}^N c_{i\sigma}^\dagger c_{i\sigma}=N/2
  \label{eq:constraintF}
\end{equation}
Rokhsar   showed\cite{rokhsar90}  that   in  the   $N\to\infty$  limit
``dimerized states'' (or  Peierls states) becomes exact  ground-states
of Eq.~\ref{eq:HSUNf} for a large class of models.\footnote{ Let $J_0$
be  the largest value of  the $J_{ij}$. Each  dimerization (no site is
left   empty) where only bonds  where  $J_{ij}=J_0$ are  occupied is a
ground-state.  Here a  ``dimer''   between to neighbors  $i$  and  $j$
consists of a  $SU(N)$  singlet made  with $N$  fermions (one of  each
flavor) hoping back and forth between $i$ and  $j$.  It is constructed
from            $\prod_{\sigma=1}^N          \left(c_{i\sigma}^\dagger
+c_{j\sigma}^\dagger\right)  \left|0\right>$.   by projecting  out the
components  which do  not  satisfy  Eq.~\ref{eq:constraintF}.   Notice
however that in  the $N\to\infty$ limit  the relative  fluctuations of
the total number of fermion on each site are of order $1/\sqrt{N}$ and
can  be neglected.}  Quite  naturally, $1/N$ corrections will induce a
dynamics into this subspace of  dimerized states; it can be  described
by a  QDM (with kinetic  energy terms only  at this order).  At lowest
order, on the square lattice, a kinetic term identical to the $J$ term
in Eq.~\ref{eq:sQDM} is generated and  naturally favors a columnar  or
resonating-plaquette  crystal (in agreement with  a number of works on
the spin-$\frac{1}{2}$ model).  The  same arguments were discussed for
the kagome    lattice.\cite{mz91}  In  that  case the    leading $1/N$
corrections to  the fermionic saddle  point generate three-dimer moves
around     hexagons and      stabilize  a   crystal     of  resonating
hexagons.\footnote{To our knowledge however there is no clear evidence
of such ordering in  the spin-$\frac{1}{2}$ case.} This  formalism was
also applied to the checkerboard  model\cite{canals02} to conclude  to
the presence of a VBC phase.

\subsection{Square lattice}

The phase diagram of the  Rokhsar and Kivelson's square lattice QDM is
shown Fig.~\ref{fig:sQDM}.  Since  a change in the signs  of the basis
dimer   configurations   can   change  $J$   into   $-J$   (see
Ref.~\cite{rk88})   we  will  choose   $J>0$  without   lost  of
generality.
\begin{figure}
  \begin{center}
    \includegraphics[height=3cm]{QDMsquare}
    \caption[99]{Schematic  phase diagram  of the  the  square lattice
      QDM.  The  possible location of an  intermediate plaquette phase
      is taken from the work of Leung {\it et al}.\cite{lcr96}}
    \label{fig:sQDM}
  \end{center}
\end{figure}

\subsubsection{Transition graphs  and topological sectors}
\label{sssec:trgr}

In     order to understand the  particularities    of QDM on bipartite
lattices it  is useful to   describe how the  set of  dimer  coverings
splits into {\em  topological  sectors}. To do   so we first   have to
define transition graphs: the transition  graph of two dimer coverings
$c$ and $c'$ is obtained  by superimposing $c$ and  $c'$ on the top of
each other; it  defines a set  of non-intersecting loops covering  the
lattice.  On  each bond  where the dimers  of $c$   and  $c'$ match, a
trivial loop of  length 2 is obtained.   When the lattice is bipartite
(two sublattices $A$ and $B$) these loops can be {\em oriented} in the
following way: any dimer belonging to $c$ is  oriented from $A$ to $B$
and dimers  of $c'$ are  oriented $B\to A$.   The  transition graph is
then  made of  loops of  the   type $A\to  B\to  A\to  B\cdots$.  With
periodic  boundary     conditions    two   winding  numbers\cite{rk88}
$-L/2\leq\Omega_x,\Omega_y\leq    L/2$   are  associated  to    such a
transition graph ($L\times  L$ sites).  $\Omega_x$ (resp.  $\Omega_y$)
is the net number of  topologically non-trivial loops (clockwise minus
counterclockwise) encircling  the     torus in the    $x$ (resp.  $y$)
direction.

Dimer coverings can   be   grouped into different   {\em   topological
sectors}. By definition two dimer coverings belong  to the same sector
if they can be transformed into each other by repeated actions of {\em
local} dimer moves (the  transition graph associated to each  movement
does not  wind  around the whole  system  if it has  periodic boundary
conditions).  On the square lattice  two-dimer moves are sufficient to
connect any two  states in the  same sector;  that  is the Hamiltonian
Eq.~\ref{eq:sQDM}  is ergodic  within  each topological  sector.  In a
torus geometry, $c$ and $c'$ belongs to the same sector if and only if
their transition graph has winding numbers $\Omega_x=\Omega_y=0$.  The
different topological sectors can  be labeled by their winding numbers
with respect to some  reference columnar configuration.  Their  number
is of order $\mathcal{O}(L^2)$ for a system of linear size $L$.


\subsubsection{Staggered VBC for $V/J> 1$}
When $V$ is sufficiently large the system tries to minimize the number
of    parallel   dimers.     The    staggered   configuration    shown
Fig.~\ref{fig:sQDM}  has  no such  {\em  flippable  plaquette}. It  is
always  a zero-energy  eigenstate  of Eq.~\ref{eq:sQDM}  and becomes  a
ground-state  for $J\geq  V$.   It breaks  several lattice  symmetries
(four-fold degenerate) and is a VBC.

The  expectation value  of the  energy per  plaquette  satisfies ${\rm
min}(0,V-J)\leq E_0/N_p\leq {\rm max}(0,V+J)$.  For $V/J>1$ this gives
$0 \leq E_0/N_p$ and any  zero-energy state saturates this lower bound
and is a therefore a  ground-state.  One should however notice that it
is possible  to make  zero-energy domain walls  in this VBC  since the
state   shown  Fig.~\ref{fig:stagg2}  is   also  annihilated   by  the
Hamiltonian. No local  dimer movement can take place  in the staggered
VBC  (with or  without domain  walls).  Each  of these  states  form a
topological   sector   with   a   single   configuration   which   has
$\left|\Omega_x\right|+\left|\Omega_y\right|=L/2$  with  respect to  a
columnar state.

\begin{figure}
  \begin{center}
    \includegraphics[height=3cm]{stagg2}
    \caption[99]{Domain wall in a staggered VBC.}
    \label{fig:stagg2}
  \end{center}
\end{figure}

\subsubsection{Columnar crystal for $V<0$}

When parallel dimers  attract each other the system  tries to maximize
the number  of flippable plaquettes. Columnar  configurations as shown
on the left side of Fig.~\ref{fig:sQDM} do maximize this number.  Such
a VBC is exactly realized  for $V<0$ and $J=0$. Elementary excitations
consist of a pair of (say) vertical dimers in a background of vertical
columns  of horizontal  dimers. Such  excitations are  gapped ($\Delta
E=2|V|$) and this VBC phase will  survive to the inclusion of a finite
$J$  term.  Notice  that unlike  the  staggered VBC  presented in  the
previous paragraph  the columnar dimer  configuration is not  an exact
eigenstate  when $J\ne0$.   The exact  diagonalizations data  of Leung
{\it et al}\cite{lcr96} suggest  that the columnar phase may disappear around
a critical value $V/J\simeq -0.2$.

\subsubsection{Plaquette phase}

When the kinetic energy dominates  ($|V|\ll J$) the system will try to
maximize   the   number    of   resonating   plaquettes   $\left|   \|
\right>+\left|=\right>$. This  can be achieved  through the resonating
plaquette  crystal shown   Fig.~\ref{fig:sQDM} and  the numerical work
(exact  diagonalizations  up  to  $8\times8$ sites)  of  Leung {\it et
al}\cite{lcr96} suggest  that this phase   is realized in  an interval
$-0.2\leq V/J \leq 1$. Although  this model would  not suffer from the
fermion sign  problem we are  not aware of  any QMC simulation on this
model.

\subsubsection{Rokhsar-Kivelson point}

The point $J=V$ (Rokhsar-Kivelson (RK) point) plays a special role. As
remarked  by  Rokhsar   and  Kivelson\cite{rk88}  the  equal-amplitude
superposition of all dimerizations in a given topological sector is an
exact ground-state. When $J=V$ the Hamiltonian can be written as a sum
of projectors:
\begin{eqnarray}
  \mathcal{H}_{J=V=1}&=&\sum_{p} \left|\Psi_p\right>\left<\Psi_p\right| \label{eq:sRK}\\
  \left|\Psi_p\right>&=&\left|\vv\right> - \left|\hh\right>
\end{eqnarray}
The linear superposition  of all dimer coverings belonging  to a given
sector $\Omega$
\begin{equation}
  \left|0\right>=\sum_{c\in\Omega}\left| c\right>
\label{eq:RK}
\end{equation}
is annihilated  by Eq.~\ref{eq:sRK}  and is therefore  a ground-state.
The  argument  is the  following.   Consider  a  plaquette $p$  and  a
configuration $\left|  c\right>$. If $\left|  c\right>$ has one  or no
dimer at all  on the edges of $p$  we have $\left<\Psi_p |c\right>=0$.
If two dimers  are present, then there exists  a configuration $\left|
c'\right>$ in the same sector which only differ from $\left| c\right>$
by a  two-dimer flip on $p$.   In such a case  the combination $\left|
c\right>+\left|c'\right>$       is      again       orthogonal      to
$\left|\Psi_p\right>$. This shows that $H\left| 0\right>=0$.

When open boundary    conditions are considered    (this restricts the
topological  sector to   $\Omega_x=\Omega_y=0$)  the RK  state is  the
linear     combination      of      an    exponential    number     of
configurations.\footnote{This   is also   true for   periodic boundary
conditions provided the two   winding numbers do   not scale like  the
linear size $L$.}  This is  very different from the crystalline states
considered so far where some  periodic configurations were favored and
it closely resembles Anderson's RVB picture.   As we shall see this RK
state is not a ``true'' liquid on the square lattice since dimer-dimer
correlations are    not  short-ranged  but   algebraically    decaying
($\sim1/r^2$)  with distance.     The  calculation    of   dimer-dimer
correlations in  the  RK  state (Eq.~\ref{eq:RK})  maps  onto   a {\em
classical    }   dimer  problem   solved  by  Kasteleyn,\cite{k61,k63}
Fisher\cite{f61}    and Fisher and   Stephenson.\cite{fs63}  From this
Rokhsar  and Kivelson\cite{rk88}  constructed gapless excitations  (in
single-mode approximation) which dispersion relation vanishes as ${\bf
k}^2$ at small momentum (the origin is taken at $(\pi,\pi)$).
Quoting Rokhsar and  Kivelson,\cite{rk88}  these excitations  (dubbed
``resonons'') are the {\em Goldstone mode  of the gauge symmetry which
allows the  phases of the different   topological sectors to  be varied
without changing the energy}.  Another   mode of gapless   excitations
(around $(\pi,0)$ and  $(0,\pi)$  in the Brillouin zone),  specific to
the fact that  the  ground-state has critical (algebraic)  dimer-dimer
correlations, was recently discovered.\cite{ms03}

The QDM  on the square  lattice is thus  believed to be  ordered (VBC)
everywhere  except  at  the  RK  point  ($J=V$)  where  it  has  quasi
long-ranged (critical) dimer-dimer  correlations. 

\subsection{Hexagonal lattice}

The QDM on  the honeycomb lattice was  studied by Moessner, Sondhi and
Chandra,\cite{msc01} in particular  with Monte Carlo simulations.  The
phase   diagram is very similar  to  the square lattice-case discussed
above.  It  possesses three crystalline  phases  (Fig.~\ref{fig:hQDM})
and has algebraically decaying dimer-dimer correlations at the Rokhsar
Kivelson  point (where the ground-state  in  each sector has the  same
form  as Eq.~\ref{eq:RK}).  
 The  absence of liquid phase  (with
exponentially   decaying 2$n$-mer-2$n$-mer  correlations)   is  believed  to
prevail in  bipartite lattices.
This relation  between the absence of a
deconfined dimer  liquid  phase
\footnote{
Bipartiteness  seems to  forbid deconfinement but  not short-ranged
dimer-dimer correlations. The 4-8 lattice (squares and octogons) is an
example  where dimer-dimer correlations   are short-ranged.   We thank
R.~Moessner  for   pointing   this  to    us.  On this   lattice   the
equal-amplitude superposition of all coverings  would be similar to an
explicit VBC wave-function (thus  confining). Such situations are only
possible when the number of sites is even in the  unit cell.}
and the  bipartite character  of   the lattice  as been discussed   by
several  authors\cite{hkms03} and is related  to the  possibility of a
{\em height   representation}\cite{height_representation}    of  dimer
coverings when the lattice is bipartite.\footnote{Consider a bipartite
lattice with coordination number $z$.  For  each dimer covering we can
associate integers  (heights) on  the  dual  lattice by  the following
rule.  Set the  height to be zero  on a plaquette  at the origin.  The
height  is  then defined  on the  whole  lattice  by turning clockwise
(resp.      counterclockwise)  around  sites    of  the $A$-sublattice
(resp. $B$-sublattice) and changing the  height by $z-1$ when crossing
a  dimer and by  $-1$ when crossing  an empty bond.    It is simple to
check the difference  of  heights $\delta h(x)=h_1(x)-h_2(x)$  between
two dimerizations is constant   inside each loop of their   transition
graph.  Notice that  the loops of a  transition graph can be naturally
oriented on a bipartite lattice.   Then, $\delta h(x)$ changes by $+z$
(resp.  $-z$) when crossing a clockwise (resp.  counterclockwise) loop
of the  transition  graph.   Columnar dimerizations  have  an averaged
height  which is flat  and staggered ones have  the maximum tilt.  The
winding numbers $(\Omega_x,\Omega_y)$ correspond to the average height
difference between both sides of the sample.  The kinetic energy terms
of  Eqs.~\ref{eq:sQDM} and   \ref{eq:hQDM}  change the height   of the
corresponding  plaquette  by $\pm z$  and  the potential terms ($V>0$)
favor tilted configurations.}

\newcommand{\ha}{
  \begin{picture}(32,14)(-16,-2)
    \Hex\pA{\La}\pC{\Lc}\pE{\Le}
  \end{picture}
}
\newcommand{\hb}{
  \begin{picture}(32,14)(-16,-2)
    \Hex
    \pB{\Lb}\pD{\Ld}\pF{\Lf}
  \end{picture}
}

\begin{eqnarray}
  \mathcal{H}=&-J&\sum_h \left(
  \left|\ha\right>\left<\hb\right| +{\rm H.c.}
  \right) \nonumber \\
  &+V&\sum_h\left(
  \left|\ha\right>\left<\ha\right|+\left|\hb\right>\left<\hb\right|
  \right)
\label{eq:hQDM}
\end{eqnarray}

Fouet   {\it  et   al.}\cite{fsl01}   studied  the  spin-$\frac{1}{2}$
$J_1$--$J_2$--$J_3$    model   on the   hexagonal   lattice  by  exact
diagonalizations and  found evidences of a  staggered VBC of  the type
predicted for  $V/J>1$ by Moessner  {\it  et al.}\cite{msc01}  in  the
QDM. Other phases  (N\'eel ordered phase  and a possible short-range RVB
SL) are also present in the spin-$\frac{1}{2}$ model.\cite{fsl01}



\begin{figure}
  \begin{center}
    \includegraphics[height=3cm]{QDMhex}
    \caption[99]{  Phase  diagram of  the  hexagonal  QDM obtained  by
      Moessner  {\it et  al.}\cite{msc01} Although  the VBC  shown for
      $V<0$ do not  have all dimers parallel to  the same direction it
      is equivalent  to the columnar  VBC found in the  square lattice
      QDM because it maximizes  the number of flippable plaquettes. It
      also  corresponds  to  the  ordering pattern  predicted  in  the
      large-$N$ theory of Ref.~\cite{rs90} As for the VBC obtained for
      $V/J>1$, it is the  hexagonal counterpart of the {\em staggered}
      VBC  of  the  square  lattice  (no  flippable  plaquette,  exact
      eigenstate and maximum tilt in a height representation).}
    \label{fig:hQDM}
  \end{center}
\end{figure}

\subsection{Triangular lattice}
\label{ssec:QDMTri}

\newcommand{\rhomb}{
  \pA{\C}\pB{\C}\pZ{\C}\pC{\C}
 }
\newcommand{\rhombF}{
  \begin{picture}(18,10)(-5,-6)
    \rhomb
    \pA{\La}\pB{\Lb}\pZ{\Le}\pC{\Ld}
  \end{picture}
}
\newcommand{\rhombH}{
  \begin{picture}(22,10)(-8,-6)
    \rhomb
    \pA{\La}\pC{\Ld}
  \end{picture}
}
\newcommand{\rhombV}{
  \begin{picture}(22,10)(-8,-6)
    \rhomb
    \pB{\Lb}\pZ{\Le}
  \end{picture}
}

The most  local dimer Hamiltonian  on the triangular  lattice contains
kinetic and potential two-dimer terms  on each rhombus; it was studied
by Moessner and Sondhi:\cite{ms01}
\begin{eqnarray}
  \mathcal{H}=&-J&\sum_r \left(
  \left|\rhombV\right>\left<\rhombH\right| +{\rm H.c.}
  \right) \nonumber \\
  &+V&\sum_r\left(
  \left|\rhombV\right>\left<\rhombV\right|+\left|\rhombH\right>\left<\rhombH\right|
  \right)
\label{eq:tQDM}
\end{eqnarray}
where  the sums run over  all  rhombi $r$  of the  lattice (with three
possible orientations).   This model was  shown to  possess (at least)
three crystalline phases,  including a   columnar and staggered    one
(Fig.~\ref{fig:QDMtri}) as in the two previous examples. An additional
VBC  (with resonating diamonds plaquettes)  with a large unit cell (12
sites) was also   predicted around $V=0$.   When $V<0$  and $J=0$  the
ground-state  is highly  degenerate  since it  is   possible, from  an
ordered columnar configuration,   to shift all  the dimers  along  any
straight line   without changing the  number  of  flippable plaquettes
(contrary to the square lattice case).   However, an infinitesimal $J$
is expected to lift this degeneracy and to order the ground-state in a
columnar way.

The phenomenology of these ordered phases is that  of usual VBC and we
refer to the original paper\cite{ms01} for details.

\begin{figure}
\begin{center}
\includegraphics[width=8.5cm]{QDMTri}
\caption[99]{Phase diagram of the triangular lattice QDM
obtained by Moessner and Sondhi.\cite{ms01}}
\label{fig:QDMtri}
\end{center}
\end{figure}

\subsubsection{RVB liquid at the RK point}

The new physics of this model appears through  the existence of a {\em
liquid} phase in the interval $0.7\lesssim V/J \leq 1$. As for the two
previous  QDM  the ground-states are  exactly  known at   the RK point
$J=V$. As before dimer-dimer correlations are obtained exactly at this
point by  a Pfaffian calculation\cite{ms01,fms02,iif02} but the result
shows a   {\em  finite} correlation length.    From  their Monte Carlo
simulations Moessner {\it et al.}\cite{ms01}  argued that the spectrum
is   gapped  at the  RK   point and  that  this   gap persists down to
$V/J\simeq 2/3$,  that is  over  a  finite range   of  coupling.  This
picture   is consistent with the   exact diagonalizations performed on
this model.\cite{ifiitb02,mlms02}

\subsubsection{Topological order}

When the lattice is not bipartite the  loops of a transition graph can
no longer be oriented.   The winding numbers $\Omega_x$ and $\Omega_y$
are now  defined as the (positive)  number  of non-trivial loop around
$x$  and $y$  in the transition  graph  with a reference configuration
(say columnar).   These two integers are not  conserved by local dimer
moves. They are however {\em conserved modulo  two}, which leaves only
four   sectors.\footnote{The        two-dimer moves     included    in
Eq.~\ref{eq:tQDM} are not  sufficient to guaranty ergodicity with each
of the  four  sectors.  Staggered   states  (12-fold degenerate  - not
$\mathcal{O}(L)$ like on   the  square  lattice) have   no   flippable
plaquette  but  can be    connected  to other states with   four-dimer
moves.\cite{ms01}} Consequently {\em the  dimer liquid ground-state is
four-fold degenerate} at the RK  point.  This degeneracy holds exactly
at the RK point even on finite-size samples but it is expected to hold
in the thermodynamic limit in the whole liquid phase ($0.7\lesssim V/J
\leq 1$).

