A sufficient symmetry condition %for finite temperature topological quantum order %and gauge like structures in electronic systems
TOPOn.tex
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\begin{document}

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%Title of paper
%\title{A sufficient symmetry condition
%for finite temperature topological quantum order
%and gauge like structures in electronic systems}
\title{Sufficient symmetry conditions for Topological Quantum Order}

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\author{Zohar Nussinov$^1$ and Gerardo Ortiz$^{2}$}
\affiliation{$^1$Department of Physics, Washington University, St.
Louis, MO 63160, USA}
\affiliation{$^2$Department of Physics, Indiana University, Bloomington,
IN 47405, USA}


\date{\today}

\begin{abstract}

\end{abstract}

% insert suggested PACS numbers in braces on next line
\pacs{05.30.-d, 11.15.-q, 71.10.-w, 71.10.Pm}

% 03.67.-a Quantum information
% 03.67.Mn Entanglement production, characterization and manipulation 
% 03.65.Ud Entanglement and quantum nonlocality
% 05.30.-d Quantum statistical mechanics
% insert suggested keywords - APS authors don't need to do this
%\keywords{}
\begin{abstract}
We prove sufficient conditions for Topological Quantum Order (TQO). The
crux of the proof hinges on the existence of low-dimensional Gauge-Like
Symmetries (GLSs), thus providing a unifying framework based on a
symmetry principle. All known examples of TQO display GLSs. Other
systems exhibiting such symmetries include Hamiltonians depicting 
orbital-dependent spin exchange and Jahn-Teller effects in transition
metal orbital compounds, short-range frustrated Klein spin models, and
p+ip superconducting arrays. We analyze the physical consequences of
GLSs (including topological terms and charges) and, most importantly,
show the insufficiency of the energy spectrum, (recently defined) 
entanglement entropy, maximal string correlators, and fractionalization
in establishing TQO.  Duality mappings illustrate that not withstanding
the existence of spectral gaps, thermal fluctuations can impose 
restrictions on suggested TQO computing schemes. Our results 
allow us to go beyond standard topological field theories 
and engineer new systems with TQO. 
\end{abstract}
%\maketitle must follow title, authors, abstract, \pacs, and \keywords
\maketitle


The Landau theory of phase transitions is a landmark in physics
\cite{Landau}. Essential is an order parameter characterizing the
thermodynamic phases of the system.  A new paradigm,  Topological
Quantum Order (TQO),  \cite{wenbook} extends the Landau
symmetry-breaking framework.  At its core, TQO is intuitively
associated with insensitivity to {\it local} perturbations. As such,
TQO cannot be described by {\it local} order parameters. Interest is
catalyzed  by the prospect of  fault-tolerant quantum computation
\cite{kitaev}.
%What are the physical principles and consequences behind
%such an order? How do we mathematically characterize it? are  questions
%whose answers are shrouded in mystery. %


Several inter-related concepts are typically invoked in connection to
TQO: symmetry, degeneracy, fractionalization of quantum numbers,
maximal string correlations ({\it non-local} order), among others. The
main issue is {\it what} is needed to have a system with TQO. But a
real problem is that there is no unambiguous definition of TQO. The
current article aims to show relations between these {\it different}
concepts, by rigorously defining and establishing the equivalence
between some of them and more lax relations amongst others. Most
importantly, we (i) prove that  systems harboring generalized
$d$-dimensional (with $d=0,1,2$) Gauge-Like Symmetries ($d$-GLSs)
exhibit TQO; (ii) analyze the resulting conservation laws and the
emergence of topological terms in the action of theories in high space
dimensions; (iii) affirm that the structure of the energy spectrum is
irrelevant for the existence of TQO (the {\it devil} is in the state
itself);  (iv) establish that, fractionalization, string correlators,
and entanglement entropy are insufficient criteria for TQO; (v) report
on a general algorithm for the construction of string correlators; (vi)
suggest links between TQO and problems in graph theory. Our goal is to
provide a unifying framework allowing the creation of new physical
models displaying TQO. A very detailed review and derivation
of the results  presented here is available in \cite{Longversion}.


%Hamiltonians potentially
%capturing a few aspects  of the properties of novel materials display
%non-local string orders, e.g. \cite{tJz,BO,Jurij,OS, KMNZ, Nijs, GM, NR}. 

We focus on quantum lattice systems (and their continuum extension) 
having  $N_s= \prod_{\mu=1}^{D} L_{\mu}$  sites, with $L_\mu$ the
number of sites along each space direction $\mu$, and $D$ the 
dimensionality of the lattice $\Lambda$.  Associated to each lattice
site (or mode, or bond, etc) ${\bf i} \in \Z^{N_s}$ there is a Hilbert
space ${\cal H}_{\bf i}$ of finite dimension ${\cal D}$. The Hilbert
space is the tensor product of the local state spaces, ${\cal H} =
\bigotimes_{\bf i} {\cal H}_{\bf i}$, in the case of distinguishable
subsystems, or a proper subspace in the case of indistinguishable ones.
Statements about local order, TQO, fractionalization, entanglement,
etc., are {\it relative} to the particular decomposition used to
describe the physical system. Typically, the most natural {\it local
language} \cite{GJW} is physically motivated. 

