Quantum magnetism in two dimensions: From semi-classical \Neel order to magnetic disorder
J.~Richter\inst{1}
J.~Schulenburg\inst{2}
A.~Honecker\inst{3}
2dqm-abs-toc.tex
\contentsline {section}{\numberline {1}Introduction}{4}{section.0.1}
\contentsline {section}{\numberline {2}Archimedean Lattices}{6}{section.0.2}
\contentsline {subsection}{\numberline {2.1}Characteristics and geometry}{6}{subsection.0.2.1}
\contentsline {subsection}{\numberline {2.2}Relationships between the lattices}{8}{subsection.0.2.2}
\contentsline {section}{\numberline {3}Criteria for N\'{e}el like order}{11}{section.0.3}
\contentsline {subsection}{\numberline {3.1}Order parameter}{11}{subsection.0.3.1}
\contentsline {subsection}{\numberline {3.2}Mechanism of symmetry breaking - the Pisa tower of quasi-degenerate joint states (QDJS)}{13}{subsection.0.3.2}
\contentsline {subsection}{\numberline {3.3}Finite-size scaling}{16}{subsection.0.3.3}
\contentsline {section}{\numberline {4}Magnetic ground-state ordering for the spin half HAFM on the Archimedean lattices}{18}{section.0.4}
\contentsline {subsection}{\numberline {4.1}Semi-classical N\'{e}el ordering on bipartite lattices}{18}{subsection.0.4.1}
\contentsline {subsubsection}{The square lattice (T2)}{19}{section*.2}
\contentsline {subsubsection}{The honeycomb lattice (T3)}{21}{section*.3}
\contentsline {subsubsection}{The CaVO (T11) and the SHD (T10) lattices}{23}{section*.4}
\contentsline {subsection}{\numberline {4.2}Semi-classical LRO on frustrated lattices}{25}{subsection.0.4.2}
\contentsline {subsubsection}{The triangular lattice (T1)}{26}{section*.5}
\contentsline {subsubsection}{The maple-leaf (T4) and the bounce (T7) lattices}{27}{section*.6}
\contentsline {subsubsection}{The trellis lattice (T5)}{30}{section*.7}
\contentsline {subsubsection}{The SrCuBO lattice (T6)}{32}{section*.8}
\contentsline {subsection}{\numberline {4.3}Absence of semi-classical LRO on frustrated lattices - the {kagom\'e} (T8) and the star (T9) lattices}{34}{subsection.0.4.3}
\contentsline {subsection}{\numberline {4.4}Summary and comparison}{39}{subsection.0.4.4}
\contentsline {section}{\numberline {5}Quantum phase transitions in 2D HAFM - the CaVO $J-J'$ model and the Shastry-Sutherland model}{44}{section.0.5}
\contentsline {section}{\numberline {6}Magnetization process}{48}{section.0.6}
\contentsline {subsection}{\numberline {6.1}Square lattice}{50}{subsection.0.6.1}
\contentsline {subsection}{\numberline {6.2}Triangular lattice}{53}{subsection.0.6.2}
\contentsline {subsection}{\numberline {6.3}Kagom\'e lattice}{55}{subsection.0.6.3}
\contentsline {subsection}{\numberline {6.4}Independent magnons and macroscopic magnetization jumps}{57}{subsection.0.6.4}
\contentsline {subsection}{\numberline {6.5}Shastry-Sutherland model versus SrCu$_2$(BO$_3$)$_2$}{61}{subsection.0.6.5}
\contentsline {subsection}{\numberline {6.6}Summary of plateaux \& related topics}{64}{subsection.0.6.6}
\contentsline {section}{References}{66}{section*.9}

2dqm-abs.tex
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\def\ChapOneD{H.-J.~Mikeska and A.K.~Kolezhuk}
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\def\ChapCCM{D.J.J.~Farnell and R.F.~Bishop}


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\begin{document}

\title*{Quantum magnetism in two dimensions: From semi-classical \Neel order
to magnetic disorder 
}
\titlerunning{Quantum magnetism in two dimensions}
\author{J.~Richter\inst{1}, J.~Schulenburg\inst{2} \and
A.~Honecker\inst{3}}
\authorrunning{Richter, Schulenburg, and Honecker}
\institute{Institut f\"ur Theoretische Physik, Otto-von-Guericke-Universit\"at
Magdeburg, P.O.Box 4120, D-39016 Magdeburg, Germany
\texttt{johannes.richter@physik.uni.magdeburg.de}
\and Universit\"atsrechenzentrum, Otto-von-Guericke-Universit\"at
Magdeburg, P.O.Box 4120, D-39016 Magdeburg, Germany
\texttt{joerg.schulenburg@urz.uni-magdeburg.de}
\and Institut f\"ur Theoretische Physik, TU Braunschweig,
Mendelsohnstr. 3, D-38106 Braunschweig, Germany
\& Universit\"at Hannover, Institut f\"ur Theoretische
Physik, Appelstrasse 2, D-30167 Hannover, Germany \\
\texttt{a.honecker@tu-bs.de}}