Conventional  orders are  often associated  to a  spontaneously broken
symmetry and  lead to  ground-state degeneracies in  the thermodynamic
limit.  The  four-fold degeneracy discussed above is  the signature of
some kind of order, called topological
order.\cite{wen91}  The peculiarity  of this  order  is that it is not
associated  to any local  order  parameter: a local observable  cannot
decide whether a given dimerization is in one  sector or another.  The
existence of topological   order   is intimately associated   to   the
fractionalized nature of the  elementary excitations (see  below).  In
the case of a RVB dimer liquid these excitations have been known to be
{\em  Ising vortices}  for   a long  time\cite{k89,rc89}  (dubbed {\em
visons}  in the   recent  literature\cite{sf00,sf01}).   We  will  now
discuss these excitations in more  details in the  framework of a  QDM
which realizes the same dimer liquid phase but for  which not only the
ground-state but all the eigenstates are known exactly.


\subsection{Solvable QDM on the kagome lattice}
\label{ssec:QDMKag}

An exactly    solvable QDM on  the kagome   lattice  was introduced by
D.~Serban,  V.~Pasquier and one  of  us\cite{msp02}.  It offers a very
simple and explicit realization of  the ideas discussed above (visons,
topological order etc.).

\subsubsection{Hamiltonian}
The kagome lattice  QDM introduced in  Ref.~\cite{msp02} contains only
kinetic terms and has no external parameter. The Hamiltonian reads:
\begin{eqnarray}
  \mathcal{H}&=&-\sum_h \sigma^x(h) \label{eq:kQDM}\\
{\rm where }\;\;
  \sigma^x(h)&=&\sum_{\alpha=1}^{32}
  \left|d_\alpha(h) \right>\left< \bar{d}_\alpha(h)\right|
  +
  {\rm H.c}
  \label{eq:sigmax}
\end{eqnarray}
The sum runs over the  32 loops on  the lattice which enclose a single
hexagon    and around      which     dimers can   be     moved    (see
Table~\ref{tab:kloops}  for  the 8 inequivalent loops).   The shortest
loop is the  hexagon itself, it involves  3 dimers.  4, 5 and 6-dimers
moves are also possible  by including 2, 4  and 6 additional triangles
(the  loop length must be  even).  The largest loop is  the star.  For
each loop $\alpha$   we associate the  two  ways dimers can be  placed
along                that      loop:  $\left|d_\alpha(h)\right>$   and
$\left|\bar{d}_\alpha(h)\right>$.    Notice that $\sigma^x(h)$ measures
the relative  phases  of dimer configurations displaying  respectively
the   $d_\alpha(h)$    and $\bar{d}_\alpha(h)$   patterns   in the wave
function.


\begin{table}
   \tbl{
The 8 different classes of loops which can  surround an hexagon of the
kagome lattice.  Including all possible symmetries we find 32 possible
loops. The first column indicates the number of dimers involved in the
coherent motion around the hexagon.  \label{tab:kloops}}
{ \begin{tabular}{|c|c|c|c|}
      \hline%_______________________________
      3&
      \begin{picture}(50,34)(-24,-15)
	\pA{\La}\pB{\Lb}\pC{\Lc}\pD{\Ld}\pE{\Le}\pF{\Lf}\KagHex
      \end{picture} &  & \\
      \hline %_______________________________
      4&
      \begin{picture}(50,34)(-24,-13)
	\pA{\La}\pB{\La}\pG{\Lc}\pC{\Lc}\pD{\Ld}\pE{\Ld}\pJ{\Lf}\pF{\Lf}
	\KagHex\pG{\C}\pJ{\C}
      \end{picture} &
      % - - - -
      \begin{picture}(50,34)(-24,-13)
	\pA{\La}\pB{\La}\pG{\Lc}\pC{\Lc}\pD{\Ld}\pE{\Le}\pF{\Le}\pK{\La}
	\KagHex\pG{\C}\pK{\C}
      \end{picture} &
      % - - - -
      \begin{picture}(50,34)(-24,-13)
	\pA{\La}\pB{\La}\pG{\Lc}\pC{\Lb}\pH{\Ld}\pD{\Ld}\pE{\Le}\pF{\Lf}
	\KagHex\pG{\C}\pH{\C}
      \end{picture}\\
      \hline %_______________________________
      5&
      \begin{picture}(50,34)(-24,-13)
	\pA{\La}\pB{\La}\pG{\Lc}\pC{\Lb}\pH{\Ld}\pD{\Ld}
	\pE{\Ld}\pJ{\Lf}\pF{\Le}\pK{\La}
	\KagHex\pG{\C}\pJ{\C}\pK{\C}\pH{\C}
      \end{picture} &
      % - - - -
      \begin{picture}(50,44)(-24,-23)
	\pA{\Lf}\pB{\Lb}\pL{\Lb}\pC{\Lb}\pH{\Ld}\pD{\Ld}
	\pE{\Ld}\pJ{\Lf}\pF{\Le}\pK{\La}
	\KagHex\pL{\C}\pJ{\C}\pK{\C}\pH{\C}
      \end{picture} &
      % - - - -
      \begin{picture}(50,44)(-24,-23)
	\pA{\Lf}\pB{\La}\pL{\Lb}\pC{\Lc}\pG{\Lc}\pD{\Ld}
	\pE{\Ld}\pJ{\Lf}\pF{\Le}\pK{\La}
	\KagHex\pL{\C}\pJ{\C}\pK{\C}\pG{\C}
      \end{picture}\\
      \hline %_______________________________
      6&
      \begin{picture}(50,50)(-24,-23)
	\pA{\Lf}\pB{\La}\pL{\Lb}\pC{\Lb}\pG{\Lc}\pD{\Lc}
	\pE{\Ld}\pJ{\Lf}\pF{\Le}\pK{\La}\pH{\Ld}\pI{\Le}
	\KagStar
      \end{picture}&  & \\
      \hline %_______________________________
    \end{tabular}
}
\end{table}

\subsubsection{RK ground-state}
As  for   the  QDM discussed   previously  the  ground-state   of this
Hamiltonian   is the  equal  amplitude   superposition  of  all  dimer
coverings belonging    to   a given  topological  sector   (as  on the
triangular  lattice there  are four sectors).   This can be
readily shown by writing  $\mathcal{H}$ as a sum  of projectors:
\begin{eqnarray}
  \mathcal{H}&=&-N_h+\sum_h \sum_{\alpha=1}^{32}
  \left[\;\left|d_\alpha(h) \right>-\left| \bar{d}_\alpha(h)\right>\;\right]
  \left[\;\left<d_\alpha(h) \right|-\left< \bar{d}_\alpha(h)\right|\;\right]
\end{eqnarray}
where $N_h$ is the number of hexagons  on the lattice.  When expanding
the products the diagonal terms give a simple constant since
\begin{equation}
  \sum_{\alpha=1}^{32} \left|d_\alpha \right>\left<d_\alpha\right|
  +
  \left| \bar{d}_\alpha\right>\left<\bar{d}_\alpha\right| = 1
\end{equation}
This  reflects  the fact that,  for  any  dimerization, the  dimers on
hexagon $h$  match {\em one and only  one} of the $2\times32$ patterns
$\left\{d_\alpha,\bar{d}_\alpha\right\}$. 

Unlike  the  square    or  triangular case,    the   RK  ground-states
$|0\rangle=\sum_{c\in\Omega}|c\rangle$ are   not  degenerate with some
staggered VBC.\footnote{Because resonances loops  of  length up to  12
are present the  dynamics is ergodic in each  of  the four topological
sectors.\cite{msp02}}   This    means  that    the    Hamiltonian   of
Eq.~\ref{eq:kQDM} is not at  a phase transition to a  VBC.  As we will
explain it  is {\em inside} a  liquid RVB phase.

The RK wave-function can be viewed as dimer condensate.  It is similar
to  the ground-state  of liquid $^4$He  which   has the  same positive
amplitude      for  any       configuration     and    its    permuted
images.\cite{feynman52-53} An important  difference, however,  is that
the  QDM state is incompressible  and cannot sustain acoustic phonons.
This can be related to  the fact that the $U(1)$  symmetry of the Bose
liquid is absent in the QDM on non-bipartite lattices.  It is replaced
instead by   a  discrete  $\Z$  gauge  symmetry  (see \S\ref{sssec:Z2}
below).


\subsubsection{Ising pseudo-spin variables}

The kinetic energy operators $\sigma^x$ defined in Eq.~\ref{eq:sigmax}
commute with each  other. This is obvious when  two such operators act
on remote  hexagons but it   also  holds for  neighboring ones.   This
property can   easily be  demonstrated  with the   help of  the  arrow
representation   of  dimer     coverings  introduced    by    Zeng and
Elser.\cite{ez93} This mapping of  kagome  dimerizations to arrows  on
the   bonds     of   the  honeycomb       lattice  is      illustrated
Fig.~\ref{fig:arrow}.  Each  arrow has   two possible  directions:  it
points toward the interior of  one of  the two neighboring  triangles.
If site $i$ belongs to a dimer $(i,j)$ its arrow must point toward the
triangle the site $j$ belongs to.  A dimer covering can be constructed
from any   arrow configuration provided   that the number  of outgoing
arrows is odd (1 or 3) on every triangle.

\begin{figure}
  \begin{center}
    \includegraphics[height=4.5cm,width=6cm]{KagArrow}
    \caption[99]{A  dimer  covering  on  the kagome  lattice  and  the
	corresponding arrows. Dashed lines: honeycomb lattice.}
    \label{fig:arrow}
  \end{center}
\end{figure}

The operators $\sigma^x$ have a  particularly simple meaning in  terms
of the arrow  degrees  of freedom: $\sigma^x(h)$  flips  the  6 arrows
sitting  on $h$.\footnote{Flipping all   the arrows around  any closed
loop (such  as around an   hexagon)  preserves  the local   constraint
imposed on  arrow configurations.    Flipping   the arrows around    a
topologically non-trivial loop changes the topological sector.}  It is
then clear that the $\sigma^x$ commute and that $\sigma^x(h)^2=1$.  In
fact these operators can  be used as  Ising pseudo-spin  variables and
the  Hamiltonian now   describes  non-interacting pseudo-spins  in   a
uniform   magnetic field  pointing  in  the   $x$ direction.   In  the
ground-state we have $\sigma^x(h)=1$ on every hexagon.

\subsubsection{Dimer-dimer correlations}
The  ground-state is the most possible  disordered dimer liquid as the
dimer-dimer  correlations strictly  vanish    beyond  a few    lattice
spacings.  Such  correlations  can    be  computed by    the  Pfaffian
method. On the kagome lattice the determinant  of the Kasteleyn matrix
(which is directly related to the partition  function of the classical
dimers problem) is exactly constant in Fourier space.\cite{hw88} Since
dimer-dimer correlations  are obtained from  the  Fourier transform of
the inverse  of  this determinant, they  turn out  to be strictly zero
beyond a few lattice spacings (as soon as the two bonds  do not touch a
common  triangle).\cite{msp02} This result can also   be obtained by a
simpler  argument\cite{msp02,msp03b} using  the  $\sigma^x$ operators.
This result is related  to the kagome geometry.\footnote{The model  of
Eq.~\ref{eq:kQDM}   can  be  generalized   to any    lattice   made of
corner-sharing triangles.\cite{msp02}}  This absence   of  long-ranged
dimer-dimer correlations  demonstrates that the   RK state is  a dimer
liquid and that it breaks no lattice symmetry.

On the  triangular lattice,   even at high    temperature, dimer-dimer
correlations decay exponentially  with distance but these correlations
remain {\em finite}  at  any distance.  On   the  square lattice  such
correlations are  even  larger  because they  decay  only as  a  power
law. This means that the  infinite hard-core dimer repulsion makes QDM
non-trivial  even at infinite temperature; dimers  cannot be free when
they are fully-packed.  From this point of view we see that the kagome
lattice is particular: it is as close as possible to a free dimer gas,
except for non-trivial correlations over  a few lattice spacings. This
is a reason why  dimer coverings on the  kagome lattice can be handled
with independent pseudo-spin  variables and why  the RK state  on this
lattice is the most possible disordered RVB liquid.

\subsubsection{Visons excitations}
\label{sssec:vison}

The $\sigma^x$ operators  can be simultaneously diagonalized but  they
must satisfy the global constraint  $\prod_h \sigma^x(h)=1$ since this
product flips every  arrow {\em twice}.  It  must  therefore leave all
dimerizations unchanged.   The  lowest  excitations  have therefore an
energy $4$ above the ground-state and they are made of a {\em pair} of
hexagons $a$ and $b$  in a $\sigma^x(a)=\sigma^x(b)=-1$ state. $a$ and
$b$  are the    locations     of  two   Ising   vortices   (or    {\em
visons}\cite{sf00,sf01}).   As  remarked before  this   means that the
relative  phases   of  the  configurations   with   $d_\alpha(h)$  and
$\bar{d}_\alpha(h)$ patterns  have now changed sign. The corresponding
wave-function  is  obtained in the   following way.  Consider a string
$\Omega$ which goes from $a$ to $b$ (see Fig.~\ref{fig:Vison}) and let
$\Omega(a,b)$ be the operator which measures  the parity $\pm1$ of the
number of  dimers crossing that   string.  $\Omega(a,b)$ commutes with
all  $\sigma^x(h)$,    except  for   the     ends   of the     string:
$\sigma^x(a)\Omega(a,b)=-\Omega(a,b)\sigma^x(a)$. A dimer move changes
the sign of  $\Omega(a,b)$ if and only if  the associated loop crosses
the  string an   odd  number of  times,  which  can  only  be done  by
surrounding one  end of  the  string.  This shows  that  $\Omega(a,b)$
flips the   $\sigma^x$ in $a$ and  $b$.\footnote{Up   to a global sign
(reference         dependent)   $\Omega(a,b)$       is  equal       to
$\sigma^z(a)\sigma^z(b)$  where  the $\sigma^z$   operators are  those
introduced by  Zeng and Elser.}  As  the  RK ground state $|0\rangle$,
$\Omega(a,b)|0\rangle$  is   a  linear   combination  of   all   dimer
configurations belonging to one sector. However the amplitudes are now
$1$ and $-1$ depending on the number of dimers crossing $\Omega$. This
wave-function  therefore has nodes, it is  an excited  state of energy
$4$ with  two vortices  in $a$ and   $b$.  It is   easy to see that  a
different choice $\Omega'$ for the string connecting $a$ and $b$ gives
the same state up to a global sign which depends  on the parity of the
number of kagome sites enclosed by $\Omega\cup\Omega'$.

These vortex excitations carry a $\Z$ charge  since attempting to put
two  vortices on  the same  hexagon does  not change  the state.  Such
excitations are not local  in terms of  the dimer degrees  of freedom.
Indeed,  determining the sign of a  given dimerization in a state with
two visons which  are far apart requires  the  knowledge of the  dimer
locations along the whole string connecting the  two vortex cores.  In
this model the visons appear to be static and non-interacting. This is
a particularity of  this  solvable model  but the  existence of gapped
vison excitations  is   believed  to be  a   robust  property   of RVB
liquids. In more  realistic models the  visons will acquire a dynamics
and  a  dispersion  relation  but will   remain gapped.\footnote{It is
possible to add potential energy  terms to Eq.~\ref{eq:kQDM} to  drive
the system outside of the liquid phase and this transition corresponds
to a vison condensation.}  They will also  have some interactions with
each  other but  should remain {\em    deconfined}.  This property  is
particularly clear in the  kagome QDM: visons are  necessarily created
by pairs but the energy is independent of their relative distances.

\begin{figure}
  \begin{center}
	\includegraphics[height=2.5cm]{VisonPairA}
	\includegraphics[height=2.5cm]{VisonPairB}

  \caption[99]{A pair of visons (located in $a$ and $b$) is created by
  applying  to the  RK wave-function  a factor  $(-1)$  for each dimer
  crossing   the string $\Omega$. The  dimerization  shown there on the left
   appears  in the  linear superposition  of  the two-vison state
  with the sign $-1$ whereas the one on the right has the sign $+1$.}

  \label{fig:Vison} \end{center}
\end{figure}

The  Ising vortices  also offer a  simple picture  of the  topological
degeneracy.  Consider  a  ground-state $|+\rangle$ of  the model which
lives in the sector where the  winding number $\Omega_y$ (with respect
to  some  arbitrary  but  fixed    dimerization) is  even.     Another
ground-state $|-\rangle$ is obtained in the odd-$\Omega_y$ sector. Now
consider the combination $|0\rangle=|+\rangle+|-\rangle$ and apply the
operator $\Omega(0,L_x)$ corresponding  to  a closed loop  surrounding
the torus  in the $x$ direction. This   amounts to creating  a pair of
nearby visons at  the origin, taking  one of them  around the torus in
the $x$  direction and annihilating them.  This  can also be viewed as
the creation  of a  vison in  one hole  of  the torus (with  no energy
cost).         It       is      simple      to       check        that
$\Omega(0,L_x)|0\rangle=|+\rangle-|-\rangle$   (up to an    irrelevant
global sign).  This provides  a simple relation between the vison-pair
creation  operator and the  existence   of two topologically  distinct
ground-states $|+\rangle+|-\rangle$ and $|+\rangle-|-\rangle$.

\subsubsection{Spinons deconfinement}
\label{sssec:QDMdeconf}

We assume that dimers represent  ``dressed'' singlet valence-bonds, as
in  the overlap expansion  (\S\ref{ssec:spin2QDM}).  Since the Hilbert
space is   made   of   fully-packed dimer  coverings  the     model of
Eq.~\ref{eq:kQDM} only describes spin-singlet states.  However, as any
QDM,   it  can   be extended  to   include  static  holes or  spinons.
Configurations  with unpaired sites (spinon or  holon) are now allowed
but the kinetic terms of the original Hamiltonian which loop passes on
an empty site  gives zero.  Consider a  system with two static spinons
in $x$ and   $y$.    As  on  the square\cite{rk88}   and    triangular
lattices\cite{ms01}  at    the  RK   point the      exact ground-state
$\left|x,y\right>$ remains the sum  of   all dimer coverings and   the
ground-state energy is  independent  of the  distance between the  two
spinons (except at  very short  distance  if they belong  to a  common
hexagon).  This is  a first indication that RVB  {\em spin} liquid has
deconfined  spin-$\frac{1}{2}$ excitations   (spinons).   In  the  QDM
language these excitations are simply unpaired sites in a dimer liquid
background. Such  unpaired sites are necessarily  created by pairs but
they can then propagate freely (no attractive potential) when they are
sufficiently far apart.

Another calculation allows  to test the  deconfinement properties of a
dimer liquid.  We   consider the state   $\left|\psi\right>=\sum_{{\bf
r}\ne0}\left|0,{\bf r}\right>$ where  $\left|0,{\bf r}\right>$ is  the
(un-normalized) state with  two spinons  in $0$ and  ${\bf  r}$.   The
probability  to find a spinon  in ${\bf r}$ in the $\left|\psi\right>$
can be obtained by the relatively  involved calculation of the monomer
correlation\footnote{Ratio of number of dimer coverings with two holes
in  $0$ and ${\bf r}$  to the number  without  hole.}  with Pfaffians.
One   the  square   lattice   this  probability   goes   to   zero  as
$1/\sqrt{r}$.\cite{fs63} This shows that the second spinon is (quasi-)
confined  in the vicinity  of  the first  one  on  the square  lattice
because escaping  far away represents a large  ``entropy'' cost in the
dimer background.  On the triangular  lattice it goes exponentially to
a constant.\cite{fms02} This result  is a signature of  deconfinement.
In fact the  same  signature  can be  obtained on  the  kagome lattice
without  any technical  calculation since the   monomer correlation is
exactly $1/4$ at any distance.\cite{msp03c}

If unpaired  sites are  allowed  one can describe spinons   or holons.
Unfortunately in the presence of simple kinetic energy terms for these
objects the model can no longer be solved.  However one can consider a
static spinon  and its  interaction with  visons: when the   spinon is
adiabatically taken around a vison the dimers are shifted along a path
encircling the vison.  Because the  vison wave-function is  particularly
simple in this model it   is easy to  check  that this multiplies  the
wave-function  by  a  factor  $-1$.    This is   the signature  of   a
long-ranged statistical  interaction\cite{rc89,k89} between visons and
spinons (or holons).  In more realistic models,  as long as the visons
are gapped  excitations the spinons  are expected to be deconfined. On
the  other hand if  the  visons condense their long-ranged statistical
interaction with spinons frustrates  their  motion. This is no  longer
true if they propagate   in {\em pairs},   in which case they are  not
sensitive  any more to  visons (see Ref.~\cite{msp02} for an extension
of the present QDM with  a  vison condensation). This simple  physical
picture illustrates the relation between vison condensation and spinon
confinement.

