Quantum Dimer Model on the Kagome Lattice: Solvable Dimer Liquid and Ising Gauge Theory
G.~Misguich
We introduce quantum dimer models on lattices made of corner-sharing triangles. These lattices includes the kagome lattice and can be defined in arbitrary geometry. They realize fully disordered and gapped dimer-liquid phase with topological degeneracy and deconfined fractional excitations, as well as solid phases. Using geometrical properties of the lattice, several results are obtained exactly, including the full spectrum of a dimer-liquid. These models offer a very natural - and maybe the simplest possible - framework to illustrate general concepts such as fractionalization, topological order and relation to $\mathbb{Z}_2$ gauge theories.
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\newcommand{\Z}{\mathbb{Z}_2}
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\begin{document}

\title{
Quantum Dimer Model on the  Kagome Lattice: Solvable Dimer Liquid and
Ising Gauge Theory}

\author{G.~Misguich}
\email{gmisguich@cea.fr}

\author{D.~Serban}

\author{V.~Pasquier}

\affiliation{
Service de Physique Th\'eorique, 
CEA/Saclay
91191 Gif-sur-Yvette c\'edex, France}

\bibliographystyle{prsty}

\pacs{75.10.Jm %Quantized spin models
      75.50.Ee %Antiferromagnetics
      74.20.Mn %Nonconventional mechanisms (spin fluctuations, polarons and bipolarons, resonating
               %valence bond model, anyon mechanism, marginal Fermi liquid, Luttinger liquid, etc.)  
}
 
\begin{abstract}

We introduce quantum dimer  models on lattices made of  corner-sharing
triangles.   These  lattices includes the  kagome   lattice and can be
defined  in  arbitrary geometry.  They   realize fully  disordered and
gapped dimer-liquid  phase with  topological degeneracy and deconfined
fractional  excitations, as well   as solid phases.  Using geometrical
properties of the  lattice,  several  results are obtained    exactly,
including  the full spectrum of a  dimer-liquid.  These models offer a
very    natural -  and maybe  the   simplest  possible  - framework to
illustrate general concepts   such as  fractionalization,  topological
order and relation to $\mathbb{Z}_2$ gauge theories.

\end{abstract}
\maketitle

%_______________________________________________________________________________


Quantum dimer  models     (QDM)   were introduced    by  Rokhsar   and
Kivelson~\cite{rk88} in the context  of resonating valence-bond  (RVB)
theories for the  high-temperature  superconductors~\cite{anderson87}.
Such   models are expected to  describe  the dynamics of singlet bonds
(dimer) in   quantum   disordered spin-$\frac{1}{2}$ antiferromagnets.
They can describe two  generic  phases: spin-liquids where  the system
breaks no  symmetry at all  and dimer (or valence-bond) crystals where
long-range  dimer-dimer   correlations develops.    Recently a genuine
liquid  phase with a finite  correlation length was  found in a QDM by
Moessner and Sondhi   on  the triangular  lattice~\cite{ms01}.    Such
liquid states  have attracted a lot  of interest because  they display
both fractional excitations and topological order~\cite{wen91}.  While
fractionalization could play  an  important role  in  some theories of
high-temperature      superconductors~\cite{anderson87,sf01},      the
topological properties  of these liquid  states have  been proposed as
possible  devices   to    implement    quantum   bits    for   quantum
computations~\cite{kitaev,ioffe02}.

In this  Letter  we introduce  QDM which  realize such a  dimer-liquid
phase.   Due  to   the  geometric properties    of   lattices made  of
corner-sharing triangles  (the simplest  two-dimensional example being
the kagome  lattice~\footnote{The QDM discussed in   this work {\em do
not} aim at describing   the physics of the spin-$\frac{1}{2}$  kagome
antiferromagnet.  We are currently  investigating a different  QDM for
this later  problem.}), these models are solved  exactly.  As  for the
solvable     point  of  the   QDM   on    the  square~\cite{rk88}  and
triangular~\cite{ms01}   lattices  (see      also    Refs.~\cite{ns01}
and~\cite{bfg01}),     the  ground-state    is     the equal-amplitude
superposition   of  all dimer    coverings   in a   given  topological
sector~\footnote{Dimer coverings are grouped in topological sectors by
considering their transition graphs, that is the set of loops obtained
by superposing two coverings on top of each other.  Two coverings with
only topologically trivial loops are in the same sector.}.  Such state
has been first considered  by Sutherland~\cite{s88} (SRK state in  the
following) and is    the prototype of    resonating valence-bond (RVB)
state.    However, our model  has   several important differences with
previous analogs~: 1) Not only the ground state but all excited states
wave  functions are known.  Elementary  excitations are (pairs of) non
interacting and gapped vortices~\cite{rc89k89} (called {\em visons} in
the recent literature~\cite{sf01}).  2) The model can be solved on any
geometry: torus,   discs or spheres.   This  allows to investigate the
interplay   between  topology, ground-state  degeneracy and elementary
excitations in an very  simple  way.  3) Dimer-dimer correlations  are
strictly  zero  above one lattice spacing.   This  makes this SRK wave
function  the most  possible disordered  dimer liquid  state.  4) This
state is {\em inside} the liquid  phase, it is not  lying at the phase
boundary with  a crystalline phase.  5)  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.    Here we show a
complete  equivalence with   a  $\Z$ gauge   theory.  This allows   to
investigate   the confinement  transition   which goes  with  a  dimer
crystallization  in a  simple QDM  which  exhibits a  liquid to  solid
transition accompanied by a vison condensation.

For all these reasons the model we introduce is  more than a toy model
but the simplest possible  RVB liquid. It  is a ``free dimer  liquid''
point in the short-ranged RVB  phase. It shows the generic  properties
which characterize that state of matter but almost any quantity can be
computed exaclty.  It is  a  natural starting  point  for perturbative
expansions toward more realistic models.