To determine what is needed for TQO, we start by defining it. Given a
set of $N$ orthonormal ground states (GSs) $\{|g_{\alpha}
\rangle\}_{\alpha=1, ..., N}$ and a (uniform) gap to excited states, TQO
exists iff for any bounded operator $V$ with compact support (i.e. 
any  {\it quasi-local operator}),
\begin{eqnarray}
\langle g_{\alpha} | V | g_{\beta} \rangle = v \ \delta_{\alpha \beta} +
c ,
\label{def.}
\end{eqnarray}
where $v$ is a constant and $c$ is a correction that it is either zero
or vanishes exponentially  in the thermodynamic limit. We  will also
examine a finite temperature ($T >0$) extension for the diagonal
elements of Eq. (\ref{def.}),
\begin{eqnarray}
\langle V \rangle_{\alpha}  \equiv \tr(\rho_{\alpha} V)= v + c
~~~~(\mbox{independent of~} \alpha)
\label{vt} ,
\end{eqnarray} 
with $\rho_{\alpha} = \exp[-H_{\alpha}/(k_{B} T)]$  a density matrix
corresponding to the Hamiltonian $H$ endowed with an infinitesimal
symmetry-breaking field favoring order in the state  $|g_{\alpha}
\rangle$. A system displays finite-$T$ TQO if it satisfies both Eqs.
(\ref{def.}), and  (\ref{vt}).

A $d$-GLS of a theory given by $H$ (or action $S$) is a group of
symmetry transformations ${\cal{G}}_d$ such that the minimal non-empty
set of fields $\phi_{\bf i}$ changed by the group operations spans a
$d$-dimensional subset  ${\cal C} \subset \Lambda$.  These
transformations can be expressed as \cite{BN}:
%\begin{equation}
${U}_{{\bl\bk}} = \prod_{{\bf i} \in {\cal C}_{\bl}} {\bf g}_{{\bf
i}{\bk}}$,
%\label{tran}
%\end{equation}
where ${\cal C}_{\bl}$ denotes the subregion $\bl$, and $\Lambda=
\bigcup_{\bl} {\cal C}_{\bl}$. (The extension of this definition to
the  continuum is straightforward.) Gauge (local) symmetries correspond
to $d=0$, while in global symmetries the region influenced by the
symmetry operation is $d=D$-dimensional.  These symmetries
may be Abelian or non-Abelian. 

What are the physical consequences of having a system endowed with a 
symmetry group ${\cal{G}}_d$?  Symmetries generally imply the
existence  of conservation laws and topological charges with associated
continuity equations. We find that systems with $d$-GLSs lead to
conservation laws within $d$-dimensional regions. To illustrate,
consider the (continuum) Euclidean Lagrangian density of a complex
field $\vec{\phi}(\vec{x})=(\phi_1(\vec{x}),
\phi_2(\vec{x}),\phi_3(\vec{x}))$ ($\vec{x}=(x_1,x_2,x_3)$):
%\begin{eqnarray}
${\cal{L}} = \frac{1}{2} \sum_{\mu} |\partial_{\mu} \phi_{\mu}|^{2} 
+ \frac{1}{2} \sum_{\mu} |\partial_{\tau} \phi_{\mu}|^{2} +W(\phi_\mu) $,
%\label{soft}
%\end{eqnarray}
with $W(\phi_\mu)= u(\sum_{\mu} |\phi_{\mu}|^{2} )^{2} - \frac{1}{2}
\sum_{\mu} m^{2}(|\phi_{\mu}|^{2})$ and $\mu,\nu=1,2,3$. ${\cal L}$ 
displays the continuous $d=1$ symmetries $\phi_{\mu} \to
e^{i \psi_{\mu}(\{x_{\nu}\}_{\nu\neq\mu})}  \phi_{\mu}$.  The conserved
$d=1$ Noether currents are tensors given by  $j_{\mu \nu}= i
[\phi_{\mu}^{*} \partial_{\nu}  \phi_{\mu} - (\partial_{\nu} 
\phi_{\mu}^{*}) \phi_{\mu}]$, which satisfy $d=1$ conservation laws
$[\partial_{\nu} j_{\mu \nu} + \partial_{\tau} j_{\mu \tau}] =0$ (with
no summation over repeated indices implicit). What is special about 
$d$-GLSs is that there is a conservation law for each line associated
with a fixed value of all coordinates $x_{\nu \neq \mu}$ relating to
the topological charge  $Q_{\mu}(\{x_{\nu \neq \mu}\}) = \int dx_{\mu}
~j_{\mu \tau}(\vec{x})$.