\subsubsection{$\Z$ gauge theory}
\label{sssec:Z2} 

The forces responsible for confinement are usually associated to gauge
fields and their fluctuations.   Whereas $U(1)$ compact gauge theories
are  generically confining in  $2+1$ dimensions,\cite{polyakov87,fs79}
$\Z$    gauge    theories   are    known    to   possess    deconfined
phases.\cite{kogut79} For this reason  some attention has been paid to
the  connexions between  $\Z$  theories and  fractionalized phases  in
2D electronic systems.\cite{sf00}

It is known\cite{msf02} that QDM can be obtained as special limits of
$\Z$ gauge  theories, the gauge variable  being the dimer  number on a
bond.  However, on the kagome lattice this connexion can be made exact
and completely  explicit since  there is a  one to  one correspondence
between   dimer   coverings    and   physical   states   ({\it   i.e.}
gauge-invariant) of  a $\Z$ gauge theory.\cite{msp02}  In this mapping
the  gauge  fields are  Ising  variables living  on  the  link of  the
honeycomb lattice  ({\it i.e.} kagome sites) and  are constructed from
the  arrows described  previously.  As  for the  constraints  of gauge
invariance they correspond to the odd parity of the number of outgoing
arrows on  every triangles. The  $\sigma^x$ operator used to  define a
solvable QDM  translate into a gauge-invariant  plaquette operator for
the gauge  degrees of  freedom (product of  the Ising  gauge variables
around an hexagon). With this mapping the visons appear to be vortices
in the  $\Z$ gauge field  and the solvable model  of Eq.~\ref{eq:kQDM}
maps to the deconfined phase of the gauge theory.

\subsection{A QDM with an extensive ground-state entropy}
\label{ssec:QDMmu}

So far we   have discussed QDM that  realize  either spontaneous VBC,
critical states  or RVB liquids.   We wish to  mention here that these
three  scenarios may  not  be   the  only possible  ground-states  for
QDM. In particular,  a  QDM on the kagome  lattice  with an extensive
ground-state  entropy was  recently  discussed.\cite{msp03} This model
was  introduced from  the observation  that  the  dimer kinetic energy
terms  arising   from   an  overlap  expansion (\S\ref{ssec:spin2QDM})
generally have  non trivial {\em signs}  as soon as resonance loops of
{\em different lengths} are considered.  It was then realized that such
signs (which make  the QDM  no longer  appropriate  for QMC simulations)   can lead to  qualitatively  new phases, different
from      VBC  or RVB   liquids.     The    Hamiltonian introduced  in
Ref.~\cite{msp03}   is           similar      to       that         of
Eqs.~\ref{eq:kQDM}-\ref{eq:sigmax} except that each resonance     loop
$\alpha$   is now included with  a  {\em sign} $(-1)^{n_\alpha}$ where
$n_\alpha=3,\cdots,6$ is the number of dimers involved:
\begin{equation}
	\mathcal{H}=\sum_h (-1)^{n_\alpha} \left[
	\left|d_\alpha(h) \right>\left< \bar{d}_\alpha(h)\right|
  	+
	\left| \bar{d}_\alpha(h)\right>\left<d_\alpha(h)\right|
	\right]
	\label{eq:muQDM}
\end{equation}
These signs are precisely those appearing in the overlap expansion (at
the  order of one   hexagon)  of the  Heisenberg  model on  the kagome
lattice.  This expansion was  carried out by Zeng and Elser\cite{ze95}
in an  insightful   paper which laid  the  basis of the analysis  of the
kagome antiferromagnet in the first neighbor valence-bonds subspace.

Although  not exactly solvable,  the Hamiltonian of Eq.~\ref{eq:muQDM}
was shown to be a dimer  liquid (short-ranged dimer-dimer correlation)
and to have a huge ground-state degeneracy $\sim 2^{N/6}=1.122^N$ ($N$
is  the   number of  kagome sites).    In addition,  several numerical
indications pointed to a critical behavior of this system,\cite{msp03}
with    a     possible     algebraic     decay    of     energy-energy
correlations.\footnote{Notice   that  a   one-dimensional   analog  of
Eq.~\ref{eq:muQDM} can be defined and exactly maps onto the Ising chain
in  transverse field at its  critical point.}  It  was argued that the
effective   QDM    describing      the singlet     dynamics   of   the
spin-$\frac{1}{2}$ Heisenberg  antiferromagnet on  the kagome  lattice
could be {\em  close} (in parameter  space) to Eq.~\ref{eq:muQDM}.  If
correct, this sheds light  on the very large  density of singlet states
observed at low energy in  the numerical  spectra  of that spin  model
(see Sec.~\ref{sec:kagome}).




%_______________________________________________________________________________
\section{Multiple-spin exchange models}\label{sec:MSE}

%_______________________________________________________________________________

\subsection{Physical realizations of multiple-spin interactions}

\subsubsection{Nuclear magnetism of solid $^3$He}

Solid $^3$He was the first magnetic system in  which the importance of
MSE interactions   was recognized.\cite{thouless65,rhd83,cf85}  Due to
the large zero-point motion  of the atoms  about their mean  positions,
tunneling events during which 2, 3 or 4 atoms exchange their positions
in a cyclic way  are frequent.  These  processes generate an effective
interaction between the (nuclear) spins which can be written
\begin{equation}
  \mathcal{H}=\sum_P -J_P (-1)^P\left( P_{spin} + P_{spin}^{-1}\right)
\end{equation}
where the  sum runs over  permutations  $P$,  $J_P>0$ is the  exchange
frequency  of the associated tunneling   process (in  real space)  and
$P_{spin}$ acts on the  Hilbert space of spin-$\frac{1}{2}$ located on
the site of  the crystal. The sign $-(-1)^P$  depends of the signature
of the permutation  $P$ and is a consequence  of  the Pauli principle.
For a {\em cyclic}  permutation involving $n$ spins  this sign is just
$(-1)^n$  and  is  responsible   for  the ferromagnetic  character  of
processes involving  an odd number   of spins.  For spin-$\frac{1}{2}$
particles, two  and three-spin exchange  terms  reduce to the familiar
Heisenberg interaction:
\begin{eqnarray}
  P_{12}=2\vec{S}_1\cdot\vec{S}_2+\frac{1}{2} \label{eq:P2} \\
  P_{123}+P_{321}=P_{12}+P_{23}+P_{31}-1 \label{eq:P3}
\end{eqnarray}
but this is no longer true for $n\geq4$:
\begin{equation}
P_{1234}+P_{4321}=P_{12}P_{34}+P_{14}P_{23}-P_{13}P_{24}+P_{13}+P_{24}-1
\end{equation}
which can be  expressed (thanks to Eq.~\ref{eq:P2}) as  a sum of terms
with two and four Pauli matrices. 

$^3$He can form solid  atomic mono-layers  with a triangular  geometry
when adsorbed on a graphite substrate at ultra low temperatures (milli
Kelvin range).  This 2D  magnet has been  studied for a long time (see
Refs.~\cite{greywall,gr95}  and references therein) and the importance
of MSE   interactions  involving up    to  six  atoms has  now    been
recognized.\cite{roger84,rbbcg98} The exchange frequencies of the most
important processes have been   computed by Path Integral  Monte Carlo
(PIMC)\cite{cj87,bcl92,ceperley95,bc99}  (analytic  WKB   calculations
have  also been carried  out\cite{roger84,ah00})  as a function of the
density.    The proposed  MSE   Hamiltonian  describing the   magnetic
properties of this 2D quantum crystal reads
\begin{eqnarray}\label{Hmulti}
\mathcal{H}=(J_2-2J_3) \sum_{
        \begin{picture}(17,10)(-2,-2)
                \put (0,0) {\line (1,0) {12}}
                \put (0,0) {\circle*{5}}
                \put (12,0) {\circle*{5}}
        \end{picture}
} P_{12}
+J_4 \sum_{
\begin{picture}(26,15)(-2,-2)
        \put (0,0) {\line (1,0) {12}}
        \put (6,10) {\line (1,0) {12}}
        \put (0,0) {\line (3,5) {6}}
        \put (12,0) {\line (3,5) {6}}
        \put (6,10) {\circle*{5}}
        \put (18,10) {\circle*{5}}
        \put (0,0) {\circle*{5}}
        \put (12,0) {\circle*{5}}
        \end{picture}
} \left( P_{1\ldots 4}+{\rm H.c}\right)\\
-J_5 \sum_{
\begin{picture}(26,15)(-2,-2)
        \put (0,0) {\line (1,0) {24}}
        \put (6,10) {\line (1,0) {12}}
        \put (0,0) {\line (3,5) {6}}
        \put (18,10) {\line (3,-5) {6}}
        \put (6,10) {\circle*{5}}
        \put (18,10) {\circle*{5}}
        \put (0,0) {\circle*{5}}
        \put (12,0) {\circle*{5}}
        \put (24,0) {\circle*{5}}
        \end{picture}
} \left( P_{1\ldots 5}+{\rm H.c}\right) \nonumber
+J_6 \sum_{
\begin{picture}(26,30)(-2,-15)
        \put (6,10) {\line (1,0) {12}}
        \put (6,-10) {\line (1,0) {12}}
        \put (0,0) {\line (3,5) {6}}
        \put (0,0) {\line (3,-5) {6}}
        \put (18,10) {\line (3,-5) {6}}
        \put (18,-10) {\line (3,5) {6}}
        \put (6,10) {\circle*{5}}
        \put (6,-10) {\circle*{5}}
        \put (18,10) {\circle*{5}}
        \put (18,-10) {\circle*{5}}
        \put (0,0) {\circle*{5}}
        \put (12,0) {\circle*{5}}
        \put (24,0) {\circle*{5}}
        \end{picture}
} \left( P_{1\ldots 6}+{\rm H.c}\right)
\label{eq:H26}
\end{eqnarray}
where Eq.~\ref{eq:P3} was used to absorb the  three-spin terms into an
effective      first-neighbor      Heisenberg     exchange   $J_2^{\rm
eff}=J_2-2J_3$. At high density the hard-core potential between Helium
atoms only leaves three-body  exchanges possible  ($J_3\gg J_{n\ne3}$)
and Eq.~\ref{eq:H26} reduces    to a first neighbor   Heisenberg  {\em
ferromagnet},\cite{drh80} as observed   experimentally for  the  first
time by Franco {\it et al.}\cite{frg86} in high-density layers. On the
other hand the  second layer solidifies  at  lower density and  higher
order exchange terms  cannot be  ignored.\footnote{The first layer  is
then so  dense that exchange is strongly  suppressed.  The first layer
can  also   be replaced  by   an   $^4$He  or  HD  mono-layer.}   PIMC
simulations\cite{bc99} and  high-temperature  fits of the experimental
data\cite{rbbcg98}  showed that   the  relative strength   of two- and
four-spin terms   if  roughly  $J_2^{\rm    eff}/J_4\sim -2$  in   the
low-density second layer solid.  The $J_2$--$J_4$ model was studied by
exact diagonalizations in this region of parameter space and evidences
for  a   short-ranged RVB SL   phase   with no  broken  symmetry  were
obtained.\cite{mblw98,mlbw99,lmsl00}   The    ultra-low    temperature
measurements      of   specific     heat\cite{kmyf97}    and   uniform
susceptibility\cite{cthrbbg01} are  not incompatible  with such a spin
liquid phase but the spin gap,  if any, has not  yet been observed 
and should be smaller than 100$\mu$K.


\subsubsection{Wigner crystal}

The   Wigner crystal is  another    fermionic solid with a  triangular
geometry where MSE  interactions can play an  important role.  At very
low density the  Coulomb   energy  dominates, the crystal   is  almost
classical and MSE interactions  are very small.   Exchange frequencies
$J_P$   can be  computed in  this   regime by  a  semi-classical (WKB)
approximation\cite{roger84,kh00,hk01}  and,  as  for  the high density
solid of  $^3$He,  three-body  exchanges dominate  and  give  rise  to
ferromagnetism. However, at higher  density and close to melting, PIMC
calculations  of the  exchange  frequency\cite{bcc01} showed  that the
magnetism  may be described by  a MSE model with parameters ($J_2^{\rm
eff}$  and $J_4$)  close to those   where the triangular  MSE model is
expected to be a RVB  SL.  Unlike the  $^3$He case, the  particles
(electrons) are  charged and  an external magnetic  field has  also an
orbital effect, it introduces complex phases in the exchange energies:
$P+P^{-1}\to   e^{i\alpha}P+e^{-i\alpha}P^{-1}$  where    the    angle
$\alpha=2\pi
\phi/\phi_0$  is  proportional to  the  magnetic  flux $\phi$  passing
through the area  enclosed by the exchange trajectory  and $\phi_0$ is
the  unit  flux  quantum.  This  can  give  rise  to very  rich  phase
diagrams\cite{bcc01,hk01}  where complex  MSE terms  compete  with the
Zeeman  effect  (see  Ref.~\cite{ok98}  for  some  early  experimental
attempts to explore this physics).

\subsubsection{Cuprates}

The possibility  of significant  four-spin exchange  around  square Cu
plaquettes in copper oxide compounds  was first suggested by Roger and
Delrieu.\cite{rd89} They interpreted  the  anomalously large width  of
Raman scattering spectra as a signature of  four-spin exchange in this
copper oxide superconductor.  The importance of these MSE interactions
in CuO$_2$ planes  ($J_4\sim 0.25 J_2$) has  then been emphasized by a
number   of  groups and   in   different  materials  and  by different
experimental                      and                      theoretical
approaches.\cite{sugai90,coldea01,mr02,kk03} Four-spin  plaquette ring
exchange    also   plays      a     significant   role     in   ladder
compounds.\cite{mkebm00,bmmnu99,sku01,gkt03}   For  instance, exchange
parameters    with  values    $J_{rung}=J_{leg}=110$~meV           and
$J_{ring}=16.5$~meV  were  proposed  for La$_6$Ca$_8$Cu$_{24}$O$_{41}$
based     on    the       dispersion      relation    of      magnetic
excitations.\cite{mkebm00,bmmnu99}

\subsection{Two-leg ladders}
\label{ssec:ladderMSE}

Numerous  works   were  devoted   to  ladder  models   with  four-spin
interactions.  These include  general bi-quadratic interactions as well
as models  with ring-exchange  terms.  We will  only discuss  here the
simplest of these MSE models:
\begin{eqnarray}
  \mathcal{H}=&J&\sum_n \left(
    \vec{S}_{n,1}\cdot \vec{S}_{n,2}
    +\vec{S}_{n,1}\cdot \vec{S}_{n+1,1}
    +\vec{S}_{n,2}\cdot \vec{S}_{n+1,2}
    \right) \nonumber \\
    +&K&\sum_\Box \left(
    P_{1234} +H.c
    \right)
    \label{eq:ladder}
\end{eqnarray}
Thanks  to several  studies\cite{bmmnu99,mvm02,hmh03,lst03}  the phase
diagram of  this Hamiltonian  is now rather  well understood  and five
different phases were identified.
\begin{itemize}

\item   {\bf   Ferromagnetic  phase}.   The   ground-state  is   fully
polarized. This phase includes  the $(J=-1,K=0)$ and the  $(J=0,K=-1)$
points.

\item {\bf  Rung-singlet phase}.  This phase  includes the ground-state of
the ladder without  MSE term $(J=1,K=0)$.  The  spectrum is gapped and
the   ground-state  is  unique.    A moderate  $K/J\gtrsim0.23\pm0.03$
destroys this  phase\cite{bmmnu99,hh01,hn02,lst03} in favor of the VBC
below.

\item {\bf Staggered VBC} with dimers  on the legs.  In one of the two
degenerate ground-states the dimerized bonds are $(2n,1)-(2n+1,1)$ and
$(2n+1,2)-(2n+2,2)$.       The      VBC     disappears             for
$K/J\gtrsim0.5$.\cite{lst03} Such  a staggered VBC was first predicted
in  the framework   of   a ladder  with  bi-quadratic interaction   by
Nersesyan   and   Tsvelik.\cite{nt97} Using     Matrix-Product Ansatz,
Kolezhuk    and  Mikeska\cite{km98}    constructed  models which   are
generalizations    of  Eq.~\ref{eq:ladder}  and   which    have  exact
ground-state with long-ranged  staggered dimer correlations.   In this
phase the   magnetic excitations are  very  different from  the magnon
excitations of the rung-singlet  phase above. Here the  excitations do
not form well-defined quasi-particles but a  continuum made of pairs of
domain walls connecting two dimerized ground-states.\cite{nt97,km98}
 
\item  {\bf   Scalar  chirality   phase}.   The  order   parameter  is
$\langle\vec{S}_{n,1}\cdot(\vec{S}_{n,2}\times\vec{S}_{n+1,2})\rangle$
and it spontaneously breaks the time-reversal symmetry and translation
invariance.  The  ground-state is two-fold  degenerate up to  the next
transition at $K/J\simeq2.8\pm0.3$.\cite{lst03} There exists a duality
transformation\cite{hmh03,mhnh03} which  maps the  scalar    chirality
order parameter onto   the   dimer   order   parameter of   the    VBC
above.\footnote{The Hamiltonian of Eq.~\ref{eq:ladder} is self-dual at
$2K=J$.}    Applying    such a  transformation     to  the  exact  VBC
ground-states   mentioned   above,  models   with  an  exactly   known
ground-state       and   scalar      chirality     LRO   can        be
constructed.\cite{hmh03,mhnh03} Although  chiral  SL  have  been  much
discussed  in  the  literature, this is   to  our knowledge  the first
realization of such a phase in a SU(2) symmetric spin-$\frac{1}{2}$ model.


\item   {\bf  Short-ranged   ordered   phase  with   vector-chirality}
correlations.        The         strongest        correlations     are
$\langle(\vec{S}_{n,1}\times\vec{S}_{n,2})\cdot(\vec{S}_{n',1}\times\vec{S}_{n',2})\rangle$
but they  remain short-ranged.    The   spectrum is gapped  and    the
ground-state  is unique.  This   phase includes  the pure $K=1$  model
where  $J=0$.  This phase   is  related by the duality  transformation
discussed    above to   the   rung-singlet   phase.\cite{mhnh03}  This
transformation    indeed relates   the  N\'eel  correlations  $\langle
(\vec{S}_{n,1}-\vec{S}_{n,2})\cdot(\vec{S}_{n',1}-\vec{S}_{n',2})\rangle$
(which  are  the strongest  ones  in the   rung-singlet phase)  to the
vector-chirality correlations.  Close    to the ferromagnetic    phase
($J<0$) one observes   a crossover to  a  region where the   strongest
correlations are  ferromagnetic spin-spin correlations along  the legs
and antiferromagnetic along the rungs.\cite{lst03}
\end{itemize}
\subsection{MSE model on the square lattice}
The phase  diagram  of  the Hamiltonian  \ref{eq:ladder}  on the  {\em
two-dimensional    square-lattice}  has   been   recently   studied by
L\"auchli\cite{lauchli03}  by  exact      diagonalizations.   N\'eel,
ferromagnetic, columnar VBC and staggered  VBC phases were identified,
as   in the  ladder   model  above.    In  addition,  a nematic  phase
characterized by  long-ranged vector chirality correlations (alternating
spin currents)  was  found  around  the $K=1$, $J=0$  point.   To  our
knowledge this is the first  microscopic realization of a nematic order
in a two-dimensional spin-$\frac{1}{2}$ model.