{\em Medial  lattice  construction}.--- The dimer  models described in
this Letter can be defined on lattices made of corner-sharing triangle,
constructed  in  the following way.  Let  $H$  be  a trivalent lattice
(each site has  three neighbors).  The  lattice $K$,  where the dimers
live, is the medial lattice  of $H$, {\it  i.e.}  the sites of $K$ are
the midpoints of the bonds of  $H$ (Fig.~\ref{medial}).  If $H$ is the
honeycomb   lattice,        $K$      is    the      kagome     lattice
(Fig.~\ref{KagHexVison}a).    In   the   following, unless   mentioned
otherwise, we use kagome  for simplicity, where  plaquettes of $H$ are
hexagons.   We  stress however  that  all  results  can be generalized
straightforwardly  to other lattices (squagome~\cite{sg01} lattice for
instance, as well as one-dimensional examples).

\begin{figure}
	\begin{center}
	\resizebox{!}{1.25cm}{\includegraphics{medial.eps}}
	\vspace*{-0.3cm}
	\end{center}
	\caption[99]{Medial  lattice construction.    From a trivalent
	lattice $H$ (full lines)    we construct a lattice $K$   whose
	sites (black dots) are centers of the bonds.  The sites of $K$
	are  linked  together   (dashed   lines) to   form  triangular
	plaquettes.  The pseudospin variables  live on the dual of $H$
	(h and h').}\label{medial}
\end{figure}

{\em Pseudospin representation}.--- Let us begin with the definition of
a simple dimer  model.  For each hexagon  $h$, we  define  an operator
$\sigma^x(h)$  as the   sum  of  all   possible kinetic  energy  terms
involving $h$ only:
\begin{equation}
	\sigma^x(h)=\sum_{\alpha=1}^{32}
	\left|d_\alpha(h) \right>\left< \bar{d}_\alpha(h)\right|
	+
	\left| \bar{d}_\alpha(h)\right>\left<d_\alpha(h)\right|
	\label{eq:sigmax}
\end{equation}
The sum runs over  the  $32$ loops  on  kagome which enclose  a single
hexagon and around which dimers  can be moved (see Ref~\cite{ze95} for
an explicit list).   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 $\left|\bar{d}_\alpha(h)\right>$.


For a given dimer covering $\left|D\right>$, all the kinetic operators
in  the sum but  one  annihilate $\left|D\right>$.  As a  consequence,
$\sigma^x(h)^2=1$.  One can  further check  that these operators  flip
the  pseudospin    variables $\sigma^z(h)$  introduced  by   Elser and
Zeng~\cite{ez93,ze95}  (EZ)  to  label    dimer coverings~\footnote{EZ
pseudospins variables    are different from the  well-known   mapping
[M.~E.~Fisher, J. Math. Phys. {\bf 7},  1776 (1966)] used to solve the
Ising   model on a  planar  lattices  in terms   of  dimers.}.  It  is
important to note that $\sigma^z$ operators depend on  the choice of a
reference $\re$  and     are not  local,   unlike   $\sigma^x$.    The
$\sigma^x(h)$ {\em commute with each other}.  This is not obvious from
Eq.~\ref{eq:sigmax} and it is most easily demonstrated in terms of the
arrow representation that we introduce below.

{\em   Arrow   representation}.---  A   correspondence  between  dimer
coverings on   the  kagome  lattice and {\em     sets  or arrows}   as
illustrated in   Fig.~\ref{KagHexVison}a was  introduced  by Elser and
Zeng~\cite{ez93}.  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.  Consider a triangle without  any
dimer,  this   arrow rule implies that   it  will have  three outgoing
arrows.   Other  triangles   will  have  two incoming  arrows  and one
outgoing arrow.   In other words,  the  number of  outgoing arrows  is
constrained to be odd.

The number of dimer  coverings is $2^{N/3+1}$ where  $N$ is the number
of sites~\cite{hw88elser89,ez93}.  $K$ has  $N$ sites and $N$  arrows,
$2N/3$  triangles and  one   constraint per  triangle.   However  only
$2N/3-1$ constraints are  independent  because their product  for  all
triangles is equal to one.  We are left with  $N/3+1$ Ising degrees of
freedom.   The existence of   this arrow representation  is a  central
reason for  which  QDM considerably simplify  on  these lattices.  For
example   $\sigma^x(h)$   translates    very simply   in    the  arrow
representation: it flips the  six  arrows sitting around hexagon  $h$,
which clearly conserves the  constraint for all triangles and commutes
from hexagon to hexagon.

\begin{figure}
	\begin{center}
	\resizebox{2.5cm}{3.2cm}{\includegraphics{KagHexArrow2.eps}}
	\resizebox{3.9cm}{3cm}{\includegraphics{VisonPairSmall.eps}}
	\end{center}
	
	\caption[99]{{\bf a)} A  dimer covering on  the kagome lattice
	(fat  bonds).    The corresponding   arrow  representation  is
	displayed.  The  Kagome lattice is the  medial  lattice of the
	hexagonal  lattice (dashed bonds).  {\bf b)}  A pair of visons
	is  created by  applying  to the   SRK wave-function a  factor
	$(-1)$ for each  dimer crossing the   cut $\Omega$.  It  is an
	exact             eigenstate        of   Eq.~\ref{eq:Hsigmax}.
	}\label{KagHexVison}
\end{figure}

{\em Rokhsar-Kivelson point}.--- Consider the following Hamiltonian:
\begin{equation}
	\mathcal{H}_0=-\Gamma\sum_{h} \sigma^x(h)
	\label{eq:Hsigmax}
\end{equation}
where  the sum  runs over  hexagons  (sites   of the dual   of $H$  in
general).    Although very simple  in   the pseudospin variables, this
Hamiltonian  is not  obviously     solvable when written    with dimer
operators.  In the pseudospin variables  the  ground-state is a  fully
polarized ferromagnet in the  $x$ direction, which  is the sum of  all
pseudospin configurations in the EZ $\sigma^z$ basis.  Back to dimers,
this  is nothing but  the sum of  all dimer configurations  in a given
topological sector, that    is a SRK  wave-function~\cite{rk88}.   The
ground-state  appears to be unique  in  each topological sector, which
gives a global 4-fold degeneracy on the torus.

{\em Correlations}.---  Correlations in  the SRK  state of  the kagome
lattice are particularly  simple: irreducible dimer-dimer correlations
are strictly  zero  when their  corresponding triangles  do not touch.
The arrows on two bonds are independent provided they are not involved
in a common constraint, that is a common triangle.  As a result, dimer
on the  kagome  lattice are the most    possible disordered: they  are
independent above a finite distance.