Can we spontaneously break $d$-GLSs?  The absolute values of quantities
not invariant under ${\cal{G}}_d$ are bounded from above by the
expectation values that they attain in a $d$-dimensional Hamiltonian
$\bar{H}$  (or corresponding action $\bar{S}$) which is globally
invariant under ${\cal{G}}_d$ and preserves the range of the
interactions of the original systems \cite{BN}. As the expectation
values of local observables vanish in low-$d$ systems, this bound
strictly forbids spontaneous symmetry breaking (SSB) of 
non-${\cal{G}}_d$ invariant local quantities in systems with
interactions of finite range and strength whenever $d=0$ (Elitzur's
theorem), $d=1$ for both discrete and continuous ${\cal{G}}_d$, and (as
a consequence of the Mermin-Wagner-Coleman theorem) whenever $d=2$ for
continuous symmetries \cite{BN}. Discrete $d=2$ symmetries may be
broken (e.g. the finite-$T$ transition of the $D=2$ Ising model, and
the $d=2$ Ising GLS of $D=3$ orbital compass systems). In the presence
of a finite gap in a system with continuous $d \le 2$ symmetries, SSB
is forbidden even at $T=0$ \cite{BN}. The absence of SSB of non-GLSs
invariant quantities is due to the presence of low-dimensional 
topological defects such as: domain walls/solitons in systems with 
$d=1$ discrete symmetries, vortices in systems with $d=2$ $U(1)$
symmetries, hedgehogs for $d=2$ $SU(2)$ symmetries. Transitions and
crossovers can only be discerned by quasi-local symmetry invariant
quantities (e.g. Wilson-like loops) or, by probing global topological 
properties (e.g. percolation in  lattice gauge theories
\cite{percolation}). Extending the bound of \cite{BN} to $T=0$, we  now
find that if $T=0$ SSB is precluded in systems with $d$-GLSs then it
will also be precluded in the higher-dimensional system for quantities
not invariant under exact or $T=0$ {\em emergent} $d$-GLSs. Exact
symmetries refer to $[U,H]=0$; in {\em emergent} symmetries \cite{GJW}
unitary operators $ U \in {\cal{G}}_{\sf emergent}$ are not bona fide
symmetries ($[U,H] \neq 0$) yet  become exact at low energies: when
applied to any GS, the resultant state must also  reside in the GS
manifold,  $U| g_{\alpha} \rangle = \sum_{\beta} u_{\alpha  \beta} |
g_{\beta} \rangle$. Thus, SSB of $T=0$, $d=0$ discrete or $d \le 1$
continuous symmetries, in systems of finite  interaction strength and
range, cannot occur. In gapped systems, $T=0$ SSB of $d \le 2$
continuous symmetries is prohibited. 
 
%Symmetries do not necessarily mandate the existence of degeneracies at
%all energies. However, 
Degeneracies imply the existence of symmetries which effect general
unitary transformations within the degenerate manifold and act as the
identity operator outside it. In this way, GS degeneracies act as exact
GLSs. The existence of such unitary symmetry operators (generally a
subset of $SU(N)$) allows for fractional charge  (``$N$-ality" in the
$SU(3)$ terminology of quantum chromodynamics) implying  that
degeneracy allows for fractionalization defined by the center of the
symmetry group. The ($m$-)rized Peierls chains constitute a typical
example of a system with universal ($m-$independent) symmetry operators
\cite{Longversion}, where fractional charge quantized in units of 
$e^{*} = e/m$ with  $e$ the electronic charge is known to occur 
\cite{fraction}. The bounds above generalize to these symmetries.
Different Peierls chain GSs break  discrete $d=1$ symmetries (violating
Eq. (\ref{def.}) in this system with  fractionalization). The Fermi
number $N_{f}$  in the Peierls chain and related Dirac-like theories 
is an integral over spectral functions \cite{NS};  the fractional
portion of $N_{f}$ stems from  soliton contributions and is invariant
under local background deformations.



%In systems with  $d-$dimensional gauge-like symmetries, 
%many operators linking various GSs operate in $d$ dimensional
%regions only as a consequence of the original  $d$ dimensional
%symmetries (exact or emergent) 
%of $H$. {\bf{Please check the following sentence- 
%we had in mind the example SSH chain
%at T=0 in which the $Z_{m}$ symmetry
%may be borken.}} Given its general structure, fractionalization 
%might occur in systems displaying global symmetry breaking.



%{\em Absence of quasiparticle excitations.} 
When the bound of \cite{BN} is applied anew  to correlators and
spectral functions, it dictates  the absence of quasiparticle (qp)
excitations in many instances \cite{NBF}.
%\cite{BN1} 
Here we elaborate on this: The bound of \cite{BN} mandates that the
absolute values of non-symmetry invariant  correlators $|G| \equiv
|\sum_{\Omega_{{\bf j}}} a_{\Omega_{{\bf j}}} \langle \prod_{{\bf i}
\in \Omega_{{\bf j}}}  \phi_{{\bf i}} \rangle|$ with $\Omega_{{\bf j}}
\subset {\cal C}_{{\bf j}}$, and  $\{a_{\Omega_{{\bf j}}}\}$ 
$c$-numbers, are bounded from above (and from below for $G\geq0$ (e.g.
that corresponding to $\langle |\phi({\bf k}, \omega)|^{2} \rangle$))
by absolute values of the  same correlators $|G|$ in a  $d$-dimensional
system defined by ${\cal C}_\bj$. In particular, $\{a_{\Omega_{{\bf
j}}}\}$ can be chosen to give the Fourier transformed  pair-correlation
functions. This leads to stringent bounds on viable qp weights and
establishes the absence of qp excitations in many cases.   In
high-dimensional systems, retarded correlators $G$ generally exhibit a
resonant (qp) contribution. %e.g. \cite{spectral},\cite{vns}.  
Here, $G= G_{{\sf res}}({\bf k}, \omega)  + G_{{\sf non-res}}({\bf k},
\omega)$ with $G_{{\sf res}}({\bf k}, \omega)  = \frac{Z_{{\bf
k}}}{\omega - \epsilon_{{\bf k}} + i 0^{+}}$. In low-dimensional
systems, the qp weight  $Z_{{\bf k}} \to 0$ and the poles of $G$ are
often replaced by weaker branch cut behavior. If the momentum ${\bf k}$
lies in a lower $d$-dimensional region ${\cal{C}}_{\bf j}$  and if no
qp resonant terms   appear in the corresponding lower-dimensional
spectral functions  in the presence of non-symmetry breaking fields
then the upper bound \cite{BN} on the correlator $|G|$  (and on related
qp weights  given by $\lim_{\omega \to \epsilon_{{\bf k}}} (\omega -
\epsilon_{{\bf k}}) G({\bf k}, \omega)$) of non-symmetry invariant
quantities mandates the absence of normal qps. Putting all of the 
pieces together we see that if fractionalization
occurs in the lower-dimensional system then its higher-dimensional
realization follows.