\subsection{RVB phase of the triangular $J_2$--$J_4$ MSE}
\label{ssec:RVBMSE}

Because  of its relevance  to solid $^3$He  films and Wigner crystals,
the MSE model on the  triangular lattice has been  the subject of many
studies.\cite{km97,mkn97,mbl98,ksmn98,mblw98,mlbw99,lmsl00}  We   will
discuss here  some properties  of the  simplest MSE  model with up  to
four-spin cyclic exchange interactions ($J_2  -2J_3$ and $J_4$ only in
Eq.~\ref{eq:H26}).         The   classical           phase     diagram
(Fig.  \ref{fig:phasediagMSE}) of this  model has been studied by Kubo
and   collaborators\cite{km97,mkn97}  and the  quantum   one has  been
roughly scanned  in  Ref.\cite{lmsl00}: we  will mainly focus   on the
short-ranged RVB  spin liquid (see Fig.~\ref{fig:phasediagMSE}), which
is probably  {\it the first RVB  SL encountered in  an SU(2)-symmetric
spin model}.

\begin{figure} 
\begin{center}
\includegraphics[width=8cm]{MSE}
\end{center}
\caption[99]{
Classical (top)  and  quantum (bottom)  phases   of the $J_2-J_4$  MSE
Hamiltonian.      The classical  model   was    studied  by  Kubo  and
Momoi\cite{km97} and is based on a variational approach.  The quantum
phase diagram  is  the  simplest scenario  compatible with  the  exact
diagonalizations data of Refs.~\cite{mblw98,mlbw99,lmsl00}.  While not
completely understood, in  the type-II spin-liquid  region the spectra
are characterized by a very  large number of singlet excitations below
the first triplet state. This is not the case in the RVB phase.}
\label{fig:phasediagMSE}
\end{figure}

\subsubsection{Non-planar classical ground-states}

It     is  well-known that      an  Heisenberg  model  (with  possible
second-neighbors,  third-neighbors, ...   interactions)  on a  Bravais
lattice  always admits a planar  helical ground-state at the classical
level. This is no longer  true when  MSE  are present and finding  the
classical  ground-state for arbitrary  $J_2$  and $J_4$ is an unsolved
problem.  A  mean-field phase diagram was  obtained  for the classical
model\cite{km97} but   very   few exact  results  are  known.   In the
neighborhood of $J_4=1$ $J_2=0$ the classical ground-state is known to
be  a  four-sublattice  configurations   with  magnetizations pointing
toward the  vertices of  a  regular tetrahedron.\cite{km97}  This is a
quite interesting model where the  ground-state spontaneously breaks a
discrete Ising  symmetry associated to the sign  of the triple product
$\vec{S}_1\cdot(\vec{S}_2\times\vec{S}_3)$   around a triangle.   This
broken symmetry  gives rise to  a finite-temperature  phase transition
which has  been observed in  Monte Carlo simulations.\cite{mkn97} This
phenomena is similar to    the transition predicted in  the  $(\pi,0)$
phase of the $J_1$--$J_2$ model on the square lattice.\cite{ccl90a}

\subsubsection{Absence of N\'eel LRO}


The   classical  ground-states   at   $J_4=1,J_2=0$  are   tetrahedral
configurations.  Although  this phase appears to be  stable within the
framework    of   linear    spin-wave    calculations\cite{mkn97}   or
Schwinger-Boson mean-field  theory,\cite{mbl98} exact diagonalizations
indicate   that  the   magnetic   LRO  is   washed   out  by   quantum
fluctuations.\cite{mkn97,mlbw99}   The   chiral  order  predicted  to
survive at  long distances and finite  temperatures\cite{mkn97} in the
classical  system for  $J_2=0$  is also  likely  to be  washed out  by
quantum fluctuations.\cite{mlbw99}

When $J_2=1$ a relatively small  amount of $J_4\sim 0.1$ is sufficient
to   destroy  the   three-sublattice  N\'eel   LRO  realized   by  the
first-neighbor Heisenberg model.\cite{lmsl00}  The nature of the phase
on  the  other  side  of  this  transition  is  not  settled  but  the
finite-size spectra display a  large density of singlet excitations at
low   energy    which   could   be   reminiscent    of   the   kagome
situation.\cite{lmsl00}

From  exact diagonalizations (up  to 36 sites) no  sign  of N\'eel LRO
could be found at $J_4=1,J_2=-2$.\cite{mblw98,mlbw99} In addition, the
finite-size analysis showed  that the spin-spin  correlation length is
quite short  at $J_2=-2$ and  $J_4=1$ and a spin  gap of  the order of
$\Delta\sim 0.8$ exists at this  point.  Much of the numerical  effort
to  elucidate the nature of the  MSE  ground-state was concentrated on
this   point  because it    is  close to   the parameters  realized in
low-density $^3$He films (when higher order exchanges are neglected).


\subsubsection{Local singlet-singlet correlations  -
  absence of  lattice   symmetry breaking}

Having  excluded the possibility of  a  N\'eel ordered ground-state at
$J_4=1,J_2=-2$ it is  natural to look for  a possible VBC.  Because of
the complexity of the  MSE Hamiltonian it   is not clear what kind  of
spatial  order should  be favored.  From   the analysis of dimer-dimer
correlations (see Fig.~\ref{fig:ddcorrel})  it  appears that  parallel
valence-bonds repel each-other at  short distance.  This is similar to
what is observed in the staggered phase of the $J_2$--$J_4$ MSE ladder
and  square-lattice   models.  For  this   reason  it appears  that  a
plausible VBC would be the staggered VBC encountered in the triangular
QDM  for  $V>J$  (\S\ref{ssec:QDMTri}).   However this  scenario seems
difficult   to    reconcile   with  the    weakness     of dimer-dimer
correlations.\cite{mlbw99} In  addition, the low-energy singlet states
and their quantum numbers\cite{StagVBC}  do  not reflect the   12-fold
quasi-degeneracy   that should  be present    if  the  system  was  to
spontaneously break some lattice symmetry according to a staggered VBC
pattern.  Small systems usually  {\em  favor} ordered  phases  because
low-energy and long-wavelength  fluctuations that could destabilize an
ordered state are reduced  compared to larger  systems.  From the fact
that the finite-size spectra do not show the signatures of a staggered
VBC symmetry breaking it is unlikely that the  MSE model could develop
a VBC of this kind in the thermodynamic limit.

 
\begin{figure} 
\begin{center}
\includegraphics[width=8cm]{DD28_m210}
\end{center}
\caption[99]{Dimer-dimer correlations in the ground-state
of  the $J_2$--$J_4$ MSE  model on the triangular  lattice  (28 sites) at
$J_2=-2$, $J_4=1$ (from  Ref.~\cite{mlbw99}). Numbers are proportional
to          $\langle\hat{d}_0\hat{d}_x\rangle        -\langle\hat{d}_0
\rangle\langle\hat{d}_x\rangle$  where the  operator       $\hat{d}_x$
projects onto the singlet space of the bond $x$ and $\hat{d}_0$ refers
to the reference  bond $(1,28)$. These  results shows a clear tendency
for repulsion between parallel dimers.}
\label{fig:ddcorrel}
\end{figure}

\subsubsection{Topological degeneracy and Lieb-Schultz-Mattis Theorem}

Because no  VBC   phase could be  identified   in  the  MSE model  at
$J_4=1,J_2=-2$ the numerical  data were compared  with the predictions
of an RVB liquid scenario.

In one  dimension    a  famous   theorem due   to  Lieb,   Schultz  and
Mattis\cite{lsm61} (LSM) states  that in   a one-dimensional spin system
with an half-integer spin in the unit cell there is at least on excited
state collapsing to the ground-state in the thermodynamic limit
(periodic boundary conditions).
There are in fact several arguments suggesting that this theorem might,
at least to some extent,
also  apply to higher dimensions.\cite{al86,o00,mlms02,hastings03,o03}
If  that is the case  a gapped system with  an odd integer spin in the
unit cell must have a degenerate  ground-state.  The simplest scenario
to  explain this degeneracy is a  translation  symmetry breaking.  One
could   think that   this  would  rule  out  the  possibility of   any
(translation invariant) RVB liquid  in such models.  This is incorrect
because a ground-state degeneracy  can have a  topological origin on a
system with  periodic boundary  conditions,  as  we discussed in   the
framework of QDM (\S\ref{sec:QDM}).  Such  a phase is characterized by
a four-fold topological ground-state degeneracy  when the system is on
a torus.   That    degeneracy  allows the  system   to  fulfill  LSM's
requirement   without     any spontaneous       translation   symmetry
breaking.\cite{mlms02}

On a finite-size system the topological degeneracy is only approximate
but  some constraints   exist for the  quantum  numbers  (momentum  in
particular)  of the quasi-degenerate  multiplet.\cite{mlms02} A system
with periodic boundary conditions with an even  number of sites but an
{\em odd  number of rows}  is expected to  have two ground-states with
differ by a momentum $\pi$ in  the direction parallel  to the rows, in
close analogy to the  the LSM theorem in  dimension one. The numerical
spectra of the MSE model exhibit a set  of three singlet energy levels
collapsing  onto     the  ground-state  when  the   system     size is
increased\cite{mlbw99} and their    quantum number  turn  out  to   be
consistent with  the   constraints   derived from the      general RVB
picture.\cite{mlms02}

\subsubsection{Deconfined spinons}

The  SL phase described above is   expected to have deconfined spinons
($S=\frac{1}{2}$ excitations).  These excitations should show up as an
incoherent   continuum    in   the   spin-spin    dynamical  structure
factor.  However such a feature would  probably be rather difficult to
observe on small 2D lattices, in particular due to the small number of
inequivalent ${\bf k}$-vectors  in  the Brillouin zone.  On  the other
hand, the binding energy of two spinons can  be evaluated by comparing
the ground-state  energy and the first   magnetic excitation energy on
even and odd samples.  In the case  of the MSE model at $J_4=1,J_2=-2$
the results show the existence of  a bound-state (it is more favorable
to put  two spinons in the  same  small sample than  in separate ones,
which is not surprising) but this {\em does  not mean that the spinons
are confined} (contrary  to  the conclusions  of  Ref.~\cite{mlbw99}).
Interestingly this binding energy seems to  go to zero for the largest
available sizes~(Fig.~\ref{spinon}):  this might  indicate the absence
of attraction between   spinons for  large  enough  separation and  an
asymptotic deconfinement.

\begin{figure}
\begin{center}
\includegraphics[height=4cm,clip=true]{spinonmse}
\caption[99]{Spinons binding energy as a function of the system size in
the MSE model at $J_2=-2$ and  $J_4=1$. The vertical bars correspond to
the  range    of    values found     for  different  sample    shapes.
}\label{spinon}
\end{center}
\end{figure}

It is important to  stress here that  the RVB SL phase discussed  here
(and    its   QDM    counterparts    of     \S\ref{ssec:QDMTri}    and
\S\ref{ssec:QDMKag})  is not the only  way to  spinon deconfinement in
2D. There is  at least another scenario, inherited from  one
dimension, which is the  {\em sliding Luttinger liquid}.  Indeed,  the
Luttinger liquid behavior and the one-dimensional critical behavior of
magnetic  chains  seem  to  be  robust  to  small (or  moderate)  {\em
frustrating transverse couplings between chains},  as observed both in
theoretical\cite{nge98,ahllt98,efkl00,be01,vc01,sp02,ssl02}        and
numerical  approaches.\cite{sfl02}  This  regime between  one  and two
dimensions which may have been observed in Cs$_2$CuCl$_4$\cite{cttt01}
is the subject of a number of recent studies.\cite{be01,vc01}


\subsection{Other models with MSE interactions}

Multiple-spin interactions are present in a number of models that were
found   to exhibit fractionalization   or  an RVB liquid ground-state.
Well known examples of MSE interactions with an Ising symmetry are $\Z$
gauge theories, where the gauge  invariant plaquette term is a product
of    Pauli  matrices $\prod_i   \sigma^z_i$.    Such  theories have a
deconfined phase in 2+1 dimension\cite{kogut79} and their relevance to
fractionalized phases of  2D  electronic systems has  been
pointed  out by  Senthil and Fisher.\cite{sf00}  The connexion between
$\Z$ gauge theories and QDM was mentioned in
\S\ref{sssec:Z2}. Some MSE  spin models with  an Ising  symmetry and a
fractionalized  ground-state were  discussed by Kitaev,\cite{kitaev97}
Nayak and   Shtengel.\cite{ns01} In the other  limit  of a  $U(1)$ (or
$XY$) symmetry several models  have been studied.  Recent examples are
based  upon  the   spin-$\frac{1}{2}$  four-spin   XY  ring   exchange
interaction\cite{bfg02,pbf02,sdss02,sm02}
\begin{equation}
  \mathcal{H}=-K\sum_{\langle ijkl\rangle} \left(
  S_i^+S_j^-S_k^+S_l^- + {\rm H.c}
\right)
\label{eq:4XY}
\end{equation}
which is the $XY$ analog of the $SU(2)$ MSE interaction $P_{1234}+{\rm
H.c}$. 

%______________________________________________________________________________

\section{Antiferromagnets on the kagome (and related) lattices}
\label{sec:kagome}


In RVB SL phases described in \S\ref{ssec:QDMTri}, \S\ref{ssec:QDMKag}
and \S\ref{ssec:RVBMSE} {\em singlet excitations (visons) are gapped}.
In the  $\Z$ gauge  theory  approach this  gap  is essential  for  the
consistency  of the   theory.\cite{sf00,sf01} The   gauge-field  quasi
particles  are  vortices   of the    gauge field  and   carry  a  unit
$\mathbb{Z}_2$ gauge flux but no spin.  As illustrated in
\S\ref{sssec:QDMdeconf} in the framework  of a simple QDM, visons have
long-ranged interaction    with spinons.  If   the   spectrum of these
visons has a gap  then the spinons  are unconfined and the phase is
``fractionalized''.  If  they   condense, the  long range  interaction
between them and the spinons frustrate the motions of the latter which
remained confined.  The  gap    in  the singlet sector     (above  the
topological degeneracy)   is thus a  crucial  ingredient of  these RVB
SL. At  the  end  of this   section we  will  show how    the spin-$\frac{1}{2}$
first-neighbor Heisenberg model   on the kagome lattice  represents an
enigma with  respect to this scheme  (as well as  the QDM described in
Sec.~\ref{ssec:QDMmu}).  We  begin  the section by   a review  of  the
properties of various models on  the kagome lattice, which topology is
the source of an extreme  frustration, and, in  many models, source of
an extensive degeneracy of the ground-state.

\subsection{Miscellaneous models on the kagome lattice}

There  has been  a large    number  of studies  devoted to   different
antiferromagnetic  models  on the  kagome  lattice.  The next neighbor
Ising model on such a  lattice is disordered, its  entropy per site is
very large $S_{\rm   kag}^{\rm  Ising}= 0.502$,  more   than half  the
independent spin value, much  larger than the triangular lattice value
$S_{\rm tri}^{\rm Ising}=  0.323$   and   of  the order of     Pauling
approximation      for    independent       triangles          $S_{\rm
Pauling}=0.501$.\cite{p38} This suggests that the correlations in this
system   are very  weak:   the   model   remains disordered at     all
temperatures.\cite{kn53,hr92} Moessner  and Sondhi have  studied  this
Ising model   in  a transverse magnetic   field  (the simplest way  to
include some quantum fluctuations  in the model):  the model  fails to
order for any transverse field, at any temperature.\cite{msc00,ms01b}

The nearest neighbor classical Heisenberg  model on the kagome lattice
has  also a  huge ground-state degeneracy.    This property is easy to
understand and  holds on different lattices  with corner sharing units
such as the checkerboard  lattice or the three  dimensional pyrochlore
lattice (Moessner and Chalker\cite{mc98,mc98a}). On all these lattices
the nearest neighbor Heisenberg Hamiltonian  can be written as the sum
of the square of the total spin $\vec{S}_{\alpha}$ of individual units
$\alpha$ (a   tetrahedron in the  2-d and  3-d pyrochlore cases  and a
triangle   for the     kagome  lattice),   which   share only      one
vertex. Classical ground-states are obtained whenever $\forall\alpha\;
\vec{S}_{\alpha}=\vec{0}$.    This     condition fixes   the  relative
positions of the three classical spins of a  triangle at $120$ degrees
from each   other  in a  plane.   But  it does  not  fix  the relative
orientation of the plane of   a triad with  respect  to the planes  of
triads on corner sharing triangles:  the model has a continuous  local
degeneracy\cite{chs92,hr92}    at  $T=0$.\footnote{ Counting  the {\em
planar}  ground-states amounts to determine in  how  many ways one can
associate  one of the three  letters $A$, $B$ and  $C$ to each site so
that each triangle  has spins along  the three different orientations.
This  already  represents  an extensive  entropy.\cite{baxter70,hr92}}
Thermal                fluctuations        select             coplanar
configurations.\cite{chs92,hr92,rcc93}  The possibility of  long-range
order in  spin-spin  correlations at   very low temperature   has been
discussed without any definitive conclusion.\cite{hr92,rb93} The order
parameter of  the planar  phase   is defined  by the   local  helicity
(sometimes called vectorial chirality)~:
\begin{equation}
\vec{\zeta}=
	\vec{S}_1\times\vec{S}_2
	+\vec{S}_2\times\vec{S}_3
	+\vec{S}_3\times\vec{S}_1
\end{equation}
where the  three sites   define a triangle.   This   kind of order  is
sometimes called  {\em nematic} by  analogy  to liquid  crystals.  The
existence   of  such an  order    parameter might  be related   to the
instability of the classical Heisenberg model on the kagome lattice to
Dzyaloshinsky-Moriya interactions.\cite{e02,ecl02} The classical model
has    a    large  density    of    low-lying  excitations    at   low
temperature.\cite{k94}


\subsection{
Spin-$\frac{1}{2}$  Heisenberg  model on    the   kagome lattice:  an    extreme
play-ground for ``quantum fluctuations''}

The  nearest neighbor spin-$\frac{1}{2}$ quantum  Heisenberg model on the kagome
lattice   has        equally    been     the     object      of   many
studies\cite{e89,ce92,s92,le93,ze95,lblps97,web98,m98,smlbpwe00,mm01,dmnm03}
From these studies the following important facts have emerged :

\subsubsection{Ground-state energy per spin}
The Heisenberg model on the kagome lattice has an extremely low energy
per  bond ($\langle{\bf  S}_i.{\bf  S}_j\rangle\simeq  -0.44)$ $  \sim
87\%$ of the energy per bond in  an isolated triangle. On this lattice
the energy per bond  of  the spin-$\frac{1}{2}$  system  is much lower than  the
classical energy $\frac {E_{qu.}}{E_{cl.}}  \sim  1.74$, a ratio  much
larger than   in any other  2D magnet,  that can  only be
compared to the value obtained for the Bethe chain (1.77).  The kagome
lattice   is  the 2D lattice   which  offers the largest
stabilization due to quantum fluctuations.