{\em  Gap}.--- The whole  spectrum is known.  The $\sigma^x$ operators
commute from hexagon to hexagon but physical dimer states must satisfy
$\prod_h\sigma^x(h)=1$.   This   constraint  comes    from the   arrow
representation  since $\prod_h\sigma^x(h)$ flips   all the arrows {\em
twice} and therefore  keeps  all dimerizations unchanged.   The  first
excited state appears not to be  a single but a  {\em pair} of flipped
hexagons with energy cost $\Delta=4\Gamma$.

{\em Visons}.---  Despite   of the  simplicity  of  the model  in  the
pseudospin variables, its excited  states are not local when expressed
with dimer degrees of freedom.  A $\sigma^x(h)=-1$ hexagon is a vortex
excitation (also  called  {\em vison}~\cite{sf01}).  Consider a string
which   goes  from    an  hexagon  $a$  to    and   hexagon  $b$  (see
Fig.~\ref{KagHexVison}b) 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 $x$ component of  the pseudospin in $a$ and $b$~\footnote{Up
to   a global sign   (reference dependent)  $\Omega(a,b)$  is equal to
$\sigma^z(a)\sigma^z(b)$.}    and   $\Omega(a,b)\left|\psi\right>$  is
precisely  the excited  state  of  energy $4\Gamma$  discussed  above.
Visons appear to be perfectly localized in this model.

{\em Spinons}.--- One can consider  the system  with two {\em  static}
unpaired sites  (spinons or   holes).  As for   others  QDM at  a  SRK
point~\cite{rk88,ms01}, the sum of all dimerizations in a given sector
remains an exact eigenstate.   The energy turns  out to be independent
of the relative distance between spinons, which is a strong indication
that   spinons   would be  also  deconfined     if they  had   kinetic
energy. Eventually, we  note that taking a  spinon around a vison will
change the wave function by a $-1$ factor, as expected~\cite{rc89k89}.


{\em Visons and  topology}.--- On a closed surface  of genus $g$,  the
spectrum has a degeneracy given by the  number of topological sectors
$2^{\rm 2g}$ and  excitations are {\em pairs}   of visons, as  already
mentioned.     When   the    sample  has     edges,   the   constraint
$\prod_h\sigma^x(h)=1$ is not  valid anymore.  To  handle this case we
introduce $\sigma^x(\tilde{h})$ which  flips the arrows along the edge
and   which  restores $\sigma^x(\tilde{h})\prod_h\sigma^x(h)=1$.   The
excitations are still pairs of visons but  one vison can be located in
the hole~\cite{sf01}. In this case the gap reduces to $2\Gamma$. In an
cylinder geometry, as in Ref.~\cite{ioffe02} for instance, we have two
sectors and  a doubly degenerate  spectrum.  It is interesting to note
that this dimer liquid has no low-energy edge states.


{\em Liquid - solid transition}.---
We  consider   a    new    QDM    which   is  a    generalization   of
Eq.~\ref{eq:Hsigmax}:
\begin{equation}	
	\mathcal{H}_1=\mathcal{H}_0%-\Gamma \sum_h \sigma^x(h)
	- J \sum_{\left<h,h'\right>}\sigma^z(h)\sigma^z(h')
	\label{eq:ising}
\end{equation}
where the  second sum runs over  pairs  of neighboring  hexagons.  The
$\sigma^z$   operators  are those defined  by  EZ,  they depend on the
choice       of  a    reference       dimerization  $\re$.    A   term
$\sigma^z(h)\sigma^z(h')$ is  local, it is  $=1$ if the arrow which is
in between $h$  and $h'$ is  in the same  position as in the reference
state  and  $-1$   otherwise.   Since   $\sigma^z(h)\sigma^z(h')$   is
equivalent to   $\Omega(h,h')$,  we see  that such  a  term allows  to
create, annihilate and move visons  in the system.  From another point
of view,  the $J$  term in  Eq.~\ref{eq:ising}  counts the   number of
arrows to  be flipped to recover the  reference state.  If $J$ goes to
$+\infty$, this Hamiltonian obviously  selects the reference  state as
the ground-state and dimers are completely  frozen.  In the pseudospin
language $\mathcal{H}_1$ is an Ising  ferromagnet in transverse field,
which displays a second order phase  transition at a critical value of
$\Gamma$  separating  two   phases:     a  ferromagnetic phase    with
$\left<\sigma^z\right>>0$    and     paramagnetic       phase     with
$\left<\sigma^z\right>=0$.  From what we   know of the $\Gamma=0$  and
$\Gamma\to+\infty$ limits  we can identify the  first one with a dimer
solid and  the second one with  the liquid.  From  the Ising  point of
view the  solid phase  is characterized  by $\left<\sigma^z\right>>0$.
The two   ferromagnetic  Ising states   ($\left<\sigma^z\right>=\pm1$)
correspond  to  the same dimer state  on  a  closed surface but differ
along   the edge   for  open     systems.  The   Ising   magnetization
$\left<\sigma^z(h)\right>$  provides a  non-local order  parameter for
the dimer solid,   $\left<\Omega(h,\infty)\right>$,  which involves  a
string going to infinity.

Up to a sign, $\sigma^z(h)$ is the sum  of a creation and annihilation
operators  of a  vison.    The dimer solidification can   therefore be
interpreted    as the onset   of  off-diagonal  order and  macroscopic
occupation  number (of the zero-momentum  state) for the visons.  This
QDM     realizes    a      condensation     of  topological    defects
(``kinks''~\cite{fs78,kogut} or visons~\cite{sf01}) at the confinement
transition.

Notice that although we call it a  solid, the large-$J$ phase does not
go  with a {\em spatial}  spontaneous symmetry breaking  since the $J$
part of  $\mathcal{H}_1$    depends on an   arbitrary  reference state
through the $\sigma^z$  operators;   this  term acts as  an   external
potential which tends to pin the dimers along the reference state.