\maketitle

\vspace*{4mm}
Published version: Chapter 2 in
\href{http://www.springerlink.com/link.asp?id=jp2ehtqa7e7h}%
{\it Quantum Magnetism},
ed by U.~Schollw\"ock, J.~Richter, D.J.J.~Farnell, R.F.~Bishop,
\href{http://www.springerlink.com/link.asp?id=ccv22wcn1jnrxfhu}%
{Lecture notes in physics {\bf 645}, 85-153}
(Springer-Verlag, Berlin Heidelberg 2004)
\vspace*{6mm} \\
\input{abstracte.tex}
\newpage
\setcounter{tocdepth}{2}
\section*{Contents\hfill (preprint version)}
\input 2dqm-abs-toc.tex

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\vspace*{6mm}
\noindent
Complete preprint version of this article:
\vspace*{1mm}

\centerline{PostScript at:
\href{http://www.tu-bs.de/~honecker/papers/2dqm.ps.gz}%
{\tt http://www.tu-bs.de/\~{ }honecker/papers/2dqm.ps.gz}}

\centerline{PDF at:
\href{http://www.tu-bs.de/~honecker/papers/2dqm.pdf}%
{\tt http://www.tu-bs.de/\~{ }honecker/papers/2dqm.pdf}}

\vspace*{3mm}
\noindent
Complete published version of this article available from:
\vspace*{1mm}

\centerline{\href{http://www.springerlink.com/link.asp?id=ccv22wcn1jnrxfhu}%
{\tt http://www.springerlink.com/link.asp?id=ccv22wcn1jnrxfhu}}

\vspace*{1mm}

PDF of published version also at:

\centerline{\href{http://www.tu-bs.de/~honecker/papers/lnp645.pdf}%
{\tt http://www.tu-bs.de/\~{ }honecker/papers/lnp645.pdf}}

\vspace*{4mm}
\noindent
Additional features of preprint version:
\begin{itemize}
\item Extended Abstract,
\item References updated,
\item Separate Table of Contents,
\item Value of the square of the order parameter
$N=36$ \kagome lattice corrected: $(m^+)^2= 0.059128$.
\end{itemize}


\end{document}

abstracte.tex
{\bf Abstract:}  

In this article we focus on the ground state  and the low-lying excitations
of the $s=1/2$ Heisenberg antiferromagnet (HAFM) on the 11 two-dimensional (2D) 
uniform Archimedean lattices.

Although we know from the Mermin-Wagner theorem that 
thermal fluctuations are strong enough to
destroy magnetic long-range order (LRO) for Heisenberg spin systems at 
any finite temperature 
in one  and two dimensions, 
 the role of quantum fluctuations is less understood.
 While  the ground state of the
 one-dimensional (1D) quantum HAFM is not long-range ordered,  
the quantum HAFM e.g.\ on the 2D square  and  triangular lattices exhibits
semi-classical N\'eel like LRO.
However, in two dimensions there are many other lattices with different coordination
numbers and topologies, and there is no general statement concerning
zero-temperature N\'eel-like LRO. Recent experimental results on
CaV$_4$O$_9$ and SrCu$_2$(BO$_3$)$_2$
demonstrate the possibility of non-N\'{e}el
 ordered ground states and signal that the $s=1/2$ HAFM  on 
2D lattices with appropriate 
topology may have a ground state without semi-classical LRO.



Based on extensive large-scale exact diagonalization studies of the ground
state and the low-lying excitations for the spin-1/2 HAFM on the Archimedean
lattices we compare and discuss the ground-state features of all 11
lattices. In this manner we obtain some insight in the influence of
lattice topology on magnetic ordering of quantum antiferromagnets in
two dimensions. From our results we conclude that
the ground state of the spin-1/2 HAFM on most of the Archimedean lattices 
(in particular the four bipartite ones) turns out to be semi-classically
N\'eel-like ordered. However, we find that the interplay of competition
of bonds (geometric frustration and non-equivalent nearest neighbor bonds)
and quantum fluctuations gives rise to a quantum paramagnetic ground state 
without semi-classical LRO 
for two lattices. The first one is the famous kagom\'e lattice, for
which this statement is  well-known by numerous studies during the last
decade. Remarkably, we find one additional lattice
among the 11 uniform Archimedean lattices, the so-called star lattice, with
a quantum paramagnetic ground state.
For both these  Archimedean lattices the ground state is
highly degenerate in the classical limit $s \to \infty$, although notably their
quantum ground states are fundamentally different.


Furthermore, we present numerical results for the
magnetization curve of the HAFM on all
11 Archimedean lattices. The magnetization process is discussed
in some detail for the square, triangular and \kagome lattices.
One focus are plateaus appearing in the magnetization curve
due to quantum fluctuations and geometric frustration.
In particular, the kagom\'e lattice
exhibits a rich spectrum of magnetization plateaus.
Another focus are magnetization jumps arising on the \kagome
and the star lattice just below the saturation field.
These magnetization jumps may be understood analytically
by using independent local magnon excitations.

Some related $s=1/2$ models are also discussed briefly.
Particular attention is given to the 2D Shastry-Sutherland model.
For this model, we discuss quantum phase transitions and discuss
the magnetization curve in comparison with experiments
on SrCu$_2$(BO$_3$)$_2$.