\subsubsection{Correlations}

The  ground-state  is disordered.\cite{ze90}  Within  accuracy of  the
finite-size numerical computations, spin-spin correlations,\cite{le93}
dimer-dimer          correlations          (Fig.~\ref{dim-dimkagMSE}),
chirality-chirality correlations\cite{ce92} are short-ranged, which is
consistent   with   the    previous   point   and   series   expansion
results.\cite{sh92}
\begin{figure}
\begin{center}
\includegraphics[height=6cm]{dimer-dimerMSEKAG}
\caption[99]{Dimer-dimer correlations in the ground-state of the
kagome  Heisenberg   model (black  triangles)  and   in the  MSE model
discussed    in     \S\ref{ssec:RVBMSE}    (open     symbols)   versus
distance. Although the decrease of these correlations is weaker in the
KH  model  than  in the  MSE  model,   it seems  nevertheless  roughly
exponential in the  first two decades,  as  the spin-spin correlations
are.\cite{ce92,le93}}\label{dim-dimkagMSE}
\end{center}
\end{figure}

\subsubsection{Spin-gap}
There is plausibly a spin-gap  of the order of  1/20$^{\rm th}$ of the
coupling  constant.\cite{web98} In view of     the smallness of   this
spin-gap with regards to the available sizes caution is necessary. The
above  conclusion  was drawn from  the  value of the {\em microscopic}
spin  gap    $E_0(S=S_{min} +1)-E_0(S=S_{min})$ on    exact spectra of
samples with up to 36 spins,\cite{web98} where  $E_0(S)$ is the lowest
energy in the $S$ sector, and $S_{min}=$  0 or $\frac{1}{2}$ depending
on the  parity of the total  number of spins.  The finite-size effects
on these results are an order of magnitude  smaller than in a N{\'e}el
ordered antiferromagnet. Nevertheless   they are still  not negligible
for these sizes. An  alternative determination of  the spin-gap can be
obtained  along the following  line.   The lowest exact eigenstate  in
each total  spin sector $S$ of a  $N$ spins  sample defines the energy
per spin  $e$  at  $T=0$  as  a function  of   its magnetization  $m =
S/(N/2)$.  For  low positive values of  the magnetization, one can fit
$e(m)$ to    the phenomenological  law:\footnote{This phenomenological
form cannot  extend  below $m=0$  and beyond  $m=1/3$ because  at both
magnetizations an angular point appears in $e(m)$ with a discontinuity
of    the      first    derivative    signaling     a    magnetization
plateau.\cite{h01,cghp02} This   is discussed in  more details  in our
lectures notes.\cite{lm02}}
\begin{equation}
e(m) = e(0) + a m + b m^2/2 +{\cal O}(m^3)
\label{eofm}
\end{equation}
$a$ and  $b$ are intensive  quantities which depend  on $N$ but should
converge to some (possibly zero) value when $N\to\infty$.  This point
of view is important because for an  infinite system the thermodynamic
function  $e(m)=\lim_{N\to\infty}E(S=mN/2)/N$ can be measured (through
the zero    temperature        magnetization    curve),    but      not
$E(S\sim\mathcal{O}(1))$.  The physical significance of Eq.~\ref{eofm}
is clear: $a$ measures half  the spin gap  (in a thermodynamic  sense,
that is  the magnetic field $H_c$  required to magnetize the system at
zero temperature) and
\begin{equation}
b=\frac{\partial^2 e}{\partial m^2} = \chi^{-1}
\end{equation}
where  $\chi$ is  the   homogeneous susceptibility of  the  medium for
fields larger than the  critical field $H_c=a$.  This determination of
the {\em thermodynamic} spin gap leads to a renormalization of the raw
data\cite{web98} for small  sizes (see Fig.~\ref{gapkago}).  A  linear
extrapolation  versus $1/N$  (which should give  a lower  bound of the
spin-gap)  leads  to   the     value  $0.06$  for   the     spin   gap
($e_{\infty}=-0.4365$  and $\chi_{\infty}=0.34$).  This  determination
is in  agreement with the direct extrapolation  of the microscopic gap
$E_0(S=S_{min}  +1)-E_0(S=S_{min})$.    For   $N=36$ at  the  smallest
non-zero  magnetization,  the linear  term of equation~(\ref{eofm}) is
$90\%$  of the quadratic term: this  is an  estimate  of the degree of
confidence on the existence of a spin-gap.
 
\begin{figure}
\begin{center}
\includegraphics[height=5cm]{gapkago}
\caption[99]{Finite size scaling of the spin gap in the spin-$\frac{1}{2}$
Heisenberg model ${\cal H} =  \sum_{<i,j>} {\bf S}_i.{\bf S}_j$ on the
kagome lattice.  Triangles (resp. squares) are  the raw results of the
microscopic   spin  gap in the  even    (resp.  odd) samples.  Bullets
represent the  ``thermodynamic'' spin gap,   they are obtained by  the
procedure described in   the text  (errors  bars  come  from  the  rms
uncertainty in the fits).}\label{gapkago}
\end{center}
\end{figure}



\subsubsection{An exceptional density of low lying excitations
 in the singlet sector}

Whatever the ultimate fate of the spin gap a big surprise emerges from
the exact spectra:  the probable absence of gap  in the singlet sector
and  the   anomalous density  of  low  energy  states  adjacent to the
ground-state.  Let us first comment  the second point in details: even
on the smallest size spectra the low lying states appear contiguous to
the  ground-state and the spectra are  extremely dense.  The number of
singlet  levels in the spin-gap  (taken as a natural energy band-width
of the problem) increases exponentially fast with $N$ as $\sim 1.15^N$
as    far   as  the     $N\leq36$    systems   are   concerned    (see
Fig.~\ref{numberostate}). This property remains  (and can be checked on
larger systems) when   the  Hilbert space   is restricted  to that  of
first-neighbor valence-bond coverings.\cite{mm01,msp03}

\begin{figure}
\begin{center}
\includegraphics[height=5cm]{numberostate1}
\caption[99]{Logarithm of the number of singlet states in the spin
gap versus sample size (black   squares).  The short dashed and   long
dashed curves display the  theoretical law (Eq.~\ref{law}  with $d=2$)
(short dashes: $p=1$, long dashes: $p=2$).  }\label{numberostate}
\end{center}

\end{figure}


Some remarks  are necessary to  fully appreciate  this  property.  The
$2^N$  states of the  system are stretched on an   energy scale of the
order  of  $NJ$    where  $J$ is    the   coupling  constant   of  the
Hamiltonian. This implies that on most of the  spectrum the density of
states increases exponentially with $N$. If we specialize to the $S=0$
sector as we will  do below, the  picture  is not very different:  the
number of states is $ C^N_{\frac{N}{2}}-C^N_{\frac{N}{2}-1} \sim {\cal
O}(\frac{2^N}{N})$ and here too,  in most of  the spectrum the density
is exponentially increasing with  $N$. But in  all the phases  that we
have studied up to now, the nature of the  ground-state and of the low
lying excitations leads to  a different behavior  at the bottom of the
spectrum.  The ground-state  degeneracy  is ${\cal O}(1)$ in
VBC,  VBS  and in the  RVB  SL  (of the  type  discussed in
\S\ref{ssec:QDMKag} and \S\ref{ssec:RVBMSE} for  instance) and  it is
${\cal O}(N^p)$  in N{\'e}el ordered  states with $p$ sublattices.  In
all these situations the low-lying  excitations are described as modes
or quasi-particles.
%In an  energy range of  order  $\mathcal{O}(J)$ one counts $N^{\beta}$
%levels associated with single-particle excitations.  This always leads
%to density of states  increasing as a power  law as a function of $N$.
%Inclusion of multi-particle excitations can be done in an average way:
Let us
suppose that single-particle excitations have a dispersion law:
\begin{equation}
\omega({\bf k}) \sim k^p.
\label{dispersionlaw}
\end{equation}
In a   d-dimensional system the total   internal energy increases with
temperature as $E \sim N T^{(p+d)/p}$, the specific heat as $C_v\sim N
T^{d/p}$ as   well as  the  entropy  ${\cal S}\sim   N T^{d/p}$.  In a
micro-canonical point of view   the density of  states $\rho(E)$  or a
large  enough  system is simply  related  to   the entropy  by  ${\cal
S}=\log(\rho(E))$. Since ${\cal S}\sim N T^{d/p}=N (E/N)^{d/(p+d)}$ we
obtain
\begin{equation}
\log (\rho(E)) \propto N^{\frac{p}{p+d}}
\label{law}
\end{equation}

As an example, let us consider the Rokhsar  Kivelson QDM on the square
lattice at   the  quantum critical    point (Sec.~\ref{sec:QDM}):  the
dispersion law  of the so-called resonons is  quadratic
around ${\bf Q}=  (\pi,\pi)$,  the logarithm  of the number  of states
increases as $N^{1/2}$ (long dashes of Fig.~\ref{numberostate}).

Even with such many-particle excitations  one would expect a number of
levels increasing  more    slowly  than in   the   numerical  spectra.
Infinitely     soft low-energy modes     ($p \rightarrow \infty$), are
necessary  to recover a density  of low-lying levels  growing as $\sim
\alpha^N$.  It is still unclear if we can  do a connection between the
``zero modes'' of the classical model at $T=0$ and  this picture.  And
we cannot  completely   indulge   ourselves in   saying   that quantum
fluctuations are unable  to  lift the classical degeneracy  as quantum
fluctuations seem to open a spin gap.

A  physical consequence  of   this exceptional density  of  low  lying
singlets can be observed in the  specific heat: at low temperature the
specific heat of this  spin system is  unusually large, with  a double
peak    structure,\cite{ze90,ey94,nm95,tr96} insensitive to relatively
large magnetic fields.\cite{smlbpwe00} This is easily understood if we
suppose that in this energy range there is  a large density of singlet
states.\cite{smlbpwe00}   This result   is   to  be compared   to  the
experimental   results of   Ramirez    {\it   et   al.}\cite{rhw00} on
SrCr$_{9p}$Ga$_{12-9p}O_{19}$   (notice   however    that  it   is   a
spin-$\frac{3}{2}$   compound  and  that  numerical calculations  were
performed on the  spin-$\frac{1}{2}$ model)  where the specific   heat
around $5$~K has an extremely low sensitivity to magnetic fields up to
$10\;$~Tesla, whereas the homogeneous  susceptibility in this range of
temperature  is probably  very  low if  we notice  that it turns  down
around $50 K$.\cite{mklmch00}

\subsubsection{Absence of gap in the singlet sector}

Up  to  $N=36$ sites there is   no evidence of   a possible gap in the
singlet    spectrum: this is    an exceptional  phenomenon in  quantum
mechanics of small systems where discretization is usually the rule.

It  has been   advocated  in recent  papers\cite{sma02,ns03}  that the
ground-state of this model could  break the translational symmetry and
be a VBC.   The first proposed crystal\cite{sma02}  is made of resonating  {\em
stars} with 6 dimers.  The corresponding unit  cell has 12 spins.  The
second  VBC\cite{ns03}   is  made   of  resonating  (trimerized)  {\em
hexagons} and was already discussed as the most reasonable crystal (if
any is realized in this system, which we think is not clear at all) by
Marston and  Zeng\cite{mz91} and Zeng  and Elser.\cite{ze95} This later
VBC has a unit cell with 36 sites. In both  scenarios it is the energy
gain obtained by {\em local  resonances} (involving respectively 6 and
3 valence-bonds) which drives the system toward a  VBC. From the energy
point  of view  the  star  VBC is   (by far)  less realistic  since it
involves a much longer  resonance loop.\footnote{Notice also that  the
gap associated to the star VBC is predicted to be of the order of 1/10
of $J$,\cite{sma02} which is 10 times larger than the largest distance
between two consecutive levels in  the spectrum of the $N=36$ sample.}
This  resonance  loop involving 6 valence-bonds  around  a star  has a
vanishing amplitude at  the  lowest non-trivial  order of the  overlap
expansion  in   the   RVB  subspace,  as  was    shown  by Zeng    and
Elser\cite{ze95}.
To our  opinion there is no  reason to think  that the physics  of the
kagome model can be described from the limit of weakly coupled stars.
On the   other  hand, in  the approximation  where  only the  shortest
resonance  loops are present,  the model was  indeed found to be in the
{\em  hexagon} VBC phase,  as re-discovered  recently from a different
point   of view.\cite{ns03} A crucial  (numerical)  result of Zeng and
Elser\cite{ze95}  is  however that  this VBC {\em   melts} when higher
order resonances loops are included.


We have studied  the 36-sites sample  which can accommodate these  two
VBC: the low-lying levels of the spectrum  do not give a clear picture
of the supposed  to be VBC.   The eigenstates with the quantum numbers
corresponding to these two VBC are not the lowest energy levels in the
spectrum.   No   gap is seen  in   the spectrum, the  largest distance
between two consecutive states  is $10^{-2}$ and seems distributed  at
random, whereas the average distance between two consecutive states in
the 50 first  states is: $2.5 10^{-3}$.  It  has been argued that long
wave-length  quantum fluctuations (almost  absent in the $N=36$ sample
which contains only  two  resonating hexagons  or three  stars)  could
eventually restore the order.  We think  that this is incorrect.  In a
VBC it  is {\em local resonances}  which favor the crystal  and longer
resonances  which tend to  reduce the order  parameter and which could
eventually destabilize it.  Long-wavelength   will tend to restore   a
larger ground-state symmetry (reducing   the degeneracy).   From  this
point of view it is unlikely that a VBC  pattern will appear in larger
systems if it  is not apparent  in the smallest systems (provided that
boundary conditions do not frustrate the  corresponding VBC).  We thus
consider that up to now numerical results do not support the claims of
a star or hexagon VBC ground-state in this model.

If the  hexagon VBC was however  realized the important question would
probably be why the associated gap is  so tiny, certainly smaller than
$10^{-3}$. A natural scenario would be  that of a very close proximity
to  a quantum critical  point, but  that critical  point remains to be
identified.  A third possibility would be that of a system with a true
extensive entropy at zero  temperature.   This is  less likely in  our
opinion since (up to now)  only  models with  simple (if not  trivial)
local symmetries were found to  have such ground-state entropy  (Ising
antiferromagnet on the triangular lattice for instance or the QDM of
\S\ref{ssec:QDMmu}) and this entropy is usually sent to (possibly
small but) finite temperatures by almost any perturbation.
 
\subsubsection{Anomalous density of states in other spin sectors}
An  anomalous density of low lying  states is  equally observed in the
spin $\frac{1}{2}$ sector (where the law could be fitted  to $N 1.15^N$), in the
spin 1 sector as well  as in other sectors  with larger total spin. It
should be noticed that such a density of states implies the absence of
an intrinsic energy scale for the low  lying excitations: a phenomenon
that    has  been   observed    in   inelastic    neutron   scattering
(Ref.~\cite{mmpfa00} and  Refs.   therein)  and theoretically  in  the
imaginary  part of  the  dynamic susceptibility  calculated within the
dynamical   mean field theory (Georges  {\it  et al.}\cite{gsf01}).  A
high spin susceptibility just above the spin  gap is not excluded
in the spin-$\frac{1}{2}$.\cite{ls02}

The global picture of  this phase is thus that  of a SL with
no  long-ranged correlations in   any local observable,  and  an large
entropy of singlets   at $T\ll J$, which is   a manifestation  of  the
extraordinary large density of states in each $S$ subspace.

\subsection{
Next-neighbor   RVB   description  of  the  spin-$\frac{1}{2}$  kagome
antiferromagnet} Considering a supposed-to-be large spin-gap, Zeng and
Elser\cite{ze95}    proposed a description     of the ground-state and
low-lying excitations  of  the  kagome model  in  the  basis  of  next
neighbor  valence bonds.   They analyzed  in  this  context the  dimer
dynamics and showed on a $N=36$ sample that  the hexagon VBC --favored
by  the shortest (three-)dimer moves--  melts  when introducing higher
order tunneling.  Mila and Mambrini\cite{m98,mm01} confirmed that this
reduced Hilbert space of next neighbor  valence bonds captures some of
the most  perplexing features  of this   magnet and  specifically  the
absence of (measurable) gap in  the singlet sector and the exponential
number of singlets.   One of us, D.~Serban and V.~Pasquier\cite{msp03}
have  elaborated on  this  work and on   Zeng and Elser's approach and
proposed a  QDM  with an  extensive  zero-point entropy   and critical
(energy-energy\footnote{Dimer-dimer correlations are  short-ranged  in
this  model.})  correlations  (see   \S\ref{ssec:QDMmu}).    All these
results  point to an absence of   an intrinsic low-energy scale.  This
feature  is typical of a   critical state, but  as  seen in the  above
discussion, the simple RK  picture does not  seem to fit nicely to the
exact diagonalization  data: may be the  available sizes are too small
or the behavior of  this    quantum system corresponds to    something
completely new.   Some recent  numerical results\cite{dmnm03} (in  the
full spin-$\frac{1}{2}$ Hilbert space as well as  in the RVB subspace)
showed that  (static)  non-magnetic impurities  (holes)  experience an
unexpected {\em   repulsion} in this   system and that  no significant
magnetic moment is created  in the vicinity of  the impurities. It has
been argued\cite{sv00} that static  non-magnetic impurities are useful
to detect   a    possible spinon  deconfinement  in    two-dimensional
antiferromagnets. From this point  of view the results mentioned above
suggest such a deconfinement.


\subsection{Experiments in compounds with kagome-like lattices}

The  low  temperature specific heat  of SrCr$_9$Ga$_{12}$0$_{19}$ (the
magnetic chromium  ion has   a spin 3/2)  is apparently   dominated by
singlet states.\cite{rhw00}  The magnetic excitations of this compound
as seen by muons can be  described as spins $\frac{1}{2}$ itinerant in
a sea of singlets.\cite{ukkll94} The non-linear spin susceptibility of
SrCr$_9$Ga$_{12}$0$_{19}$ exhibits a very large increase at about 5 K,
reminiscent of  spin glasses,\cite{rec90} but  neutrons and muons show
that  a very significant fraction  of the spins  are  not frozen below
this   temperature     and       exhibit      still   very       rapid
fluctuations.\cite{lbar96} The same  phenomena  have been observed  in
two jarosites that are equally  good models of kagome antiferromagnets
with half-odd-integer spin per unit cell.\cite{kklllwutdg96,whmmt98}

\subsection{ "Haldane's conjecture"}
Whereas the classical Heisenberg model on the kagome, checkerboard and
pyrochlore  lattices    share  the  properties   of  local  continuous
degeneracy and disorder at $T=0$, their quantum counterparts are quite
different.   As it   has   been explained  in Sec.~\ref{sec:VBC},  the
Heisenberg model  on the checkerboard  lattice has an ordered VBC with
gaps     to  all excitations.  In    contrast    to the   case of  the
$S=\frac{1}{2}$ Heisenberg model on the kagome lattice, recent results
from Hida\cite{h00} show that there is a large  gap to all excitations
in the $S=1$ Heisenberg model on this same  lattice, in agreement with
experiments.\cite{aoyhimw94,wkyoya97,wwyoaonn98}

Less  is known on  the ground-state   of the Heisenberg  model on  the
three-dimensional pyrochlore  lattice:   Canals and Lacroix\cite{cl98}
have shown that the spin-spin  correlations are short ranged and  they
have observed  that on a  16-sites spectrum the first  excitations are
singlet ones.  Fouet  {\it et al.}   have  computed the  spectrum of a
32-sites pyrochlore sample~\cite{fsl03},  this work confirms  that the
first excitations are still  singlets     for this size.  There     is
plausibly a degeneracy of the ground-state  in the thermodynamic limit
but    no evidence of  a  closing  of the   gap  above: the system may
therefore  be a VBC.  A  first description of  the  singlet sector was
proposed by Harris, Berlinsky and Bruder\cite{hbb91} and was developed
further by Tsunetsugu\cite{t01,t02}.   It  starts  from the  limit  of
weakly-coupled tetrahedron  (and thus breaks some spatial symmetries).
The ground-state of an isolated tetrahedron is two-fold degenerate and
can  be described by an Ising  pseudo-spin.   An effective Hamiltonian
describing  the   interactions   between    these   pseudo-spins   was
written\cite{hbb91} and  analyzed  in  a semi-classical approximation.
It  was  concluded that  the system maybe    a VBC. It   was argued by
Tsunetsugu  that  a    soft    mode  could exist   in    the   singlet
sector.\cite{t02}.  The CORE approach of Berg {\it et al.}\cite{baa03}
seems more appropriate  to deal with  these systems where dimerization
is probably the dominant  phenomenon. They treated larger units (block
of four tetrahedron) and  concludes to the existence  of a VBC  with a
larger unit cell than the one predicted before\cite{hbb91,t01,t02} and
a small singlet gap.


All these results  seem  to    confirm a  2D version  of     Haldane's
conjecture:  among  these  frustrated  systems with  local  continuous
degeneracies  in the  classical  limit,  the spin-$\frac{1}{2}$ kagome
antiferromagnet is the only system to have an half-odd integer spin in
the unit cell.  It is maybe not by chance that it is the only one with
gapless excitations.\footnote{The SCGO compound  with 7 spins 3/2  by
unit   cell  belongs to   the  same  category    and  as  also gapless
excitations.\cite{rhw00}} The spin-$\frac{1}{2}$ Heisenberg model   on
the checkerboard lattice or on  the pyrochlore lattice and the  spin-1
Heisenberg model on the kagome lattice  have integer spins in the unit
cell and  quantum fluctuations lead to  gapful excitations. It is also
interesting to note  that     these results are  consistent    with  a
generalization     of     the      LSM   theorem       in    dimension
two.\cite{al86,o00,mlms02,hastings03,o03}

An  interesting    analysis of Von    Delft  and Henley\cite{vdh92-93}
supports this   conjecture.   These  authors  studied   the collective
tunneling of a small cluster of spins  between two spin configurations
that  are degenerate in   the  classical limit.   They found  that for
half-odd-integer  spins the tunneling  amplitude for  a cluster of six
spins around an  hexagon (and on other  larger loops)  is exactly zero
because      of    destructive    interferences    between   different
symmetry-related instantons.  For small integer spins the interference
is constructive and the tunnel amplitude  and the tunnel splitting are
large: this is consistent with  numerical results which gives a  large
gap for the $S=1$ kagome  antiferromagnet, and small  gaps (if any) in
the spin-$\frac{1}{2}$ system.
%_____________________________________________________________________
%_____________________________________________________________________
\section{Conclusions}
\setcounter{footnote}{0}
We  conclude  by summarizing  some   properties  --and  related  open
questions-- of the different phases  discussed in  this review.

The      properties  of       these   phases  are     summarized    in
Table~\ref{tab:conclusion}.  Semi-classical  phases with   N\'eel long
ranged order, magnons as gapless  excitations, do exist in spin-$\frac{1}{2}$ 2D
systems  with  moderate  frustration:  the  Heisenberg   model on  the
triangular lattice is  the most  explicit  example, with a  sublattice
magnetization about one  half of the classical value.\cite{bllp94} The
ground-state degeneracy is in the thermodynamic limit a power of $N$.