{\em $\Z$ gauge  theory}.---  The model  of Eq.~\ref{eq:ising}  is the
dual  of  a   $\Z$  gauge  theory~\cite{wegner}   (in  its Hamiltonian
formulation) where  the gauge degrees of   freedom $\tau^z(i)$ live on
the bonds of $H$ ({\it i.e.} sites of $K$).  By definition $\tau^z(i)$
is the operator which flips the arrow at  site $i$ and gauge-invariant
observable  are     made of  products   of    $\tau^z$  around  closed
loops. Spatial gauge  transformations require the $x$ component, which
we define with respect to the  reference state $\re$: $\tau^x(i)=1$ if
the arrow $i$ has the same orientation as  in $\re$ and $\tau^x(i)=-1$
otherwise.  For every  site of    $H$  (every triangle  of $K$)    the
constraint   reads $\tau^x(i_1)\tau^x(i_2)\tau^x(i_3)=1$ where  $i_1$,
$i_2$ and $i_3$ are the  bonds of $H$ emanating  from that site.  This
expresses the fact that physical states must be gauge invariant.  This
shows a one-to-one correspondence between  physical state of the gauge
theory  and dimer coverings  of $K$  and the  redundancy in the  gauge
theory  is solved   by  the  dimer  coverings.   We  wish   to express
$\mathcal{H}_0$ with the gauge degrees of freedom. Since $\sigma^x(h)$
operator flips  all the  arrows around  $h$, it  becomes the plaquette
operator:
\begin{equation}
	\sigma^x(h)=\prod_{i} \tau^z(i)
	\label{eq:tauz}
\end{equation}
where the product runs over the bonds of  $H$ surrounding $h$.
The Ising interactions can be written
\begin{equation}
	\sigma^z(h)\sigma^z(h')=\tau^x(i)
	\label{eq:taux}
\end{equation}
where $i$   is   the  common    link between     $h$ and   $h'$   (see
Fig.~\ref{medial}).  As  a   result   the QDM  of   Eq.~\ref{eq:ising}
translates into the Hamiltonian of a $\Z$  gauge theory (in continuous
time), which is a manifestation of the well-known duality between $\Z$
gauge theories and Ising models~\cite{wegner,kogut}.
Such  a $\Z$ gauge  theory  has a  confined  and deconfined phases.  A
classical  result  is that   they  can be  distinguished through   the
expectation value of a gauge-invariant Wilson loop~\cite{wegner,kogut}
\begin{equation}
	W(\omega)=< \prod_{i\in \partial\omega} \tau^z(i) >
	=
	< \prod_{i\in \omega} \sigma^x(i) >
	\label{eq:W}
\end{equation}
where the first product runs over a  close loop $\partial\omega$ which
surrounds  the area  $\omega$.    $W(\omega)$  changes  from a    {\em
perimeter law}  $\sim\exp(-|\partial\omega|)$   to an {\em   area law}
$\sim\exp(-|\omega|)$ when going from the deconfined at large $\Gamma$
to   the  confined  phase  at large  $J$.      The right-hand  side of
Eq.~\ref{eq:W} expresses $W(\omega)$ as a correlation function for the
dimer   problem.   $W(\omega)$   moves    dimers    along  the    loop
$\partial\omega$ and the mapping to the gauge theory tells us that its
long-distance behavior characterizes the liquid and frozen phases.

{\em  Liquid   -   crystal transition}.---  $\mathcal{H}_1$  has  the
disadvantage of  depending on a  specific reference state.   To remedy
for that we introduce  a QDM which  is reference free and restores the
full lattice symmetry:
\begin{equation}	
	\mathcal{H}_2=\mathcal{H}_0
	- J \sum_{h}s^z(h)
	\label{eq:h2}
\end{equation}
where $s^z(h)=\pm1$ is diagonal in the dimer basis and counts a factor
$-1$  per  anticlockwise arrow   around   the  hexagon $h$.    In  the
$J\to\infty$ limit the system selects $s^z(h)=1$ everywhere.  This can
be   achieved  if the   system spontaneously   breaks  the translation
symmetry and crystallizes in an ordered pattern of six-dimer ``stars''
described in Ref.~\cite{sm02}.   These  ordered states are  degenerate
with  others ($\sim2^{3L}$ where $L$ is  the  linear size) obtained by
shifting the dimers along any straight line.

We have looked numerically (diagonalizations up to 54 kagome sites) at
$\mathcal{H}_2$ and  found  evidence   for a  single   and  continuous
transition from   the      liquid   to    the  star  crystal        at
$J\simeq\Gamma$. This appears as a collapse of the first excitation of
the  liquid, which then transforms  into  a degenerate ground-state of
the crystal phase.   We claim that the critical   point is exactly  at
$J=\Gamma$ from a duality  argument.  From the arrow representation it
is  clear that $\sigma^x(h)$ and $s^z(h')$  commute except  if $h$ and
$h'$  are neighbors,  in  which  case  they  anticommute.   Indeed the
algebraic relations of  $\sigma^x$ and $s^z$ are completely symmetric.
In particular, we observed numerically that the ground-state energy is
{\em exactly}  symmetric with respect  to the exchange of $\Gamma$ and
$J$ (this is not true for all the  eigenstates) and the critical point
must lie at the self-dual point $J=\Gamma$.

{\em Conclusions}.--- The  kagome lattice has the  remarkable property
that dimer  coverings correspond to  the physical  states of  an Ising
model  on  the triangular  lattice  and, by duality,  of  a $\Z$ gauge
theory on the hexagonal one.  Exploiting  this, we introduced QDMs for
which exact results  are derived. In particular,  we obtained for  the
first time the full spectrum of a  QDM realizing a dimer liquid phase.
It  explicitly  realizes fractionalized  excitations  and  topological
order.  Through   several models we   showed that  QDM  on the  kagome
lattice  are    very simple and  natural     tools to investigate  the
connexions between  frustrated  magnets, RVB  physics  and spin-charge
separation.

{\em Acknowledgments}.--- We   are grateful to M.~Gaudin,  K.~Mallick,
R.~Moessner,  C.~Lhuillier  and  M.~Feigel'man  for  several  fruitful
discussions. Numerical diagonalizations of QDM models were done on the
Compacq alpha server of the CEA under project 550.