%(fractionalized
%excitations propagate in $d_{s} = D-d$ dimensional regions) \cite{NBF}.

Our central contention is that in all systems known to harbor TQO (and
in new examples), $d$-GLSs are present. Old examples include: Quantum
Hall systems, $\Z_{2}$ lattice gauge theories, the Toric-code model
\cite{kitaev} and other systems.  In all cases of TQO, we may cast
known ``topological" symmetry operators as general low-dimensional $d
\le 2$  GLSs (e.g. in the Toric code model,  there are $d=1$ symmetry
operators  spanning toric cycles). The presence of these symmetries
allows for the existence of freely-propagating decoupled
$d$-dimensional topological defects  (or instantons in $(d+1)$
dimensions of Euclidean space time)   which eradicate local order.
These  defects enforce TQO. We now state a central result: 
%\begin{theorem}
%\label{thm:1}


{\bf{Theorem.}} {\em When, in a system of finite interaction range  and
strength which satisfies Eq. (\ref{def.}), all GSs may be linked by 
discrete $d \le 1$ or by  continuous $d \le 2$ GLSs $U \in
{\cal{G}_{\it d}}$, then  the system displays finite-$T$ {\rm TQO}. } 
%\end{theorem}

\proof  
Let's start by decomposing $V = V_{0} + V_{\perp}$. Here, $V_{0}$ is 
the portion that transforms as a singlet under ${\cal G}_d$ i.e.,
$[U,V_{0}]=0$.  To prove the finite-$T$ relation of  Eq. (\ref{vt}),   we
write the expectation values over a complete  set of orthonormal states
$\{ |a \rangle \}$,
\begin{eqnarray}
\!\!\!\!\!\!\!\! \langle V_{0}  \rangle_{\alpha}  &=& \lim_{h \to 0^{+}} 
\frac{\sum_{a} \langle a| V_{0}| a \rangle  e^{-\beta (E_{a} + \phi^{a}_{\alpha}
%|\langle a|
%g_{\alpha} \rangle|^{2} 
h)}}{\sum_{a} e^{-\beta (E_{a} + \phi^{a}_{\alpha}
%|\langle a|
%g_{\alpha} \rangle|^{2} 
h)}} \nonumber \\ 
&=& \lim_{h \to 0^{+}} \frac{ \sum_{a} \langle a| U^{\dagger} V_{0} U | a
\rangle  e^{-\beta (E_{a} + \phi^{a}_{U^{\dagger} \beta}
%|\langle a| U| g_{\beta} \rangle|^{2}
h)}}{\sum_{a} e^{-\beta (E_{a}  + \phi^{a}_{U \beta}
%|\langle a| U | g_{\beta}
%\rangle|^{2} 
h)}} \nonumber \\ 
&=& \lim_{h \to 0^{+}}  \frac{\sum_{b} \langle b| V_{0}| b \rangle 
e^{-\beta (E_{b} +  \phi^{b}_{\beta}
%|\langle b| g_{\beta} \rangle|^{2} 
h)}} {\sum_{b}
e^{-\beta (E_{b} + \phi^{b}_{\beta}
%|\langle b| g_{\beta} \rangle|^{2} 
h)}} = \langle V_{0}
\rangle_{\beta}.
\label{long} 
\end{eqnarray}
Here, we invoked $U|a \rangle \equiv |b \rangle,$ and $E_{a} = E_{b}$
(as $[U,H]=0$). The term $\phi^{a}_{\alpha}$ monitors the order
parameter content of the state $|a \rangle$ vis a vis that preferred by
the GS $|g_{\alpha} \rangle$. In the above derivation, 
$\phi^{a}_{\alpha} = \phi^{Ua}_{U \alpha}  = \phi^{b}_{\beta}$, which
is evident by the application of a simultaneous  unitary transformation
in going from $|a \rangle \to^{U}  |b \rangle$ and  $|g_{\alpha}
\rangle \to^{U} |g_{\beta} \rangle$. $V_{\perp}$ is not invariant 
under ${\cal{G}_{\it d}}$ ($[U,V_{\perp}]  \neq 0$) then by the theorem
of \cite{BN}, $\langle V_{\perp} \rangle_\alpha =0$,  i.e. for systems
with low-dimensional GLSs SSB is precluded. Eq. (\ref{long})  is valid
whenever $[U,V_{0}] = 0$ for {\it any} symmetry $U$. However, 
$[U,V_{\perp}] \neq 0$ implies $\langle V_{\perp}  \rangle_\alpha =0$
only if $U$ is a low-dimensional GLS.  In systems in which not all GS
pairs can be linked by the use of low-dimensional GLSs $U \in
{\cal{G}}_{d}$ ($U| g_{\alpha} \rangle = | g_{\beta} \rangle$),
finite-$T$ SSB may occur. We conclude this proof with two remarks 
\cite{Longversion}: (i) $T=0$ TQO holds whenever all GSs may be linked
by continuous  $d \le 2$ GLSs. (ii) In many systems (whether gapped or
gapless),  $T=0$ TQO states may be constructed by  employing
Wigner-Eckart type selection rules for  GLSs.   