An increased  frustration,  lower coordination number  or smaller spin
lead  to  quantum phases, with a  ground-state  of higher symmetry, no
long  ranged  order  in  spin-spin correlations,   a spin gap  and the
restored $SU(2)$ symmetry. Two  main alternatives are then opened: the
VBC or VBS  phases on one hand,  the RVB SL  on the other.  RVB SL and
VBC   (as  well as   VBS) first  requires     the formation  of  local
singlets.  When a particular local   resonance  pattern dominates  the
dynamics of the Hamiltonian the system will try to maximize the number
of occurrence of this pattern.  This is usually  achieved by a regular
arrangement, that is a VBC. When  no such pattern dominates the system
may form   a  translation invariant RVB   SL.  In  the  first case the
ground-state  can  be     qualitatively described   by   one   ordered
configuration of singlets dressed by small fluctuations. In the RVB SL
the  amplitudes   of   the  wave-function   are  distributed  over  an
exponentially large number of configurations. These ground-states lead
to  very  different excitations:  $\Delta  S=1$ gapped  magnons in the
first case (and  $\Delta S=0$ domain-wall excitations), gapped $\Delta
S=0$ visons  and  gapped $\Delta S=\frac{1}{2}$  unconfined spinons in
the   second case.

    In    agreement  with  the   large-$N$   results
(Sec.~\ref{sec:largeN}), VBC or VBS  phases appear in general
in quantum situations where the large-$S$ classical limit  displays collinear order
\footnote{We have discussed some possible counter examples in section 3. 6},
 whereas up to
now RVB SL phases have only be encountered in range of parameters were
the classical solutions are non collinear (MSE\cite{mblw98,mlbw99}
 and $J_1$--$J_2$ on the
honeycomb lattice\cite{fsl01}).

These states obey the 2D extension  of LSM theorem:  if $2S$ is odd in
the unit  cell and if excitations are  gapped there is  a ground-state
degeneracy in the thermodynamic limit (with periodic boundary
conditions). However  the origin of the degeneracy  differs  in the two
types   of  quantum phases.    In the   VBC  phases  the degeneracy is
associated to spontaneously broken translation symmetry whereas in the
RVB SL the   degeneracy  has a   topological origin.  In  the VBS  (or
explicit VBC) the ground-state is unique but  $2S$ is even in the unit
cell. As in  one dimension the LSM theorem  seems to play an important
role and a  formal proof of its  validity in 2D  is perhaps very close,
 if
not   achieved.\cite{hastings03} A  RVB SL     may have been   observed
numerically  in  a   spin-$\frac{1}{2}$   model  on   the    hexagonal
lattice.\cite{fsl01} From  the  point of   view  of large-$N$ and  QDM
approaches  a   topological  degeneracy   is   expected  in  such    a
SL. Interestingly  that degeneracy is not  imposed by the 2D extension
of LSM's theorem and has not been detected.\cite{fsl01}


These paradigms    are relatively well   understood,  at least  on the
qualitative level.  They also appear naturally  in the broader context
of the   classification  of  Mott  insulators.\cite{sachdev03} However
several kinds of  2D frustrated  magnets do  not fall in  these simple
classes and many open  questions remain.

%As a first example, we mention the possibility of  a state RVB SL with
%an {\em integer spin} in the unit cell.\footnote{Such a state may have
%been observed numerically in the spin-$\frac{1}{2}$ $J_1$--$J_2$ model
%on the hexagonal lattice.\cite{fsl01}}   The 2D extension of  the  LSM
%argument fails to predict a (topological) degeneracy in that case.


This review was restricted to $SU(2)$ invariant Hamiltonians.  Whereas
the Ising limit has been much studied, the differences between quantum
XY and Heisenberg models have received much less attention.

Chiral  SL   have  not been   discussed  in   this review.   They  are
characterized by a broken time-reversal symmetry. This possibility has
been    studied               intensely                   since    the
80's\cite{wwz89,kl87-89,ywg93,fradkinbook}.     To  the  best   of our
knowledge   it  has not    yet  been  identified  in  a   realistic 2D
model.\footnote{An example was  recently discovered in  one dimension:
the scalar-chirality  phase of the spin ladder  model with  4-spin MSE
interactions (see \S\ref{ssec:ladderMSE}).}

The issue of  quantum phase transitions in  frustrated antiferromagnet
is also an active topic  that is not presented  in this review.   Many
properties of these critical points are  still unknown, not to mention
the fascinating problems associated with (quenched) disorder.

Limited by place (and competence) we have not discussed in details the
works done   on spatially anisotropic  models. This  field which is in
between one (review   by P.~Lecheminant  in  this  book) and 2D    is
extremely flourishing tackled by bosonization and large-$N$ methods.

To conclude  we  would like  to emphasize  that  new analytical and/or
numerical methods  are highly desirable to  proceed in the analysis of
the two emblematic  problems by which  we have opened  and closed this
review:  the $J_1$--$J_2$    model  on the  square   lattice   and the
spin-$\frac{1}{2}$  Heisenberg  model on   the kagome (and pyrochlore)
lattices.   In  both  of these  problems   a consensus  remains  to be
obtained.


\begin{sidewaystable}
\tbl{Different phases encountered in $SU(2)$-symmetric frustrated models in 2D
\label{tab:conclusion}
}{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline %__________________________________________________________________________________________________________________________________________________________
		&		&			&			&		&		&			&			\\
Phase		& $2S$/cell	& Order			& Degeneracy 		& Broken sym.	& Excitations 	& Thermo. 		& Examples		\\
		&		&			&			&		&		&			&			\\
\hline %__________________________________________________________________________________________________________________________________________________________
N\'eel AF	&		&			&			& $SU(2)$	&Gapless magnons& $C_v\sim T^2$	& Spin-$\frac{1}{2}$ triangular	\\
p-sublattice	& any		& spin-spin LRO		&$\mathcal{O}(N^p)$	& Translations	&(spin waves)	& $\chi\sim {\rm cst}$	& Heisenberg AF 	\\
		&		&			&			& Point group	&		&			& 			\\
\hline %__________________________________________________________________________________________________________________________________________________________
VBC
(\S\ref{sec:VBC})& odd 		& singlet-singlet  LRO	& $>1$    		& Translations  &Gapped magnons	& $C_v$ and $\chi$ 	& Honeycomb $J_1$--$J_2$\\
(spontaneous) 	&     		& {\tiny colli.
					spin-spin SRO}	&			&  Point group  &           	& activated		& Checkerboard		\\
		&		&			&			&		&		&			& Square $J_1$--$J_2$ ?	\\
\hline %__________________________________________________________________________________________________________________________________________________________
VBC (\S\ref{sec:VBC})
		& even		&   None		& 1			& None 		&Gapped magnons	& $C_v$ and $\chi$ 	& SrCu$_2$(BO$_3$)$_2$	\\
(explicit)	&		&			&        		&      		&		& activated  		& CaV$_4$O$_9$        	\\ 
\hline
VBS (\S
\ref{ssec:VBS})	& even		& ``String'' LRO   	& 1      		& None 		&Gapped magnons	& $C_v$ and $\chi$ 	& AKLT	Hamiltonians	\\
		&       	&			&        		&      		&Edge excitations& activated   		& $S=1$ kagome AF ?	\\ 
\hline %__________________________________________________________________________________________________________________________________________________________
	  	& 	    	& Topological         	& 4       		&		&Gapped spinons	& $C_v$ and $\chi$ 	& MSE (\S\ref{sec:MSE})\\
RVB SL		& odd    	& {\tiny non-colli. SRO}& (torus) 		& None		&Gapped visons  & activated        	& QDM on triangular	\\
(\S\ref{ssec:QDMTri}
\S\ref{ssec:QDMKag},
\S\ref{ssec:RVBMSE})
		&		&			&			&		&		&			& and kagome lattices	\\
%\hline
%RVB SL		&  even		& 			& 1			& None          &Gapped spinons	& $C_v$ and $\chi$ 	&          \\
%		&		& {\tiny non-colli. SRO}&			&               &Gapped visons	& activated        	&          \\
\hline %__________________________________________________________________________________________________________________________________________________________
Kagome (\S\ref{sec:kagome})
		& 3		& None ?		& $\sim 1.15^N$ ?	& None ?	&Gapped
												  triplets ?	& $C_v\sim T^\alpha$ ? &			\\
Heisenberg AF	&		&			&			&		&Gapless
												  singlets	& $\chi$ activated	&			\\
\hline %__________________________________________________________________________________________________________________________________________________________
\end{tabular}}
\begin{tabnote}
This  table   summarizes  the  properties  of   some  important phases
encountered in 2D frustrated magnets.  $S$ is the value of the spin on
each site.  ``Order''    refers  to the   nature of  the   long-ranged
correlations (if any).  The ground-state degeneracy in the limit of an
infinite system  (with periodic boundary   conditions) is indicated in
the  fourth column,  except  for   RVB  SL   it  is  related  to   the
spontaneously  broken  symmetries  mentioned    in the  next   column.
Elementary   excitations  and  the  low-temperature   behavior of  the
specific heat ($C_v$) and uniform susceptibility ($\chi$) are given in
column six.  The  last column gives  some  examples of theoretical  or
experimental realizations of these phases. The six families of systems
presented here of course do not exhaust all possibilities. The results
which   are    plausible     but  still   debated    (concerning   the
spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the kagome lattice in
particular) are  indicated by question  marks.  Some  authors classify
all the systems  with gapped    excitations  in a loose category    of
``quantum  disordered systems'',  alluding to  the  absence of  N\'eel
long-ranged order.  It is a rather  unhappy appellation for VBC (which
obviously have some order) and in fact for most of the quantum systems
with   a   gap.  In   classical   statistical  physics ``disorder'' is
associated to  entropy, which is not  the case in these gapped systems
at $T=0$.
\end{tabnote}
\end{sidewaystable}
%_____________________________________________________________________
%_____________________________________________________________________
\section{Acknowledgments}

It  would have been  difficult   to thank  individually  many of   our
colleagues cited in references with whom we  had numerous and fruitful
discussions -- many thanks to all of  them.  It is  also a pleasure to
thank our close collaborators   B.~Bernu, V.~Pasquier, D.~Serban   and
P.~Sindzingre, this review owes much to them.  We are also grateful to
C.~Henley, A.~Honecker,   R.~Moessner  and  O.~Tchernyshyov for  their
insightful comments on the manuscript.

%_____________________________________________________________________
%_____________________________________________________________________

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%Frustrated Quantum Heisenberg Antiferromagnet

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R. Melzi  {\it et   al.}, \href{http://link.aps.org/abstract/PRB/v64/e024409}{Phys. Rev. B {\bf 64}, 024409 (2001)}.

%Magnetic and
%thermodynamic  properties of Li2VOSiO$_4$:  A  two-dimensional S = 1/2
%frustrated antiferromagnet on a square lattice

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H.   Rosner    {\it et   al.},
\href{http://link.aps.org/abstract/PRL/v88/e186405}{Phys. Rev. Lett. {\bf 88}, 186405 (2002)}.

%Realization  of a  Large $J_2$  Quasi-2D  Spin-Half Heisenberg System:
%Li2VOSiO$_4$.

\bibitem{rosner03}
H. Rosner   {\it et   al.},   \href{http://link.aps.org/abstract/PRB/v67/e014416}{Phys. Rev. B {\bf 67}, 014416 (2003)}.

%High-temperature   expansions  for   the  $J_1$-$J_2$  Heisenberg    models:
%Applications to ab  initio  calculated models for  Li$_2$VOSiO$_4$ and
%Li$_2$VOGeO$_4$.

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D. J. Klein,
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%Exact ground states for a class of antiferromagnetic Heisenberg models
%with short-range interactions.

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%On next-nearest-neighbor interaction in linear chain. II.


\bibitem{sgh88}
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%Ground States of Low-Dimensional Quantum Antiferromagnets.

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%Perturbation expansions for quantum many-body systems. 

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Zheng Weihong, J. Oitmaa, and C. J. Hamer,
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%Square-lattice Heisenberg antiferromagnet at T=0.

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J. Oitmaa and Zheng  Weihong, \href{http://link.aps.org/abstract/PRB/v54/e3022}{Phys. Rev. B {\bf 54}, 3022 (1996)}.

%Series expansion for   the $J_1-J_2$ Heisenberg antiferromagnet on   a
%square lattice.
    
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M. P. Gelfand, R. R. P.  Singh and D. A.  Huse,
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%Zero-temperature   ordering   in two-dimensional  frustrated   quantum
%Heisenberg antiferromagnets.

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M. P. Gelfand.  \href{http://link.aps.org/abstract/PRB/v42/e8206}{Phys. Rev. B {\bf 42}, 8206 (1990)}.

%Series investigations  of  magnetically  disordered ground   states in
%two-dimensional frustrated quantum antiferromagnets.

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R.~R.~P.~Singh, Zheng Weihong, C.~J.~Hamer, and  J.~Oitmaa,
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%Dimer  order with  striped   correlations in the  $J_1-J_2$ Heisenberg
%model.

\bibitem{kosw00}
V.~N.~Kotov, J.~Oitmaa, O.~Sushkov  and  Zheng Weihong,
Phil. Mag. B  {\bf 80}, 1483 (2000).

%\href{http://arxiv.org/abs/cond-mat/9912228}{cond-mat/9912228}
%Spontaneous  dimer   order, excitation  spectrum,   and  quantum phase
%transitions in the $J_1$--$J_2$ Heisenberg model.

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M. E.   Zhitomirsky  and  K.  Ueda,
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%Valence-bond crystal  phase  of a  frustrated  spin-1/2 square-lattice
%antiferromagnet.

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L.  Capriotti,      F.     Becca,  A. Parola,    and      S.  Sorella,
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%Resonating  Valence Bond Wave  Functions for  Strongly Frustrated Spin
%Systems.

\bibitem{sow02}
O.~P.~Sushkov, J.~Oitmaa, and   Zheng Weihong,
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% Critical  dynamics of singlet and  triplet excitations
%in strongly frustrated spin systems

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O. P. Sushkov, J. Oitmaa, and Zheng Weihong,
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%Quantum phase transitions in the two-dimensional $J_1-J_2$ model.

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\bibitem{fkksrr90}
%F.~Figueirido,  A.~Karlhede, S.~Kivelson,  S.~Sondhi, M.~Rocek  and D.~S.~Rokhsar
F.~Figueirido {\it et al.}, \href{http://link.aps.org/abstract/PRB/v41/e4619}{Phys. Rev. B {\bf 41}, 4619 (1990)}.


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%Static    and   dynamical  correlations   in   a  spin-1/2  frustrated
%antiferromagnet.
D.~Poilblanc, E.~Gagliano, S.~Bacci and E.~Dagotto,
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\bibitem{schulz}
%\bibitem{sz92}
H.~J.~Schulz, T.~A.~L.~Ziman, \href{http://publish.edpsciences.com/abstract/EPL/v18/p355}{Europhys. Lett. {\bf 18}, 355 (1992)}.
%\bibitem{es95}
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%Direct  Calculation  of    the  Spin  Stiffness in   the  $J_1$--$J_2$
%Heisenberg Antiferromagnet.

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%Magnetic  order and disorder   in  the frustrated quantum   Heisenberg
%antiferromagnet in two dimensions.


\bibitem{nlsm}
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%Two-dimensional quantum Heisenberg antiferromagnet at low temperatures.
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%Effective-action approach to the frustrated Heisenberg antiferromagnet
%in two dimensions.
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%Phase diagrams of the   two-dimensional Hubbard and  t-J models  by  a
%variational Monte Carlo method

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%\bibitem{nt97}
%A.~A. Nersesyan and A.~M. Tsvelick, \href{http://link.aps.org/abstract/PRL/v78/e3939}{Phys. Rev. Lett. {\bf 78}, 3939 (1997)}.

%\bibitem{km98}
%A.~K. Kolezhuk and H.-J. Mikeska, \href{http://link.aps.org/abstract/PRL/v80/e2709}{Phys. Rev. Lett. {\bf 80}, 2709 (1998)}.

\bibitem{nt97}
A.~A.~Nersesyan and A.~M.~Tsvelik,
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%One-Dimensional Spin-Liquid without Magnon Excitations.

\bibitem{km98} A.~K.~Kolezhuk and H.-J.  Mikeska,
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%Non-Haldane Spin-Liquid Models with Exact Ground States
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%Models with  exact ground states  connecting smoothly the  S=1/2 dimer
%and S=1 Haldane phases of one-dimensional spin chains


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T. Jolic{\oe}ur, private communication.

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K. Hida, \href{http://jpsj.ipap.jp/link?JPSJ/69/4003}{J. Phys. Soc. Jpn. {\bf 69}, 4003 (2000)}.

\bibitem{yk00}
%Spin-Driven Jahn-Teller Distortion in a Pyrochlore System
Y.~Yamashita and K.~Ueda, 
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\bibitem{CaV4O9}
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N.~Katoh and M.~Imada, \href{http://jpsj.ipap.jp/link?JPSJ/63/4529}{J. Phys. Soc. Jpn. {\bf 63}, 4529 (1994)}.
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T.~Ohama, H.~Yasuoka, M.~Isobe, and Y.~Ueda,
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K. Ueda, H. Kontani, M. Sigrist, and P.~A. Lee, \href{http://link.aps.org/abstract/PRL/v76/e1932}{Phys. Rev. Lett. {\bf 76}, 1932 (1996)}.

\bibitem{am96b}
%Spin gap in CaV4O9: A large-S approach
M.~Albrecht and F. Mila, \href{http://link.aps.org/abstract/PRB/v53/e2945}{Phys. Rev. B {\bf 53}, 2945 (1996)}.

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M. Troyer, H. Kontani, and K. Ueda, \href{http://link.aps.org/abstract/PRL/v76/e3822}{Phys. Rev. Lett. {\bf 76}, 3822 (1996)}.



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H. Kageyama {\it et~al.}, \href{http://link.aps.org/abstract/PRL/v84/e5876}{Phys. Rev. Lett. {\bf 84}, 5876 (2000)}.

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A. Koga and N. Kawakami, \href{http://link.aps.org/abstract/PRL/v84/e4461}{Phys. Rev. Lett. {\bf 84}, 4461 (2000)}.

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E. M\"uller-Hartmann, R.~R.~P. Singh, C. Knetter, and G.~S. Uhrig,
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\bibitem{kthb02}
%K. Kodama, M. Takigawa , M. Horvatic, C.
%Berthier,  H. Kageyama, Y. Ueda, S. Miyahara, F. Becca
%and F. Mila
K. Kodama {\it et al.}, Science {\bf 298},395 (2002).

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W. Zheng, J. Oitmaa, and C. J. Hamer
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\bibitem{am96a}
%First-order transition between  magnetic order and valence  bond order
%in a 2D frustrated Heisenberg model
M.~ Albrecht and F. Mila, \href{http://publish.edpsciences.com/abstract/EPL/v34/p145}{Europhys. Lett. {\bf 34}, 145 (1996)}.

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C.~H.~Chung, J.~B.~Marston, and S.~Sachdev,
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%Quantum  phases of the Shastry-Sutherland antiferromagnet: Application
%to SrCu2(BO3)2


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Y. Fukumoto and A. Oguchi, \href{http://jpsj.ipap.jp/link?JPSJ/68/3655}{J. Phys. Soc. Jpn. {\bf 68}, 3655 (1999)}.

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T. Momoi and K. Totsuka, \href{http://link.aps.org/abstract/PRB/v61/e3231}{Phys. Rev. B {\bf 61}, 3231 (2000)}.

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T. Momoi and K. Totsuka, \href{http://link.aps.org/abstract/PRB/v62/e15067}{Phys. Rev. B {\bf 62}, 15067 (2000)}.

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S. Miyahara and K. Ueda, \href{http://link.aps.org/abstract/PRB/v61/e3417}{Phys. Rev. B {\bf 61}, 3417 (2000)}.

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G. Misguich, T. Jolic{\oe}ur, and S.~M.~Girvin,
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C. Lhuillier and G. Misguich, in {\em High Magnetic Fields}, edited by
C.   Berthier,  L. Levy, and  G.  Martinez  (Springer,  Berlin, 2002),
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P.~W. Leung and N. Lam, \href{http://link.aps.org/abstract/PRB/v53/e2213}{Phys. Rev. B {\bf 53}, 2213 (1996)}.

\bibitem{fsl01}  J.-B.~Fouet,  P.~Sindzingre,   and  C.~Lhuillier,
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%An investigation  of the quantum $J_1-J_2-J_3$ model  on the honeycomb
%lattice

\bibitem{canals02}
B.~Canals, \href{http://link.aps.org/abstract/PRB/v65/e184408}{Phys. Rev. B {\bf 65}, 184408 (2002)}.

\bibitem{fsl03}
J.-B.~Fouet,     M.~Mambrini,    P.~Sindzingre,  and     C.~Lhuillier,
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\bibitem{sfl02}
P. Sindzingre, J.~B. Fouet, and C. Lhuillier,
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W. Brenig  and  A. Honecker,
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\bibitem{baa03}
E.~Berg, E.~Altman, and A.~Auerbach,
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\bibitem{tsma03}
%Bond order from disorder in the planar pyrochlore magnet
O.~Tchernyshyov, O.~A.~Starykh, R.~Moessner and A.~G.~Abanov,
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\bibitem{mts01}
R. Moessner, Oleg Tchernyshyov and S.~L.~Sondhi,
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%Planar pyrochlore, quantum ice and sliding ice

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S. Palmer and J.~.T.~Chalker, \href{http://link.aps.org/abstract/PRB/v64/e094412}{Phys. Rev. B {\bf 64}, 094412 (2002)}.