%___________________________________________________________________

\begin{thebibliography}{100}

\bibitem{rk88}
D. S. Rokhsar and S. A. Kivelson.
%Superconductivity and the quantum hard-core dimer gas.
Phys. Rev. Lett. {\bf 61},  2376  (1988).
%We discuss the short-range  resonating-valence-bond system as realized
%by   a  quantum  hard-core   dimer  gas  of arbitrary    density  on a
%two-dimensional  square lattice. When  the dimers completely cover the
%lattice, we argue that there is a first-order  transition from a dimer
%crystal     to an insulating  quantum    liquid  state which possesses
%low-energy, neutral,  spinless excitations which we call ``resonons.'' 
%For    less  than  close-packed   densities, the   ground   state is a
%superfluid. In   addition  to  the usual   Goldstone  mode, there  are
%low-energy, spinless zone-boundary excitations.

\bibitem{anderson87}
P.~W.~Anderson,
Science {\bf 235}, 1196 (1987).
%The resonating  valence   bond   state in  La/sub  2/CuO/sub    4/ and
%superconductivity

%The oxide superconductors, particularly those recently discovered that
%are  based on La/sub 2/CuO/sub  4/, have a   set of peculiarities that
%suggest a common,  unique mechanism: they tend in  every case to occur
%near a metal-insulator  transition into an odd-electron insulator with
%peculiar magnetic properties. This insulating phase  is proposed to be
%the   long-sought  'resonating-valence-bond'  state  or  'quantum spin
%liquid' hypothesized in    1973.  This insulating   magnetic phase  is
%favored by low spin, low dimensionality, and magnetic frustration. The
%preexisting  magnetic singlet pairs of   the  insulating state  become
%charged superconducting pairs when the insulator is doped sufficiently
%strongly.  The mechanism for  superconductivity is hence predominantly
%electronic  and magnetic, although  weak phonon interactions may favor
%the state. Many unusual  properties  are predicted, especially of  the
%insulating state

\bibitem{ms01}
R. Moessner and S.~L. Sondhi, Phys. Rev. Lett. {\bf 86},  1881  (2001).

\bibitem{wen91}
X. G. Wen, Phys. Rev. B {\bf 44}, 2664 (1991).

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

%The  mean-field  theory of a T-   and P-symmetric spin-liquid state is
%developed. The quasiparticle excitations  in the spin-liquid state are
%shown   to be spin-1/2  neutral  fermions  (the spinons)  and charge e
%spinless bosons  (the holons). The  spin-liquid state  is shown  to be
%characterized    by a   nontrivial   topological  order.  Although our
%discussions are based  on the mean-field  theory,  the concept of  the
%topological order and the associated  universal properties (e.g.,  the
%quantum number of the quasiparticles) are  expected to be valid beyond
%the mean-field theory. We also discuss  the dynamical stability of the
%mean-field theory.



\bibitem{sf01}
T.~Senthil and M. P. A. Fisher.
Phys. Rev. Lett. {\bf 86}, 292 (2001).
%Fractionalization in the Cuprates: Detecting the Topological Order.
%The precise theoretical characterization  of a fractionalized phase in
%spatial dimensions   higher  than  one   is through the    concept  of
%"topological order."  We describe a physical effect  that is  a robust
%and a direct consequence of  this  hidden order  that should enable  a
%precise experimental  characterization  of  fractionalized phases.  In
%particular,  we   propose    specific  "smoking-gun" experiments    to
%unambiguously  settle the issue   of electron fractionalization in the
%underdoped cuprates.
%URL: http://link.aps.org/abstract/PRL/v86/p292
%Senthil and Matthew P. A. Fisher.
%Fractionalization, topological order, and cuprate superconductivity.
Phys. Rev. B {\bf 63}, 134521 (2001).

%This  paper   is concerned  with   the   idea   that the  electron  is
%fractionalized  in  the cuprate  high-Tc  materials.  We show  how the
%notion  of  topological   order  may  be  used  to  develop  a precise
%theoretical  characterization of  a  fractionalized  phase in  spatial
%dimension  higher  than 1. Apart   from the  fractional particles into
%which  the electron   breaks   apart,  there  are nontrivial    gapped
%topological excitations-dubbed "visons."  A cylindrical sample that is
%fractionalized exhibits two disconnected topological sectors depending
%on whether a  vison is trapped in  the  "hole" or not. Indeed,  "vison
%expulsion"   is to fractionalization what   the Meissner effect ("flux
%expulsion") is to superconductivity.  This understanding enables us to
%address a  number of conceptual issues  that need to  be confronted by
%any theory of the cuprates based  on fractionalization ideas. We argue
%that whether or  not the electron fractionalizes  in the cuprates is a
%sharp and well-posed question with a definite  answer. We elaborate on
%our recent proposal  for an  experiment  to unambiguously settle  this
%issue.

\bibitem{kitaev}
A. Kitaev, quant-ph/9707021 (unpublished).
%Fault-tolerant quantum computation by anyons

%A  two-dimensional quantum  system  with  anyonic excitations  can  be
%considered as a  quantum   computer. Unitary transformations  can   be
%performed by moving the  excitations around each  other.  Measurements
%can be  performed by  joining  excitations in pairyaks and  observing the
%result of fusion.  Such computation is  fault-tolerant by its physical
%nature.

\bibitem{ioffe02}
L. B. Ioffe, M.~V.~Feigel'man, A.~Ioselevich, D.~Ivanov, M.~Troyer and
G.~Blatter.
%Topologically protected quantum bits from Josephson junction arrays.
Nature {\bf 415}, 503 (2002).


%All physical implementations  of quantum  bits (qubits), carrying  the
%information and  computation in a  putative quantum  computer, have to
%meet the  conflicting  requirements of  environmental decoupling while
%remaining  manipulable  through designed  external signals.  Proposals
%based   on quantum optics naturally   emphasize  the aspect of optimal
%isolation,  while those  following the  solid  state route exploit the
%variability  and    scalability    of  modern   nanoscale  fabrication
%techniques. Recently, various designs using superconducting structures
%have been successfully tested for quantum coherent operation, however,
%the ultimate  goal of  reaching coherent  evolution over thousands  of
%elementary operations  remains a   formidable task.  Protecting qubits
%from decoherence by exploiting  topological stability, a qualitatively
%new proposal due  to Kitaev,  holds the  promise for  long decoherence
%times, but its practical physical  implementation has remained unclear
%so far. Here, we  show how strongly  correlated systems  developing an
%isolated two-fold degenerate  quantum dimer liquid groundstate  can be
%used  in the construction of topologically   stable qubits and discuss
%their implementation using Josephson junction arrays.