%{\em Gauge like symmetries in known (and new) 
%examples of topological order.} Apart from 
%Quantum Hall systems, several other prominent 
%systems known to display topological order, do in 
%fact possess a low dimensional gauge like symmetry
%from which the existence topological order immediately
%follows. In all examined cases of TQO, we were 
%able to cast known
%``topological" symmetry operators as general low 
%dimensional gauge like symmetries. As proven above, 
%the presence of these symmetries
%mandates TQO. As simple examples, 
%we discuss in Appendix \ref{app5} (a) 
%{\em{Kitaev's toric code model}}, \cite{kitaev} 
%(b) $\Z_{2}$ lattice
%gauge theories, \cite{wenbook} \cite{wegner}, \cite{Kogut} 
%and (c) Wen's plaquette model
%\cite{wen_plaq}
%in terms of their "topological" $d=1$ gauge like 
%symmetries. In Appendix \ref{app6}, we discuss topological order
%in very physical two and three dimensional systems which were heavily 
%investigated in the literature as systems harboring topological order. 
 
A related consequence of systems with $d$-GLSs is that topological
terms which appear in $d+1$-dimensional theories also appear in higher
$D+1$-dimensional systems ($D>d$). These topological terms appear in
actions $\bar{S}$ which bound quantities not invariant under the 
$d$-GLSs. For instance, in the isotropic $D=2$ (or 2+1) dimensional 
general spin $t_{2g}$ KK model \cite{KK, Longversion},  of exchange
constant  $J>0$, the corresponding continuum Euclidean action is of the
1+1 form
%\begin{eqnarray}
$\bar{S} = \frac{1}{2g} \int dx d \tau  \Big[ \frac{1}{v_{s}}
(\partial_{\tau} \hat{m})^{2} - v_{s} (\partial_{x} \hat{m})^{2} \Big] 
+ i \theta {\cal{Q}}  + S_{{\sf tr}},~~ 
{\cal{Q}} = \frac{\theta}{8 \pi} \epsilon_{\mu \nu} \cdot
(\partial_{\mu} \hat{m} \cdot \partial_{\nu} \hat{m})$,
%\end{eqnarray}
with  $\mu, \nu \in \{ x, \tau \}$ and $\epsilon_{\nu \mu}$ the rank
two Levi Civita symbol.  Here, as in the non-linear-$\sigma$ model of a
spin-$S$ chain, $\hat{m}$ a normalized slowly varying  staggered field,
$g=2/S$, $v_{s} = 2 JS$, $\theta = 2 \pi S$, and $S_{{\sf tr}}$ a
``transverse-field" action term which does  not act on the spin degrees
of freedom along a given chain \cite{BN}.  ${\cal{Q}}$ is the
Pontryagin index  corresponding to the mapping between the 1+1 
dimensional space-time $(x,\tau)$ plane and the two-sphere on which
$\hat{m}$ resides. This 1+1 dimensional topological term appears  in
the 2+1 dimensional KK system even for arbitrary large positive
coupling $J$. This, in turn, places bounds on the spin correlations and
implies, for instance, that in $D=2$ integer-spin $t_{2g}$  KK systems,
a finite correlation length exists. 

What characterizes TQO? We will see that neither the spectrum of $H$,
nor the entanglement entropy \cite{entropy}, nor string correlations
are sufficient criteria. That the spectrum, on its own, is insufficient
is established by  counter-examples, e.g. that of the $D=3$ $\Z_{2}$
lattice gauge theory  which is dual to a $D=3$ Ising model
\cite{wegner}. Albeit sharing the same energy spectrum, the gauge
theory displays TQO while the Ising model harbors a local order. A
spectral equivalence exists between other TQO inequivalent system
pairs. For example, Kitaev's  \cite{kitaev} and Wen's models
\cite{wen_plaq} are equivalent and have a spectrum which
is identical to that of an Ising spin chain with nearest neighbor
interactions \cite{Longversion}. The mappings between these systems to
an Ising chain demonstrate that despite  the spectral gap in these
systems,  the toric operator expectation values may vanish once thermal
fluctuations are present. These mappings also illustrate that the
quantum states themselves in a particular (operator language)
representation encode TQO and that the duality mappings, being
non-local in the original representation, disentangle the order. Thus,
we cannot, as coined by Kac, ``hear the shape of a  drum". The
information is in the eigenvectors.  

It was recently suggested \cite{entropy} that a non-local borne
deviation $\gamma>0$ from an asymptotic area law scaling for the
entanglement entropy constitutes a sharp measure of TQO. We find, for
example in Klein spin models which do not host TQO (i.e. do  not
satisfy Eq. (\ref{def.}) for the set of all GSs),  an arbitrary --
contour shape dependent --  entanglement entropy which deviates from an
area law \cite{Longversion}. This suggests that the criterion of
\cite{entropy} can be violated and may require additional improvements,
such as the specification of an exact limit and/or average for large
contours. 