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R. Moessner and J.~T. Chalker, \href{http://link.aps.org/abstract/PRB/v58/e12049}{Phys. Rev. B {\bf 58}, 12049 (1998)}.

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R. Moessner and J.~T. Chalker, \href{http://link.aps.org/abstract/PRL/v80/e2929}{Phys. Rev. Lett. {\bf 80}, 2929 (1998)}.


\bibitem{pp01}
%Classical  Heisenberg  antiferromagnet  on a  garnet lattice:  A Monte
%Carlo simulation
O.~A.~Petrenko and D.~McK.~Paul,
\href{http://link.aps.org/abstract/PRB/v63/e024409}{Phys. Rev. B {\bf 63}, 024409 (2001)}.


\bibitem{ka02}
%Chiral   Kosterlitz-Thouless transition in  the frustrated  Heisenberg
%antiferromagnet on a pyrochlore slab
Hikaru Kawamura and Takuya Arimori,
\href{http://link.aps.org/abstract/PRL/v88/e077202}{Phys. Rev. Lett. {\bf 88}, 077202 (2002)}

\bibitem{henley2000}
%Effective  Hamiltonians and  dilution effects  in  Kagome and  related
%anti-ferromagnets
C.~L.~Henley, Can. J. Phys. (Canada) {\bf 79}, 1307 (2001).
[\href{http://arxiv.org/abs/cond-mat/0009130}{cond-mat/0009130}]

\bibitem{f03}
J.-B.~Fouet, Ph.D. thesis, Universit\'e Cergy Pontoise, 2003.

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\bibitem{bm02}
%Peierls-Like  Transition Induced  by Frustration in  a Two-Dimensional
%Antiferromagnet
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%Spin-Liquid   phase of the  Multiple-Spin  Exchange Hamiltonian on the
%Triangular Lattice.
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%Large-n Limit of SU(n) Quantum "Spin" Chains

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% Large-N Expansion for frustrated and doped quantum antiferromagnets

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%Quadratic quatum antiferromagnets in the fermionic large-N limit

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%Confinement of quarks

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R. Balian, J. M. Drouffe, and C. Itzykson,
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% Gauge fields on a lattice. I. General outlook

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\bibitem{ot03}
We thank O.~Tchernyshyov  for pointing us this physical interpretation
of the $U(1)$ flux.


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%identification with the O(3) nonlinear sigma model
%

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%Nonlinear   Field Theory  of  Large-Spin Heisenberg  Antiferromagnets:
%Semiclassically  Quantized Solitons  of the One-Dimensional  Easy-Axis
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%

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%O(3) Nonlinear sigma   Model and the  Topological  Distinction between
%Integer- and Half-Integer-Spin Antiferromagnets in Two Dimensions


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%N\'el and  Spin-Peierls Ground States  of Two-Dimensional SU(N) Quantum
%Antiferromagnets


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%Statistical Mechanics of Dimers on a Plane Lattice


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%Ising and dimer models in two and three dimensions
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%Resonating Valence Bond Phase in the Triangular Lattice Quantum Dimer Model


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J. Appl. Phys. {\bf 69}, 5962 (1991).

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%Columnar  dimer and  plaquette resonating-valence-bond  orders  in the
%quantum dimer model


\bibitem{fs63}
%Statistical  Mechanics  of  Dimers  on  a  Plane  Lattice.  II.  Dimer
%Correlations and Monomers
M. E. Fisher and J. Stephenson \href{http://link.aps.org/abstract/PR/v132/e1411}{Phys. Rev. {\bf 132}, 1411 (1963)}.

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concerning the square-lattice QDM .

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%Phase diagram of the hexagonal lattice quantum dimer model

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D.~A.~Huse,  W.~Krauth, R.~Moessner and S.~L.~Sondhi, \href{http://arxiv.org/abs/cond-mat/0305318}{cond-mat/0305318}
and references therein.

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L.~S.~Levitov, \href{http://link.aps.org/abstract/PRL/v64/e92}{Phys. Rev. Lett. {\bf 64}, 92 (1990)},
%Equivalence  of  the  dimer  resonating-valence-bond  problem  to  the
%quantum roughening problem

%-------
C.~L.~Henley, J. Stat. Phys. {\bf 89}, 483 (1997).

%Relaxation time for a dimer covering with height representation 

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P. Fendley, R.~Moessner and S.~L.~Sondhi,
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%Classical dimers on the triangular lattice

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A.  Ioselevich, D.~A.~Ivanov  and M.~V.~Feigelman,
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%Ground-state  properties of  the Rokhsar-Kivelson  dimer model  on the
%triangular lattice

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L.~B.~Ioffe, M.~V.~Feigel'man, A.~Ioselevich, D.~Ivanov,  M.~Troyer,
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\bibitem{mlms02}
G.~Misguich,   C.~Lhuillier,   M.~Mambrini,   P.~Sindzingre,
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%Degeneracy   of  the   ground-state   of  antiferromagnetic   spin-1/2
%Hamiltonians


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%Mean-field  theory of spin-liquid  states with  finite energy  gap and
%topological orders

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T. Senthil and M.~P.~A.~Fisher,
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%Z2 gauge  theory of electron fractionalization  in strongly correlated
%systems

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T. Senthil and M. P. A. Fisher,
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%kagome spin-1/2 antiferromagnets in the hyperbolic plane.
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A.~J.~Phares and   F.~J.~Wunderlich,
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See {\S}V.E.6 of Ref.\cite{msp03}

\bibitem{msp03c}  This  follows from  the  independence  of the  arrow
variables, see {\S}V.B of Ref.\cite{msp03}

%Quantum dimer model with  extensive ground-state entropy on the kagome
%lattice

\bibitem{polyakov87}
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\bibitem{kogut79}
%An introduction to lattice gauge theory and spin systems
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\bibitem{ze95}
C.~Zeng    and   V.~Elser.
%Quantum     dimer  calculations   on  the
%spin-$\frac{1}{2}$ kagome Heisenberg  antiferromagnet.
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D. J. Thouless, Proc. Phys. Soc. London {\bf 86}, 893 (1965).
%Exchange in solid 3He and the Heisenberg Hamiltonian.

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%Magnetism in solid 3He.

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%Magnetism in solid  3He: Confrontation between  theory and experiment.
\bibitem{greywall}   D.~S.~Greywall   and   P.~A.~Busch,
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%Heat capacity of 3He  adsorbed on graphite at millikelvin temperatures
%and near third-layer promotion.
D.~S.~Greywall,  \href{http://link.aps.org/abstract/PRB/v41/e1842}{Phys. Rev. B {\bf 41}, 1842 (1990)}.
%Heatcapacity  of  multilayers  of  3He  adsorbed on  graphite  at  low
%millikelvin temperatures.
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H. Godfrin and R.~E.~Rapp, Adv. Phys. {\bf 44}, 113 (1995).
%Two-dimensional nuclear magnets.

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%Multiple exchange in 3He and in the Wigner solid.

\bibitem{rbbcg98} M.  Roger, C. B\"auerle,  Yu.~M.~Bunkov, A.-S.~Chen, and
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%Multiple-Spin   Exchange  on  a   Triangular Lattice:   A Quantitative
%Interpretation of Thermodynamic  Properties  of Two-Dimensional  Solid
%3He.


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D.~M.~Ceperley and G.  Jacucci,    \href{http://link.aps.org/abstract/PRL/v58/e1648}{Phys. Rev. Lett. {\bf 58}, 1648 (1987)}.

%Calculation of exchange frequencies  in bcc 3He with the path-integral
%Monte Carlo method.

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D.~M.~Ceperley, \href{http://link.aps.org/abstract/RMP/v67/p279}{Rev. Mod. Phys. {\bf 67}, 279 (1995)}.
%Path integrals in the theory of condensed helium.

\bibitem{bc99} B.  Bernu and D.  Ceperley, in {\it Quantum Monte Carlo
Methods in  Physics and Chemistry},  edited by M. P.   Nightingale and
C. J. Umrigar (Kluwer, Dordrecht, The Netherlands, 1999).

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H.~Ashizawa and D.~S.~Hirashima, \href{http://link.aps.org/abstract/PRB/v62/e9413}{Phys. Rev. B {\bf 62}, 9413 (2000)}.
%WKB calculation of multiple spin exchange in monolayer solid 3He.

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%Exchange and magnetic order in hcp helium-3 and adsorbed helium-3 with
%triangular lattice.

\bibitem{frg86}   H.~Franco,   R.~E.~Rapp   and   H.~Godfrin,
\href{http://link.aps.org/abstract/PRL/v57/e1161}{Phys. Rev. Lett. {\bf 57}, 1161 (1986)}.

%Nuclear Ferromagnetism of Two-Dimensional 3He.

\bibitem{mblw98}   G.~Misguich,   B.~Bernu,  C.~Lhuillier,   and
C.~Waldtmann, \href{http://link.aps.org/abstract/PRL/v81/e1098}{Phys. Rev. Lett. {\bf 81}, 1098 (1998)}.

\bibitem{lmsl00}
W.~LiMing,  G.~Misguich, P.~Sindzingre, and  C.~Lhuillier,
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%From N\'eel  long-range order to  spin  liquids  in the  multiple-spin
%exchange model.

\bibitem{kmyf97}  K.~Ishida,  M.~Morishita, K.~Yawata,  and  H.~Fukuyama,
\href{http://link.aps.org/abstract/PRL/v79/e3451}{Phys. Rev. Lett. {\bf 79}, 3451 (1997)}.

%Low Temperature Heat-Capacity Anomalies in Two-Dimensional Solid 3He.

\bibitem{cthrbbg01}
%E.~Collin,   S.~Triqueneaux,  R.~Harakaly,   M.~Roger,   C.~B\"auerle,
%Yu.~M.~Bunkov, and H.~Godfrin
E.~Collin {\it et al.}, \href{http://link.aps.org/abstract/PRL/v86/e2447}{Phys. Rev. Lett. {\bf 86}, 2447 (2001)}.

%Quantum   Frustration in the  "Spin   Liquid" Phase of Two-Dimensional
%$^3$He.

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M. Katano and D.~S.~Hirashima, \href{http://link.aps.org/abstract/PRB/v62/e2573}{Phys. Rev. B {\bf 62}, 2573 (2000)}.
%Multiple-spin exchange in a two-dimensional Wigner crystal.

\bibitem{hk01}          D.~S.~Hirashima          and          K.~Kubo,
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\bibitem{bcc01}   B.   Bernu,  L.   Candido,   and   D.  M.   Ceperley
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%Exchange Frequencies in the 2D Wigner Crystal.

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%Magnetism in a  Wigner solid and  the Aharonov-Bohm effect: Experiment
%and theory.

\bibitem{rd89} M. Roger and J. M. Delrieu,
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%Cyclic  four-spin  exchange   on a   two-dimensional   square lattice:
%Possible applications in high-T$_c$ superconductors.

\bibitem{sugai90}
S.~Sugai {\it et al.},
%M.~Sato,  T.~Kobayashi,   J.~Akimitsu,
%T.~Ito, H.~Takagi, S.~Uchida, S.~Hosoya, T.~Kajitani and T.~Fukuda,
\href{http://link.aps.org/abstract/PRB/v42/e1045}{Phys. Rev. B {\bf 42}, 1045 (1990)}.

%High-energy spin  excitations  in the insulating  phases of high-T$_c$
%superconducting cuprates and La$_2$NiO$_4$.

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R.~Coldea  {\it  et   al.},
%S.~M.~Hayden,  G.~Aeppli,   T.~G.~Perring,  C.~D.~Frost,  T.~E.~Mason,
%S.-W.~Cheong, and Z.~Fisk,
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%Spin Waves and Electronic Interactions in La$_2$CuO$_4$. 

\bibitem{mr02}
E.~M\"uller-Hartmann and A.~Reischl,
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%Derivation  of   effective spin models  from  a  three  band model for
%CuO$_2$-planes.

\bibitem{kk03} A.~A.~Katanin, and A.~P.~Kampf,
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%Theoretical  analysis of magnetic  Raman  scattering in La$_2$CuO$_4$:
%Two-magnon intensity with the inclusion of ring exchange.

\bibitem{mkebm00}

M.   Matsuda {\it et al.},
%K.   Katsumata,   R.~S.~Eccleston,
%S. Brehmer and H.-J.~Mikeska,
\href{http://link.aps.org/abstract/PRB/v62/e8903}{Phys. Rev. B {\bf 62}, 8903 (2000)}.
%Magnetic  excitations  and  exchange  interactions in  the  spin-($\frac{1}{2}$)
%two-leg ladder compound La$_6$Ca$_8$Cu$_{24}$O$_{41}$.

\bibitem{bmmnu99}
S.~Brehmer {\it et al.},
%H.-J.~Mikeska,  M.~M\"uller, N.~Nagaosa and S.~Uchida,
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%Effects  of  biquadratic  exchange  on   the  spectrum of   elementary
%excitations in spin ladders.

\bibitem{sku01}  K.   P.  Schmidt,  C.   Knetter  and  G.   S.  Uhrig,
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%Raman response in antiferromagnetic two-leg S=1/2 Heisenberg ladders.

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%U.~Kuhlmann, C.~Thomsen A.~L\"offert, C.~Gross, and W.~Assmus,
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%A Possible Magnetic Phase with Scalar Chirality in Solid 3He Layers.


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\end{document}


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%Li$_2$VO(Si,  Ge)O$_4$,   a   Prototype   of a  Two-Dimensional
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%Magnetic and
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%High-temperature   expansions  for   the  $J_1$-$J_2$  Heisenberg    models:
%Applications to ab  initio  calculated models for  Li$_2$VOSiO$_4$ and
%Li$_2$VOGeO$_4$.

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D. J. Klein,
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%Exact ground states for a class of antiferromagnetic Heisenberg models
%with short-range interactions.

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%On next-nearest-neighbor interaction in linear chain. II.


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%Ground States of Low-Dimensional Quantum Antiferromagnets.

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%Perturbation expansions for quantum many-body systems. 

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Zheng Weihong, J. Oitmaa, and C. J. Hamer,
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%Square-lattice Heisenberg antiferromagnet at T=0.

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J. Oitmaa and Zheng  Weihong, \href{http://link.aps.org/abstract/PRB/v54/e3022}{Phys. Rev. B {\bf 54}, 3022 (1996)}.

%Series expansion for   the $J_1-J_2$ Heisenberg antiferromagnet on   a
%square lattice.
    
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M. P. Gelfand, R. R. P.  Singh and D. A.  Huse,
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%Zero-temperature   ordering   in two-dimensional  frustrated   quantum
%Heisenberg antiferromagnets.

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M. P. Gelfand.  \href{http://link.aps.org/abstract/PRB/v42/e8206}{Phys. Rev. B {\bf 42}, 8206 (1990)}.

%Series investigations  of  magnetically  disordered ground   states in
%two-dimensional frustrated quantum antiferromagnets.

\bibitem{swho99}
R.~R.~P.~Singh, Zheng Weihong, C.~J.~Hamer, and  J.~Oitmaa,
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%Dimer  order with  striped   correlations in the  $J_1-J_2$ Heisenberg
%model.

\bibitem{kosw00}
V.~N.~Kotov, J.~Oitmaa, O.~Sushkov  and  Zheng Weihong,
Phil. Mag. B  {\bf 80}, 1483 (2000).

%\href{http://arxiv.org/abs/cond-mat/9912228}{cond-mat/9912228}
%Spontaneous  dimer   order, excitation  spectrum,   and  quantum phase
%transitions in the $J_1$--$J_2$ Heisenberg model.

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M. E.   Zhitomirsky  and  K.  Ueda,
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%Valence-bond crystal  phase  of a  frustrated  spin-1/2 square-lattice
%antiferromagnet.

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L.  Capriotti,      F.     Becca,  A. Parola,    and      S.  Sorella,
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%Resonating  Valence Bond Wave  Functions for  Strongly Frustrated Spin
%Systems.

\bibitem{sow02}
O.~P.~Sushkov, J.~Oitmaa, and   Zheng Weihong,
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% Critical  dynamics of singlet and  triplet excitations
%in strongly frustrated spin systems

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O. P. Sushkov, J. Oitmaa, and Zheng Weihong,
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%Quantum phase transitions in the two-dimensional $J_1-J_2$ model.

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\bibitem{fkksrr90}
%F.~Figueirido,  A.~Karlhede, S.~Kivelson,  S.~Sondhi, M.~Rocek  and D.~S.~Rokhsar
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%Static    and   dynamical  correlations   in   a  spin-1/2  frustrated
%antiferromagnet.
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\bibitem{schulz}
%\bibitem{sz92}
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%\bibitem{es95}
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%Direct  Calculation  of    the  Spin  Stiffness in   the  $J_1$--$J_2$
%Heisenberg Antiferromagnet.

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%Magnetic  order and disorder   in  the frustrated quantum   Heisenberg
%antiferromagnet in two dimensions.


\bibitem{nlsm}
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%Two-dimensional quantum Heisenberg antiferromagnet at low temperatures.
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%Phase diagrams of the   two-dimensional Hubbard and  t-J models  by  a
%variational Monte Carlo method

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%\bibitem{nt97}
%A.~A. Nersesyan and A.~M. Tsvelick, \href{http://link.aps.org/abstract/PRL/v78/e3939}{Phys. Rev. Lett. {\bf 78}, 3939 (1997)}.

%\bibitem{km98}
%A.~K. Kolezhuk and H.-J. Mikeska, \href{http://link.aps.org/abstract/PRL/v80/e2709}{Phys. Rev. Lett. {\bf 80}, 2709 (1998)}.

\bibitem{nt97}
A.~A.~Nersesyan and A.~M.~Tsvelik,
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%One-Dimensional Spin-Liquid without Magnon Excitations.

\bibitem{km98} A.~K.~Kolezhuk and H.-J.  Mikeska,
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%Non-Haldane Spin-Liquid Models with Exact Ground States
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%Models with  exact ground states  connecting smoothly the  S=1/2 dimer
%and S=1 Haldane phases of one-dimensional spin chains


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T. Jolic{\oe}ur, private communication.

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K. Hida, \href{http://jpsj.ipap.jp/link?JPSJ/69/4003}{J. Phys. Soc. Jpn. {\bf 69}, 4003 (2000)}.

\bibitem{yk00}
%Spin-Driven Jahn-Teller Distortion in a Pyrochlore System
Y.~Yamashita and K.~Ueda, 
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\bibitem{CaV4O9}
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N.~Katoh and M.~Imada, \href{http://jpsj.ipap.jp/link?JPSJ/63/4529}{J. Phys. Soc. Jpn. {\bf 63}, 4529 (1994)}.
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T.~Ohama, H.~Yasuoka, M.~Isobe, and Y.~Ueda,
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K. Ueda, H. Kontani, M. Sigrist, and P.~A. Lee, \href{http://link.aps.org/abstract/PRL/v76/e1932}{Phys. Rev. Lett. {\bf 76}, 1932 (1996)}.

\bibitem{am96b}
%Spin gap in CaV4O9: A large-S approach
M.~Albrecht and F. Mila, \href{http://link.aps.org/abstract/PRB/v53/e2945}{Phys. Rev. B {\bf 53}, 2945 (1996)}.

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M. Troyer, H. Kontani, and K. Ueda, \href{http://link.aps.org/abstract/PRL/v76/e3822}{Phys. Rev. Lett. {\bf 76}, 3822 (1996)}.



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H. Kageyama {\it et~al.}, \href{http://link.aps.org/abstract/PRL/v84/e5876}{Phys. Rev. Lett. {\bf 84}, 5876 (2000)}.

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A. Koga and N. Kawakami, \href{http://link.aps.org/abstract/PRL/v84/e4461}{Phys. Rev. Lett. {\bf 84}, 4461 (2000)}.

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E. M\"uller-Hartmann, R.~R.~P. Singh, C. Knetter, and G.~S. Uhrig,
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\bibitem{kthb02}
%K. Kodama, M. Takigawa , M. Horvatic, C.
%Berthier,  H. Kageyama, Y. Ueda, S. Miyahara, F. Becca
%and F. Mila
K. Kodama {\it et al.}, Science {\bf 298},395 (2002).