\bibitem{ns01}
C.~Nayak and K.~Shtengel,
Phys. Rev. B {\bf 64}, 064422 (2001).

%Microscopic   models  of two-dimensional  magnets  with fractionalized
%excitations

%We demonstrate that spin-charge separation can occur in two dimensions
%and    note its  confluence   with  superconductivity, topology, gauge
%theory,   and  fault-tolerant  quantum computation.   We  construct  a
%microscopic  Ising-like model and,   at  a special coupling   constant
%value, find its exact ground state as  well as neutral spin- (spinon),
%spinless   charge e  (holon),   and   Z2  vortex  (vison)  states  and
%energies. The fractionalized excitations reflect the topological order
%of the ground state which is evinced by its fourfold degeneracy on the
%torus-a degeneracy which  is unrelated to translational  or rotational
%symmetry-and is described   by a Z2   gauge theory. A  magnetic moment
%coexists with the topological order. Our model is a member of a family
%of topologically ordered   models, one  of   which is integrable   and
%realizes the toric quantum error correction code but does not conserve
%any component of the spin. We relate our  model to a dimer model which
%could  be a spin SU(2) symmetric  realization of topological order and
%its concomitant quantum number fractionalization.

\bibitem{bfg01}
L.~Balents, M.~P.~A.~Fisher,  S.~M.~Girvin,  Phys.  Rev.  B {\bf  65},
224412 (2002). Although dealing with a spin  model on the {\em kagome}
lattice, the  solvable model  discussed  by  these authors is  closely
related  to   that  of  Ref.~\cite{ms01}  with   dimers  on   the {\em
triangular} lattice.

%Fractionalization in an Easy-axis Kagome Antiferromagnet

%We study an antiferromagnetic     spin-1/2  model with up  to    third
%nearest-neighbor couplings on the Kagome    lattice in the   easy-axis
%limit,  and show that its low-energy  dynamics are governed  by a four
%site XY ring exchange Hamiltonian. Simple ``vortex pairing'' arguments
%suggest that the model sustains a novel fractionalized phase, which we
%confirm by exactly solving a modification of the Hamiltonian including
%a further four-site   interaction. In this   limit,  the system is   a
%featureless   ``spin  liquid'', with gaps    to   all excitations,  in
%particular: deconfined S^z=1/2  bosonic ``spinons'' and Ising vortices
%or ``visons''. We use an Ising duality transformation to express vison
%correlators as non-local strings  in terms of  the spin operators, and
%calculate the string  correlators using the  ground state wavefunction
%of the modified Hamiltonian.  Remarkably, this wavefunction is exactly
%given  by      a   kind  of   Gutzwiller      projection    of  an  XY
%ferromagnet. Finally, we show  that  the deconfined spin liquid  state
%persists over a finite  range as the additional four-spin  interaction
%is reduced, and study the effect of this reduction  on the dynamics of
%spinons and visons.

\bibitem{s88}
B. Sutherland, Phys. Rev. B {\bf 37}, 3786 (1988).
%Systems with resonating-valence-bond  ground  states: Correlations and
%excitations

%We consider a resonating-valence-bond  state as a variational estimate
%for the  ground state of   a Heisenberg antiferromagnet. We  derive an
%expression  for the normalization of  this  wave function, which is  a
%superposition of a very large number  of singlet pair states, which we
%then use to show  that the singularities, correlations, and  low-lying
%excitations of    this  quantum system  can  be   calculated  from the
%partition function  of   a  related  dimerlike   classical statistical
%mechanics problem. Possible calculational schemes are discussed.

\bibitem{rc89k89}
N. Read and B.~Chakraborty, Phys. Rev. B 40, 7133 (1989).
S. Kivelson,     Phys. Rev. B 39, 259 (1989).

\bibitem{msf02}
R. Moessner, S.  L.  Sondhi, and  E.~Fradkin.
%Short-ranged  resonating valence bond  physics, quantum  dimer models,
%and Ising gauge theories
Phys. Rev. B {\bf 65}, 024504 (2002).

%Quantum dimer models are believed  to capture the essential physics of
%antiferromagnetic   phases dominated   by  short-ranged   valence bond
%configurations.  We show that these models  arise as particular limits
%of Ising (Z2)   gauge theories, but that in   these limits the  system
%develops   a  larger  local    U(1)  invariance   that  has  different
%consequences  on  different  lattices. Conversely,  we   note that the
%standard Z2  gauge  theory is  a  generalized quantum dimer  model, in
%which the  particular  relaxation of the   hardcore constraint for the
%dimers breaks the U(1)  down to Z2.   These mappings indicate that  at
%least   one      realization  of the     Senthil-Fisher   proposal for
%fractionalization is exactly the  short ranged resonating valence bond
%(RVB) scenario of  Anderson and of  Kivelson, Rokhsar and Sethna. They
%also suggest  that other realizations  will require the identification
%of   a   local low    energy, Ising  link    variable and    a natural
%constraint. We also  discuss  the notion of   topological order in  Z2
%gauge theories and its connection  to earlier ideas  in RVB theory. We
%note that  this notion is not   central to the experiment  proposed by
%Senthil  and Fisher to detect   vortices in the  conjectured Z2  gauge
%field.

%URL: http://link.aps.org/abstract/PRB/v65/e024504 

\bibitem{hw88elser89}
A.~J.~Phares and F.~J.~Wunderlich, Nuovo Cimento B {\bf 101}, 653 (1988).
V.~Elser.
%Nuclear antiferromagnetism in a registered $^3$He solid.
Phys. Rev. Lett. {\bf 62}, 2405 (1989).

\bibitem{ez93}
V. Elser and C. Zeng.
%kagome spin-1/2 antiferromagnets in the hyperbolic plane.
Phys. Rev. B {\bf 48}, 13647 (1993). 