The insufficiencies of the spectra and entropy in determining if TQO is
present have counterparts in the topology of graphs and in the Graph
Equivalence Problem (GEP) in particular. The adjacency matrix of a
graph has elements  $C_{\bi\bj} =1$ if vertices $\bi$ and $\bj$ are
linked by an edge and $C_{\bi\bj}=0$ otherwise. Vertex relabeling $\bi
\to p(\bi)$ leaves a graph invariant but changes the adjacency matrix
$C$ according to  $C \to C^{\prime} =  P^{\dagger} C P$ with $P$ an
orthogonal matrix which represents the permutation: $P =
\delta_{\bj,p(\bi)}$. The GEP is the following \cite{godsil}: ``Given
$C$ and $C^{\prime}$, can we decide if both correspond to the same
topological graph?"  The spectra of $C$ and $C^{\prime}$ are
insufficient criteria. Entropic measures \cite{godsil} are useful but
also do not suffice. 

In many systems (with or without TQO), there are non-local ``string''
correlators which display enhanced (or maximal) correlations vis a vis
standard 2-point correlation functions. We now outline an algorithm for
the construction of such non-local correlators. [In systems with
uniform global order already  present in their GS, the algorithm leads
to the usual 2-point correlators.] We seek a unitary transformation
$U_{s}$ which rotates the GSs into  a new set of states which have
greater correlations as measured by a  set of local operators
$\{V_{\bi}\}$. These new states may have  an appropriately defined {\em
polarization} (eigenvalues $\{v_{\bi}\}$) of either (i) more slowly
decaying (algebraic or other) correlations, (ii) a uniform sign ({\em
partial polarization}) or (iii) maximal  expectation values $v_{\bi} =
v_{max}$ for all $\bi$ ({\em maximal polarization}). Cases (i) or (ii)
may lead to a lower dimensional gauge like structure for the enhanced
correlator. In systems with entangled GSs (such as in many (yet not
all) GSs hosting $T=0$ TQO),  $U_{s}$ cannot be a uniform product of
locals; $U_{s} \neq \prod_{\bi \in \Lambda}  O_{\bi}$.  To provide a
concise known example where these concepts become clear, we focus below
on case (ii) within the  well-studied  AKLT  Hamiltonian \cite{AKLT}. 
Here, there is a non-trivial unitary operator $U_{s} \equiv
\prod_{\bj<\bk} \exp[i \pi S^{z}_{\bj} S^{x}_{\bk}]$, ($[U_{s}, H] \neq
0$) which maps the GSs  $\{|g_{\alpha}\rangle \}$  into linear
superpositions of states in each of which  the  local staggered
magnetization  $V_{\bi}=(-1)^{\bi}S_{\bi}^{z} \equiv  M_{\bi}$  is
uniformly non-negative (or non-positive) at every site $\bi$. As all
transformed states $|p_{\alpha} \rangle = U_s | g_{\alpha} \rangle$ are
superpositions of states with uniform sign  $v_{{\bf{i}}}$ (allowing
for two non-negative or non-positive  $v_{{\bf{i}}}$ values at every
site out of the three $S=1$ states),  $\tilde{G}_{\bi\bj}  \equiv 
\langle p_{\alpha} | V_{\bi} V_{\bj} |p_{\alpha} \rangle  > \langle
g_{\alpha}| V_{\bi} V_{\bj} |g_{\alpha} \rangle \equiv G_{\bi\bj}$.
When $U_{s}$ is written in full, $\tilde{G}_{\bi\bj}$ becomes  a
non-local ``string'' correlator $\tilde{G}_{\bi\bj} =  (-1)^{|{\bf{i}}-
{\bf{j}}|}  \langle g_{\alpha} | S_\bi^{z} \prod_{\bi < \bk < \bj}
\exp[i \pi S_{\bk}^{z}] S_{\bj}^{z} | g_{\alpha} \rangle = (2/3)^{2}$
for arbitrarily large separation $|\bi-\bj|$. In the $\otimes 
S^{z}_{\bf{i}}$ eigen-basis, there are $2^{N_{s}}$ states $\ket{\phi}$ 
with non-negative (non-positive) $v_{\bi}$ for all $\bi$ (they form a
Hilbert subspace), and all of these have the same value of
$\tilde{G}_{\bi\bj}  = \langle {\phi} | V_{\bi}  V_{\bj} | {\phi}
\rangle$, that is $4/9$. Such an exponential number of states appears
in systems with a local $\Z_{2}$ gauge structure. Here, the string
correlators  exhibit a  symmetry under local $(d=0$) gauge
transformations while $H$ itself is not invariant.  These $d=0$
$\Z_{2}$ transformations correspond (in the $\otimes S^{z}_{\bi}$
basis) to the creation (annihilation) of an $S^{z}_{\bj}=0$ state at
any site ${\bj}$  followed by a unit displacement of $S^{z}_{\bf k}$ at
all sites ${\bf k}>{\bf j}$.   The operators linking the GSs  form a 
discrete ($\Z_{2} \times \Z_{2}$) global symmetry group \cite{AKLT}
and, as such, $T=0$ SSB may occur.  The general polarization operators
are intimately tied to selection rules which  hold for all of the GSs;
these selection rules (and a particular low-energy projection of
truncated  Hamiltonians for gapped systems) enable the construction of
general string and higher dimensional ``brane''  correlators.
In the AKLT problem, the local expectation value
$\langle g_{\alpha} | S_{\bi=1}^{z} | g_{\alpha} \rangle$ depends on
$\ket{g_\alpha}$, violating Eq. (\ref{def.}) and suggesting that the
AKLT chain is not topologically ordered. 
%However, finite temperature spontaneous symmetry breaking
%is precluded in this one dimensional system,
%$\langle S^{z} \rangle =0$.