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W. Zheng, J. Oitmaa, and C. J. Hamer
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\bibitem{am96a}
%First-order transition between  magnetic order and valence  bond order
%in a 2D frustrated Heisenberg model
M.~ Albrecht and F. Mila, \href{http://publish.edpsciences.com/abstract/EPL/v34/p145}{Europhys. Lett. {\bf 34}, 145 (1996)}.

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C.~H.~Chung, J.~B.~Marston, and S.~Sachdev,
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%Quantum  phases of the Shastry-Sutherland antiferromagnet: Application
%to SrCu2(BO3)2


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Y. Fukumoto and A. Oguchi, \href{http://jpsj.ipap.jp/link?JPSJ/68/3655}{J. Phys. Soc. Jpn. {\bf 68}, 3655 (1999)}.

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T. Momoi and K. Totsuka, \href{http://link.aps.org/abstract/PRB/v61/e3231}{Phys. Rev. B {\bf 61}, 3231 (2000)}.

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T. Momoi and K. Totsuka, \href{http://link.aps.org/abstract/PRB/v62/e15067}{Phys. Rev. B {\bf 62}, 15067 (2000)}.

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S. Miyahara and K. Ueda, \href{http://link.aps.org/abstract/PRB/v61/e3417}{Phys. Rev. B {\bf 61}, 3417 (2000)}.

\bibitem{mjg01}
G. Misguich, T. Jolic{\oe}ur, and S.~M.~Girvin,
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C. Lhuillier and G. Misguich, in {\em High Magnetic Fields}, edited by
C.   Berthier,  L. Levy, and  G.  Martinez  (Springer,  Berlin, 2002),
pp.~161--190, \href{http://arxiv.org/abs/cond-mat/0109146}{cond-mat/0109146}.

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P.~W. Leung and N. Lam, \href{http://link.aps.org/abstract/PRB/v53/e2213}{Phys. Rev. B {\bf 53}, 2213 (1996)}.

\bibitem{fsl01}  J.-B.~Fouet,  P.~Sindzingre,   and  C.~Lhuillier,
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%An investigation  of the quantum $J_1-J_2-J_3$ model  on the honeycomb
%lattice

\bibitem{canals02}
B.~Canals, \href{http://link.aps.org/abstract/PRB/v65/e184408}{Phys. Rev. B {\bf 65}, 184408 (2002)}.

\bibitem{fsl03}
J.-B.~Fouet,     M.~Mambrini,    P.~Sindzingre,  and     C.~Lhuillier,
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\bibitem{sfl02}
P. Sindzingre, J.~B. Fouet, and C. Lhuillier,
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\bibitem{bh02}
W. Brenig  and  A. Honecker,
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\bibitem{baa03}
E.~Berg, E.~Altman, and A.~Auerbach,
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\bibitem{tsma03}
%Bond order from disorder in the planar pyrochlore magnet
O.~Tchernyshyov, O.~A.~Starykh, R.~Moessner and A.~G.~Abanov,
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\bibitem{mts01}
R. Moessner, Oleg Tchernyshyov and S.~L.~Sondhi,
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%Planar pyrochlore, quantum ice and sliding ice

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S. Palmer and J.~.T.~Chalker, \href{http://link.aps.org/abstract/PRB/v64/e094412}{Phys. Rev. B {\bf 64}, 094412 (2002)}.

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R. Moessner and J.~T. Chalker, \href{http://link.aps.org/abstract/PRB/v58/e12049}{Phys. Rev. B {\bf 58}, 12049 (1998)}.

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R. Moessner and J.~T. Chalker, \href{http://link.aps.org/abstract/PRL/v80/e2929}{Phys. Rev. Lett. {\bf 80}, 2929 (1998)}.


\bibitem{pp01}
%Classical  Heisenberg  antiferromagnet  on a  garnet lattice:  A Monte
%Carlo simulation
O.~A.~Petrenko and D.~McK.~Paul,
\href{http://link.aps.org/abstract/PRB/v63/e024409}{Phys. Rev. B {\bf 63}, 024409 (2001)}.


\bibitem{ka02}
%Chiral   Kosterlitz-Thouless transition in  the frustrated  Heisenberg
%antiferromagnet on a pyrochlore slab
Hikaru Kawamura and Takuya Arimori,
\href{http://link.aps.org/abstract/PRL/v88/e077202}{Phys. Rev. Lett. {\bf 88}, 077202 (2002)}

\bibitem{henley2000}
%Effective  Hamiltonians and  dilution effects  in  Kagome and  related
%anti-ferromagnets
C.~L.~Henley, Can. J. Phys. (Canada) {\bf 79}, 1307 (2001).
[\href{http://arxiv.org/abs/cond-mat/0009130}{cond-mat/0009130}]

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J.-B.~Fouet, Ph.D. thesis, Universit\'e Cergy Pontoise, 2003.

\bibitem{gwsoh96}
M.~P.~Gelfand {\it et~al.}, \href{http://link.aps.org/abstract/PRL/v77/e2794}{Phys. Rev. Lett. {\bf 77}, 2794 (1996)}.


\bibitem{bm02}
%Peierls-Like  Transition Induced  by Frustration in  a Two-Dimensional
%Antiferromagnet
F. Becca and F. Mila, \href{http://link.aps.org/abstract/PRL/v89/e037204}{Phys. Rev. Lett. {\bf 89}, 037204 (2002)}.

\bibitem{mlbw99}
%Spin-Liquid   phase of the  Multiple-Spin  Exchange Hamiltonian on the
%Triangular Lattice.
G. Misguich, C. Lhuillier, B. Bernu and C. Waldtmann,
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\bibitem{a85}
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%Large-n Limit of SU(n) Quantum "Spin" Chains

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%Large-$N$ expansion for frustrated quantum antiferromagnets.

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% Large-N Expansion for frustrated and doped quantum antiferromagnets

\bibitem{s93}  S.  Sachdev  in  {\it  Low  Dimensional  Quantum  Field
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%Quadratic quatum antiferromagnets in the fermionic large-N limit

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%Confinement of quarks

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R. Balian, J. M. Drouffe, and C. Itzykson,
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% Gauge fields on a lattice. I. General outlook

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\bibitem{ot03}
We thank O.~Tchernyshyov  for pointing us this physical interpretation
of the $U(1)$ flux.


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%Continuum   dynamics    of the   1-D    Heisenberg    antiferromagnet:
%identification with the O(3) nonlinear sigma model
%

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%Nonlinear   Field Theory  of  Large-Spin Heisenberg  Antiferromagnets:
%Semiclassically  Quantized Solitons  of the One-Dimensional  Easy-Axis
%N?el State
%

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%O(3) Nonlinear sigma   Model and the  Topological  Distinction between
%Integer- and Half-Integer-Spin Antiferromagnets in Two Dimensions


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%N\'el and  Spin-Peierls Ground States  of Two-Dimensional SU(N) Quantum
%Antiferromagnets


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%Statistical Mechanics of Dimers on a Plane Lattice


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%Ising and dimer models in two and three dimensions
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%Resonating Valence Bond Phase in the Triangular Lattice Quantum Dimer Model


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J. Appl. Phys. {\bf 69}, 5962 (1991).

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%Columnar  dimer and  plaquette resonating-valence-bond  orders  in the
%quantum dimer model


\bibitem{fs63}
%Statistical  Mechanics  of  Dimers  on  a  Plane  Lattice.  II.  Dimer
%Correlations and Monomers
M. E. Fisher and J. Stephenson \href{http://link.aps.org/abstract/PR/v132/e1411}{Phys. Rev. {\bf 132}, 1411 (1963)}.

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R. Moessner and  S.~L.   Sondhi,  \href{http://arxiv.org/abs/cond-mat/0307592}{cond-mat/0307592}.  See  Appendix   B
concerning the square-lattice QDM .

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%Phase diagram of the hexagonal lattice quantum dimer model

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D.~A.~Huse,  W.~Krauth, R.~Moessner and S.~L.~Sondhi, \href{http://arxiv.org/abs/cond-mat/0305318}{cond-mat/0305318}
and references therein.

\bibitem{height_representation}

L.~S.~Levitov, \href{http://link.aps.org/abstract/PRL/v64/e92}{Phys. Rev. Lett. {\bf 64}, 92 (1990)},
%Equivalence  of  the  dimer  resonating-valence-bond  problem  to  the
%quantum roughening problem

%-------
C.~L.~Henley, J. Stat. Phys. {\bf 89}, 483 (1997).

%Relaxation time for a dimer covering with height representation 

\bibitem{fms02}
P. Fendley, R.~Moessner and S.~L.~Sondhi,
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%Classical dimers on the triangular lattice

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A.  Ioselevich, D.~A.~Ivanov  and M.~V.~Feigelman,
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%Ground-state  properties of  the Rokhsar-Kivelson  dimer model  on the
%triangular lattice

\bibitem{ifiitb02}
L.~B.~Ioffe, M.~V.~Feigel'man, A.~Ioselevich, D.~Ivanov,  M.~Troyer,
G.~Blatter, Nature {\bf 415}, 503 (2002).

\bibitem{mlms02}
G.~Misguich,   C.~Lhuillier,   M.~Mambrini,   P.~Sindzingre,
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%Degeneracy   of  the   ground-state   of  antiferromagnetic   spin-1/2
%Hamiltonians


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X. G. Wen, \href{http://link.aps.org/abstract/PRB/v44/e2664}{Phys. Rev. B {\bf 44}, 2664 (1991)}.

%Mean-field  theory of spin-liquid  states with  finite energy  gap and
%topological orders

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N. Read and B. Chakraborty, \href{http://link.aps.org/abstract/PRB/v40/e7133}{Phys. Rev. B {\bf 40}, 7133 (1989)}.

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T. Senthil and M.~P.~A.~Fisher,
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%Z2 gauge  theory of electron fractionalization  in strongly correlated
%systems

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T. Senthil and M. P. A. Fisher,
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R.~P.~Feynman, \href{http://link.aps.org/abstract/PR/v90/e1116}{Phys. Rev. {\bf 90}, 1116 (1952)},
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%kagome spin-1/2 antiferromagnets in the hyperbolic plane.
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\bibitem{hw88}
A.~J.~Phares and   F.~J.~Wunderlich,
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\bibitem{msp03b}
See {\S}V.E.6 of Ref.\cite{msp03}

\bibitem{msp03c}  This  follows from  the  independence  of the  arrow
variables, see {\S}V.B of Ref.\cite{msp03}

%Quantum dimer model with  extensive ground-state entropy on the kagome
%lattice

\bibitem{polyakov87}
A.~M.~Polyakov, {\it Gauge Fields and Strings}, (Harwood Academic, New York, 1987).



\bibitem{kogut79}
%An introduction to lattice gauge theory and spin systems
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\bibitem{msp03} G.~Misguich,  D.~Serban and V.~Pasquier,
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\bibitem{ze95}
C.~Zeng    and   V.~Elser.
%Quantum     dimer  calculations   on  the
%spin-$\frac{1}{2}$ kagome Heisenberg  antiferromagnet.
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D. J. Thouless, Proc. Phys. Soc. London {\bf 86}, 893 (1965).
%Exchange in solid 3He and the Heisenberg Hamiltonian.

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%Magnetism in solid 3He.

\bibitem{cf85} M. C. Cross and D.  S. Fisher,
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%Magnetism in solid  3He: Confrontation between  theory and experiment.
\bibitem{greywall}   D.~S.~Greywall   and   P.~A.~Busch,
\href{http://link.aps.org/abstract/PRL/v62/e1868}{Phys. Rev. Lett. {\bf 62}, 1868 (1989)}.
%Heat capacity of 3He  adsorbed on graphite at millikelvin temperatures
%and near third-layer promotion.
D.~S.~Greywall,  \href{http://link.aps.org/abstract/PRB/v41/e1842}{Phys. Rev. B {\bf 41}, 1842 (1990)}.
%Heatcapacity  of  multilayers  of  3He  adsorbed on  graphite  at  low
%millikelvin temperatures.
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H. Godfrin and R.~E.~Rapp, Adv. Phys. {\bf 44}, 113 (1995).
%Two-dimensional nuclear magnets.

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M.~Roger, \href{http://link.aps.org/abstract/PRB/v30/e6432}{Phys. Rev. B {\bf 30}, 6432 (1984)}.
%Multiple exchange in 3He and in the Wigner solid.

\bibitem{rbbcg98} M.  Roger, C. B\"auerle,  Yu.~M.~Bunkov, A.-S.~Chen, and
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%Multiple-Spin   Exchange  on  a   Triangular Lattice:   A Quantitative
%Interpretation of Thermodynamic  Properties  of Two-Dimensional  Solid
%3He.


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D.~M.~Ceperley and G.  Jacucci,    \href{http://link.aps.org/abstract/PRL/v58/e1648}{Phys. Rev. Lett. {\bf 58}, 1648 (1987)}.

%Calculation of exchange frequencies  in bcc 3He with the path-integral
%Monte Carlo method.

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D.~M.~Ceperley, \href{http://link.aps.org/abstract/RMP/v67/p279}{Rev. Mod. Phys. {\bf 67}, 279 (1995)}.
%Path integrals in the theory of condensed helium.

\bibitem{bc99} B.  Bernu and D.  Ceperley, in {\it Quantum Monte Carlo
Methods in  Physics and Chemistry},  edited by M. P.   Nightingale and
C. J. Umrigar (Kluwer, Dordrecht, The Netherlands, 1999).

\bibitem{ah00}
H.~Ashizawa and D.~S.~Hirashima, \href{http://link.aps.org/abstract/PRB/v62/e9413}{Phys. Rev. B {\bf 62}, 9413 (2000)}.
%WKB calculation of multiple spin exchange in monolayer solid 3He.

\bibitem{drh80}  J.~M.~Delrieu, M.~Roger,  J.~H.~Hetherington,  J. Low
Temp.  Phys.  {\bf 40}, 71 (1980).

%Exchange and magnetic order in hcp helium-3 and adsorbed helium-3 with
%triangular lattice.

\bibitem{frg86}   H.~Franco,   R.~E.~Rapp   and   H.~Godfrin,
\href{http://link.aps.org/abstract/PRL/v57/e1161}{Phys. Rev. Lett. {\bf 57}, 1161 (1986)}.

%Nuclear Ferromagnetism of Two-Dimensional 3He.

\bibitem{mblw98}   G.~Misguich,   B.~Bernu,  C.~Lhuillier,   and
C.~Waldtmann, \href{http://link.aps.org/abstract/PRL/v81/e1098}{Phys. Rev. Lett. {\bf 81}, 1098 (1998)}.

\bibitem{lmsl00}
W.~LiMing,  G.~Misguich, P.~Sindzingre, and  C.~Lhuillier,
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%From N\'eel  long-range order to  spin  liquids  in the  multiple-spin
%exchange model.

\bibitem{kmyf97}  K.~Ishida,  M.~Morishita, K.~Yawata,  and  H.~Fukuyama,
\href{http://link.aps.org/abstract/PRL/v79/e3451}{Phys. Rev. Lett. {\bf 79}, 3451 (1997)}.

%Low Temperature Heat-Capacity Anomalies in Two-Dimensional Solid 3He.

\bibitem{cthrbbg01}
%E.~Collin,   S.~Triqueneaux,  R.~Harakaly,   M.~Roger,   C.~B\"auerle,
%Yu.~M.~Bunkov, and H.~Godfrin
E.~Collin {\it et al.}, \href{http://link.aps.org/abstract/PRL/v86/e2447}{Phys. Rev. Lett. {\bf 86}, 2447 (2001)}.

%Quantum   Frustration in the  "Spin   Liquid" Phase of Two-Dimensional
%$^3$He.

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M. Katano and D.~S.~Hirashima, \href{http://link.aps.org/abstract/PRB/v62/e2573}{Phys. Rev. B {\bf 62}, 2573 (2000)}.
%Multiple-spin exchange in a two-dimensional Wigner crystal.

\bibitem{hk01}          D.~S.~Hirashima          and          K.~Kubo,
\href{http://link.aps.org/abstract/PRB/v63/e125340}{Phys. Rev. B {\bf 63}, 125340 (2001)}.

\bibitem{bcc01}   B.   Bernu,  L.   Candido,   and   D.  M.   Ceperley
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%Exchange Frequencies in the 2D Wigner Crystal.

\bibitem{ok98} T. Okamoto  and S. Kawaji,
\href{http://link.aps.org/abstract/PRB/v57/e9097}{Phys. Rev. B {\bf 57}, 9097 (1998)}.

%Magnetism in a  Wigner solid and  the Aharonov-Bohm effect: Experiment
%and theory.

\bibitem{rd89} M. Roger and J. M. Delrieu,
\href{http://link.aps.org/abstract/PRB/v39/e2299}{Phys. Rev. B {\bf 39}, 2299 (1989)}.

%Cyclic  four-spin  exchange   on a   two-dimensional   square lattice:
%Possible applications in high-T$_c$ superconductors.

\bibitem{sugai90}
S.~Sugai {\it et al.},
%M.~Sato,  T.~Kobayashi,   J.~Akimitsu,
%T.~Ito, H.~Takagi, S.~Uchida, S.~Hosoya, T.~Kajitani and T.~Fukuda,
\href{http://link.aps.org/abstract/PRB/v42/e1045}{Phys. Rev. B {\bf 42}, 1045 (1990)}.

%High-energy spin  excitations  in the insulating  phases of high-T$_c$
%superconducting cuprates and La$_2$NiO$_4$.

\bibitem{coldea01}
R.~Coldea  {\it  et   al.},
%S.~M.~Hayden,  G.~Aeppli,   T.~G.~Perring,  C.~D.~Frost,  T.~E.~Mason,
%S.-W.~Cheong, and Z.~Fisk,
\href{http://link.aps.org/abstract/PRL/v86/e5377}{Phys. Rev. Lett. {\bf 86}, 5377 (2001)}.

%Spin Waves and Electronic Interactions in La$_2$CuO$_4$. 

\bibitem{mr02}
E.~M\"uller-Hartmann and A.~Reischl,
\href{http://publish.edpsciences.com/abstract/EPJB/v28/p173}{Eur. Phys. J. B {\bf 28}, 173 (2002)}.

%Derivation  of   effective spin models  from  a  three  band model for
%CuO$_2$-planes.

\bibitem{kk03} A.~A.~Katanin, and A.~P.~Kampf,
\href{http://link.aps.org/abstract/PRB/v67/e100404}{Phys. Rev. B {\bf 67}, 100404R (2003)}.

%Theoretical  analysis of magnetic  Raman  scattering in La$_2$CuO$_4$:
%Two-magnon intensity with the inclusion of ring exchange.

\bibitem{mkebm00}

M.   Matsuda {\it et al.},
%K.   Katsumata,   R.~S.~Eccleston,
%S. Brehmer and H.-J.~Mikeska,
\href{http://link.aps.org/abstract/PRB/v62/e8903}{Phys. Rev. B {\bf 62}, 8903 (2000)}.
%Magnetic  excitations  and  exchange  interactions in  the  spin-($\frac{1}{2}$)
%two-leg ladder compound La$_6$Ca$_8$Cu$_{24}$O$_{41}$.

\bibitem{bmmnu99}
S.~Brehmer {\it et al.},
%H.-J.~Mikeska,  M.~M\"uller, N.~Nagaosa and S.~Uchida,
\href{http://link.aps.org/abstract/PRB/v60/e329}{Phys. Rev. B {\bf 60}, 329 (1999)}.

%Effects  of  biquadratic  exchange  on   the  spectrum of   elementary
%excitations in spin ladders.

\bibitem{sku01}  K.   P.  Schmidt,  C.   Knetter  and  G.   S.  Uhrig,
\href{http://publish.edpsciences.com/abstract/EPL/v56/p877}{Europhys. Lett. {\bf 56}, 877 (2001)}.

%Raman response in antiferromagnetic two-leg S=1/2 Heisenberg ladders.

\bibitem{gkt03}
A.~G\"o�ling {\it et al.},
%U.~Kuhlmann, C.~Thomsen A.~L\"offert, C.~Gross, and W.~Assmus,
\href{http://link.aps.org/abstract/PRB/v67/e052403}{Phys. Rev. B {\bf 67}, 052403 (2003)}.
%Magnetic excitations

\bibitem{mvm02}
M. M\"uller, T. Vekua, and H.-J. Mikeska,
\href{http://link.aps.org/abstract/PRB/v66/e134423}{Phys. Rev. B {\bf 66}, 134423 (2002)}.
%Perturbation theories  for the  S=1/2  spin ladder with  a four-spin
%ring exchange

\bibitem{hmh03}
T. Hikihara, T. Momoi, and X. Hu,
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%Spin-Chirality Duality in a Spin Ladder with Four-Spin Cyclic Exchange.

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%A Possible Magnetic Phase with Scalar Chirality in Solid 3He Layers.


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\end{thebibliography}