%Spin-dimerized      states  are   useful   in    the   construction of
%spin-disordered wave functions but  difficult to deal with  because of
%nonorthogonality.   For the  spin-1/2   kagome antiferromagnet, a
%systematic expansion of  matrix elements of these nonorthogonal states
%is  made possible  by considering  generalizations  of the kagome
%structure in  the hyperbolic plane. The  first nontrivial term in this
%expansion  is an effective  spin Hamiltonian which describes resonance
%among  dimerized  states.  Minimum-energy   states  of  the  effective
%Hamiltonian correspond  to  a high degree of  resonance  among a small
%fraction of the dimers.

\bibitem{ze95}
C.~Zeng    and   V.~Elser.
%Quantum     dimer  calculations   on  the
%spin-$\frac{1}{2}$ kagome Heisenberg  antiferromagnet.
Phys.  Rev. B {\bf 51}, 8318 (1995).
%In analogy to the quantum dimer model on the square lattice introduced
%by Rokhsar  and Kivelson, we identify the  low-energy Hilbert space of
%the  spin-1/2  kagome Heisenberg   antiferromagnet  with  the set   of
%dimerized states (nearest-neighbor  singlets) and derive  an effective
%Hamiltonian which describes  resonance and  includes fluctuations into
%next-nearest-neighbor   singlets. Complete   diagonalization   of this
%effective Hamiltonian for systems as  large as 36 spins makes possible
%the   study of  the     low-energy thermodynamics   of this    quantum
%antiferromagnet  and strongly supports  the double peak feature in the
%heat capacity.



\bibitem{sg01}
R. Siddharthan and A. Georges.
%Square kagome quantum antiferromagnet and the eight-vertex model.
Phys. Rev. B {\bf 65}, 014417 (2002).

%We  introduce  a two-dimensional  network  of corner-sharing triangles
%with square-lattice  symmetry.  Properties  of magnetic  systems  here
%should be similar  to those  on  a  kagome  lattice. Focusing  on  the
%spin-1/2  Heisenberg quantum antiferromagnet,  we  generalize the spin
%symmetry group from SU(2)  to SU(N). In  the large-N limit, we map the
%model exactly to the  eight-vertex model solved  by Baxter. We predict
%an exponential number of low-lying  singlet states, a triplet gap, and
%a  two-peak specific heat. In  addition, the  large-N limit suggests a
%finite   temperature phase   transition   into a  phase  with  ordered
%"resonance loops" and broken translational symmetry.

% URL: http://link.aps.org/abstract/PRB/v65/e014417 

\bibitem{waldtmann}

Ch.  Waldtmann, H.~Kreutzmann, U.~Schollw\"ock, K.~Maisinger, and
H.-U.~Everts, Phys. Rev. B {\bf 62} 9472 (2000).

%Ground states   and   excitations  of a    one-dimensional kagom�-like
%antiferromagnet

%We study a Heisenberg antiferromagnet  (AF) on a one-dimensional strip
%composed   of two rows of  corner   sharing triangular plaquettes. The
%geometry  of  this  ladder-like  lattice   naturally allows  for   two
%different exchange couplings, one between  pairs of spins on the outer
%legs (J) and a different  one between the spins on  the axis and their
%nearest    neighbors  on the  legs  (J).  For   J/J0.5 the  model is a
%ferrimagnet. Our main   interest is in  the region  J/J0.5, where  the
%classical ground   state of  this system   shows the same  macroscopic
%degeneracy as the classical ground state of the antiferromagnet on the
%kagom� lattice. To explore to which extent this similarity between the
%classical ground  states of these   two models carries  over to  their
%quantum states we have   applied exact diagonalization techniques  and
%density-matrix    renormalization group    (DMRG)  methods   to    our
%model. Exact diagonalization is restricted to system  sizes of up to N
%=  30  sites. For J/J   = 1, the results   obtained by this technique,
%low-energy spectra, correlation functions and the specific heat, agree
%qualitatively with the  results  obtained  for finite samples  of  the
%kagom� AF. As in the case of the  kagom� AF, these finite size results
%suggest that our model is a spin liquid  with a gap between the ground
%state  and  the lowest spin  excitation.  However, extrapolations from
%DMRG data for strips of up to N =  120 sites point towards a vanishing
%spin gap in a wide range of couplings, 0.5J/J1.25, so that contrary to
%the  above conjecture,    our one-dimensional model  may  in   fact be
%critical in this  parameter  range. Moreover, our DMRG   data indicate
%that the model  undergoes a transition to a  gapped state as the ratio
%J/J  increases  through the value   J/J1.25. In  the vicinity  of  the
%transition point, a high density of low-lying singlets develops in the
%spin gap.

\bibitem{fs78}
E.~Fradkin and L. Susskind, Phys. Rev. D {\bf 17}, 2637 (1978).
%Order and disorder in gauge systems and magnets
%We show  how  phase transitions  in  Abelian two-dimensional  spin and
%four-dimensional     gauge systems can    be  understood  in  terms of
%condensation of topological objects. In the spin systems these objects
%are  kinks   and in  the  gauge systems   either magnetic monopoles or
%fluxoids (quantized lines of  magnetic flux). Four models are studied:
%two-dimensional Ising and  XY models and  four-dimensional Z2 and U(1)
%gauge

\bibitem{kogut}
%An introduction to lattice gauge theory and spin systems
J. B. Kogut, Rev. Mod. Phys. {\bf 51}, 659 (1979).
%This  article is an  interdisciplinary review  of lattice gauge theory
%and  spin systems. It discusses  the  fundamentals,  both physics  and
%formalism,  of these   related subjects. Spin  systems   are models of
%magnetism and  phase  transitions. Lattice gauge theories   are cutoff
%formulations  of     gauge  theories     of     strongly   interacting
%particles. Statistical mechanics  and field theory are closely related
%subjects, and the connections between them are developed here by using
%the transfer matrix. Phase diagrams  and critical points of continuous
%transitions are stressed as   the keys to understanding the  character
%and continuum limits of  lattice theories.  Concepts such as  duality,
%kink condensation,  and the existence of  a  local, relativistic field
%theory at a critical  point of a  lattice theory are illustrated  in a
%thorough discussion of the two-dimensional Ising model.  Theories with
%exact local (gauge) symmetries are introduced following Wegner's Ising
%lattice gauge theory.  Its gauge-invariant "loop" correlation function
%is   discussed  in detail.  Three-dimensional   Ising  gauge theory is
%studied thoroughly. The  renormalization group of  the two dimensional
%planar model is presented  as  an illustration  of a phase  transition
%driven by  the condensation of  topological excitations. Parallels are
%drawn to Abelian lattice gauge theory  in four dimensions. Non-Abelian
%gauge theories are introduced and the possibility of quark confinement
%is discussed. Asymptotic  freedom  of O(n) Heisenberg spin  systems in
%two  dimensions  is  verified for n  >=   and is explained   in simple
%terms. The direction of present-day research is briefly reviewed.