We may also generate maximal ($\tilde{G}_{\bi\bj} =1$) string
correlators; here, the number of states with  maximal polarization is
finite and a non-local gauge-like structure emerges.  Generally, in any
system with  known (or engineered) GSs, we may explicitly construct
polarizing transformations $U_{s}$. 

To conclude, we proved a theorem establishing how $d \le 2$ 
dimensional GLSs in gapped systems  can mandate TQO via freely
propagating low-dimensional topological defects (e.g. $d=1$ solitons). 
This result extends the set of known systems which exhibit TQO to
include new orbital, spin, and Josephson  junction array problems.  
All known TQO systems display GLSs \cite{Longversion}. We examined
physical consequences of GLSs  (such as conservation laws and
topological terms in high-$D$ theories), and  low-dimensional
gauge-like structures  in theories with entangled GSs  in non-maximal
string correlators (irrespective of TQO).  We established
insufficiencies of the Hamiltonian spectra, fractionalization, 
entanglement entropy, and maximal string correlations in determining
TQO.


We thank C. D. Batista, S. Bravyi, A. Kitaev, S. Nussinov, J.
Preskill,  and P. Wiegmann for discussions, and the DOE, NSA, CMI of WU
for support.




\begin{thebibliography}{99}
\bibitem{Landau} 
L.D. Landau and E.M. Lifshitz, {\it Statistical Physics}
(Butterworth-Heinemann, Oxford, 1980).

%\bibitem{andersonrev} 
%P.W. Anderson, {\it Basic Notions of Condensed Matter Physics}
%(Addison-Wesley, Redwood City, 1992).

\bibitem{wenbook} 
X-G. Wen, {\it Quantum Field Theory of Many-Body Systems} (Oxford
University Press, Oxford, 2004).


\bibitem{kitaev} 
A. Kitaev, Ann. Phys. {\bf 303}, 2 (2003).
%; cond-mat/0506438  (2005). 

\bibitem{Longversion} 
%Supporting material is available at:
Z. Nussinov and G. Ortiz, cond-mat/0702377.

\bibitem{GJW} 
C. D. Batista and G. Ortiz, Adv. in Phys. {\bf 53}, 2 (2004).

\bibitem{BN} 
C. D. Batista and Z. Nussinov, Phys. Rev. B {\bf 72}, 045137 (2005).

%\bibitem{Elitzur}
%S. Elitzur, Phys. Rev. D {\bf 12}, 3978 (1975).

%\bibitem{Mermin}
%N. D. Mermin and H. Wagner, Phys. Rev. Lett. {\bf 17}, 1133 (1966); S.
%Coleman, Commun. Math. Phys. {\bf 31}, 259 (1973).

\bibitem{percolation} 
Z. Nussinov, Phys. Rev. D {\bf 72}, 054509 (2005).
%A. Vestergren, J. Lidmar, and T. H. Hansson, 
%Europhys. Lett. {\bf 69}, 256 (2005);
%M. B. Hastings and Xiao-Gang Wen,
%Phys.Rev. B {\bf 72}, 045141 (2005) 

%\bibitem{satz}
%H. Satz, Comput. Phys. Commun. {\bf 147}, 46 (2002)



\bibitem{fraction}  
W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. {\bf
42}, 1698 (1979).

\bibitem{NS} 
A. J. Niemi and G. W. Semenoff, Phys. Rep. {\bf 135}, 99 (1986).

\bibitem{NBF} Z. Nussinov, C. D. Batista, and E. Fradkin,
cond-mat/0602569.

%\bibitem{BN1} C. D. Batista and Z. Nussinov, {\em in preparation}

%\bibitem{spectral} A. L. Fetter 
%and J. D. Walecka {\em Quantum Theory of Many-Particle Systems}, McGraw-Hill 
%(1971)

%\bibitem{vns} C. M. Varma, Z. Nussinov, and W. van Saarloos, Physics Reports {%\bf 361}, 267 (2002)


%\bibitem{NBNT} Z. Nussinov, C. D. Batista, B. Normand, S. A. Trugman, cond-mat/0602528

%\bibitem{BT} C. D. Batista and S. A. Trugman, 
%Phys. Rev. Lett. {\bf 93}, 217202 (2004).




%\bibitem{NBCv} Z. Nussinov, M. Biskup, L. Chayes, J. v. d. Brink,
%Europhysics Letters, {\bf 67}, 990 (2004) 

%\bibitem{BCN} M. Biskup, L. Chayes, Z. Nussinov,
% Comm.  Math. Phys., 
%{\bf 255}, 253 (2005)

%\bibitem{Mishra}
%A. Mishra et al., 
%Phys. Rev. Lett. {\bf 93}, 207201 (2004) 


%\bibitem{confine} 
%A. M. Polyakov, Nucl. Phys. B {\bf 120}, 429 (1977). 

%\bibitem{kohn} 
%W. Kohn, Phys. Rev. {\bf 133}, A171 (1964). 

%\bibitem{matt}
%M. B. Hastings and T. Koma, math-ph/0507008.  

\bibitem{KK} 
K. I. Kugel and D. I. Khomskii, Sov. Phys. JETP {\bf 37}, 725 (1973).
%Sov. Phys. Usp. {\bf 25}, 231 (1982)

\bibitem{entropy}  
A. Kitaev and J. Preskill, Phys. Rev. Lett. {\bf 96}, 110404 (2006); 
M. Levin and X-G. Wen, Phys. Rev. Lett. {\bf 96}, 110405 (2006).

\bibitem{wegner} F. J. Wegner, 
J. of Math. Phys. {\bf 12}, 2259 (1971).

\bibitem{wen_plaq} 
X-G. Wen, Phys. Rev. Lett. {\bf 90}, 016803 (2003).

%\bibitem{graph} 
%D. Cvetkovic, P. Rowlinson and S. Simic, 
%{\em Eigenspaces of graphs} 
%(Encyclopedia of mathematics and its applications, v. 66), 
%Cambridge, New York:Cambridge University Press, 258p., (1997);
%A. Renyi, 
%{\em Probability Theory}, 
%North-Holland Publishing Company-Amsterdam-London, 
%and American Elsevier Publishing Company, Inc.-New York, (1970)

\bibitem{godsil} C. D. Godsil, D. A. Holton, and B. D. McKay, 
{\em The spectrum of a graph}, in Lecture Notes in Math. 
{\bf 622}, 91 (Springer-Verlag, Berlin, 1977).
 
%\bibitem{GN} V. Gudkov and S. Nussinov,
%cond-mat/0209112

%\bibitem{GNN} V. Gudkov, S. Nussinov, and Z. Nussinov,
%cond-mat/0209419

\bibitem{AKLT}  
I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Commun. Math. Phys.
{\bf 115}, 477 (1988); T. Kennedy and H. Tasaki, Comm. Math. Phys. 
{\bf 147}, 431 (1992).





\end{thebibliography}
\end{document}

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cond-mat0605316.extracted_bib
\begin{thebibliography}{99}
\bibitem{Landau} 
L.D. Landau and E.M. Lifshitz, {\it Statistical Physics}
(Butterworth-Heinemann, Oxford, 1980).

%\bibitem{andersonrev} 
%P.W. Anderson, {\it Basic Notions of Condensed Matter Physics}
%(Addison-Wesley, Redwood City, 1992).

\bibitem{wenbook} 
X-G. Wen, {\it Quantum Field Theory of Many-Body Systems} (Oxford
University Press, Oxford, 2004).


\bibitem{kitaev} 
A. Kitaev, Ann. Phys. {\bf 303}, 2 (2003).
%; cond-mat/0506438  (2005). 

\bibitem{Longversion} 
%Supporting material is available at:
Z. Nussinov and G. Ortiz, cond-mat/0702377.

\bibitem{GJW} 
C. D. Batista and G. Ortiz, Adv. in Phys. {\bf 53}, 2 (2004).

\bibitem{BN} 
C. D. Batista and Z. Nussinov, Phys. Rev. B {\bf 72}, 045137 (2005).

%\bibitem{Elitzur}
%S. Elitzur, Phys. Rev. D {\bf 12}, 3978 (1975).

%\bibitem{Mermin}
%N. D. Mermin and H. Wagner, Phys. Rev. Lett. {\bf 17}, 1133 (1966); S.
%Coleman, Commun. Math. Phys. {\bf 31}, 259 (1973).

\bibitem{percolation} 
Z. Nussinov, Phys. Rev. D {\bf 72}, 054509 (2005).
%A. Vestergren, J. Lidmar, and T. H. Hansson, 
%Europhys. Lett. {\bf 69}, 256 (2005);
%M. B. Hastings and Xiao-Gang Wen,
%Phys.Rev. B {\bf 72}, 045141 (2005) 

%\bibitem{satz}
%H. Satz, Comput. Phys. Commun. {\bf 147}, 46 (2002)



\bibitem{fraction}  
W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. {\bf
42}, 1698 (1979).

\bibitem{NS} 
A. J. Niemi and G. W. Semenoff, Phys. Rep. {\bf 135}, 99 (1986).

\bibitem{NBF} Z. Nussinov, C. D. Batista, and E. Fradkin,
cond-mat/0602569.

%\bibitem{BN1} C. D. Batista and Z. Nussinov, {\em in preparation}

%\bibitem{spectral} A. L. Fetter 
%and J. D. Walecka {\em Quantum Theory of Many-Particle Systems}, McGraw-Hill 
%(1971)

%\bibitem{vns} C. M. Varma, Z. Nussinov, and W. van Saarloos, Physics Reports {%\bf 361}, 267 (2002)


%\bibitem{NBNT} Z. Nussinov, C. D. Batista, B. Normand, S. A. Trugman, cond-mat/0602528

%\bibitem{BT} C. D. Batista and S. A. Trugman, 
%Phys. Rev. Lett. {\bf 93}, 217202 (2004).




%\bibitem{NBCv} Z. Nussinov, M. Biskup, L. Chayes, J. v. d. Brink,
%Europhysics Letters, {\bf 67}, 990 (2004) 

%\bibitem{BCN} M. Biskup, L. Chayes, Z. Nussinov,
% Comm.  Math. Phys., 
%{\bf 255}, 253 (2005)

%\bibitem{Mishra}
%A. Mishra et al., 
%Phys. Rev. Lett. {\bf 93}, 207201 (2004) 


%\bibitem{confine} 
%A. M. Polyakov, Nucl. Phys. B {\bf 120}, 429 (1977). 

%\bibitem{kohn} 
%W. Kohn, Phys. Rev. {\bf 133}, A171 (1964). 

%\bibitem{matt}
%M. B. Hastings and T. Koma, math-ph/0507008.  

\bibitem{KK} 
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\end{thebibliography}