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

%Duality in generalized Ising    models and phase  transitions  without
%local order parameters

%It is shown that  any Ising model with  positive coupling constants is
%related to another Ising model by a duality transformation. A class of
%Ising models M/sub dn/ on  d-dimensional  lattices characterized by  a
%number n=1,2,...,d(n=1 corresponds to   the Ising model  with two-spin
%interaction)  is defined.  These  models are   related by  two duality
%transformations.  The models with   1(n(d  exhibit a phase  transition
%without local order parameter.  A nonanalyticity in the  specific heat
%and  a  different  qualitative behavior of   certain  spin correlation
%functions  in the  low and the  high temperature  phases indicate  the
%existence of  a phase transition. The  Hamiltonian of the simple cubic
%dual model contains products of  four Ising spin operators. Applying a
%star square transformation, one obtains  an Ising model with competing
%interactions exhibiting a   singularity  in the specific  heat  but no
%long-range order of the spins in the low temperature phase


\bibitem{sm02}
%Hidden long range order in Heisenberg Kagome antiferromagnets
A.~V.~Syromyatnikov and S.~V.~Maleyev, cond-mat/0202117 (unpublished).


\end{thebibliography}
\end{document}

cond-mat0204428.extracted_bib
\begin{thebibliography}{100}

\bibitem{rk88}
D. S. Rokhsar and S. A. Kivelson.
%Superconductivity and the quantum hard-core dimer gas.
Phys. Rev. Lett. {\bf 61},  2376  (1988).
%We discuss the short-range  resonating-valence-bond system as realized
%by   a  quantum  hard-core   dimer  gas  of arbitrary    density  on a
%two-dimensional  square lattice. When  the dimers completely cover the
%lattice, we argue that there is a first-order  transition from a dimer
%crystal     to an insulating  quantum    liquid  state which possesses
%low-energy, neutral,  spinless excitations which we call ``resonons.'' 
%For    less  than  close-packed   densities, the   ground   state is a
%superfluid. In   addition  to  the usual   Goldstone  mode, there  are
%low-energy, spinless zone-boundary excitations.

\bibitem{anderson87}
P.~W.~Anderson,
Science {\bf 235}, 1196 (1987).
%The resonating  valence   bond   state in  La/sub  2/CuO/sub    4/ and
%superconductivity

%The oxide superconductors, particularly those recently discovered that
%are  based on La/sub 2/CuO/sub  4/, have a   set of peculiarities that
%suggest a common,  unique mechanism: they tend in  every case to occur
%near a metal-insulator  transition into an odd-electron insulator with
%peculiar magnetic properties. This insulating phase  is proposed to be
%the   long-sought  'resonating-valence-bond'  state  or  'quantum spin
%liquid' hypothesized in    1973.  This insulating   magnetic phase  is
%favored by low spin, low dimensionality, and magnetic frustration. The
%preexisting  magnetic singlet pairs of   the  insulating state  become
%charged superconducting pairs when the insulator is doped sufficiently
%strongly.  The mechanism for  superconductivity is hence predominantly
%electronic  and magnetic, although  weak phonon interactions may favor
%the state. Many unusual  properties  are predicted, especially of  the
%insulating state

\bibitem{ms01}
R. Moessner and S.~L. Sondhi, Phys. Rev. Lett. {\bf 86},  1881  (2001).

\bibitem{wen91}
X. G. Wen, Phys. Rev. B {\bf 44}, 2664 (1991).

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

%The  mean-field  theory of a T-   and P-symmetric spin-liquid state is
%developed. The quasiparticle excitations  in the spin-liquid state are
%shown   to be spin-1/2  neutral  fermions  (the spinons)  and charge e
%spinless bosons  (the holons). The  spin-liquid state  is shown  to be
%characterized    by a   nontrivial   topological  order.  Although our
%discussions are based  on the mean-field  theory,  the concept of  the
%topological order and the associated  universal properties (e.g.,  the
%quantum number of the quasiparticles) are  expected to be valid beyond
%the mean-field theory. We also discuss  the dynamical stability of the
%mean-field theory.



\bibitem{sf01}
T.~Senthil and M. P. A. Fisher.
Phys. Rev. Lett. {\bf 86}, 292 (2001).
%Fractionalization in the Cuprates: Detecting the Topological Order.
%The precise theoretical characterization  of a fractionalized phase in
%spatial dimensions   higher  than  one   is through the    concept  of
%"topological order."  We describe a physical effect  that is  a robust
%and a direct consequence of  this  hidden order  that should enable  a
%precise experimental  characterization  of  fractionalized phases.  In
%particular,  we   propose    specific  "smoking-gun" experiments    to
%unambiguously  settle the issue   of electron fractionalization in the
%underdoped cuprates.
%URL: http://link.aps.org/abstract/PRL/v86/p292
%Senthil and Matthew P. A. Fisher.
%Fractionalization, topological order, and cuprate superconductivity.
Phys. Rev. B {\bf 63}, 134521 (2001).

%This  paper   is concerned  with   the   idea   that the  electron  is
%fractionalized  in  the cuprate  high-Tc  materials.  We show  how the
%notion  of  topological   order  may  be  used  to  develop  a precise
%theoretical  characterization of  a  fractionalized  phase in  spatial
%dimension  higher  than 1. Apart   from the  fractional particles into
%which  the electron   breaks   apart,  there  are nontrivial    gapped
%topological excitations-dubbed "visons."  A cylindrical sample that is
%fractionalized exhibits two disconnected topological sectors depending
%on whether a  vison is trapped in  the  "hole" or not. Indeed,  "vison
%expulsion"   is to fractionalization what   the Meissner effect ("flux
%expulsion") is to superconductivity.  This understanding enables us to
%address a  number of conceptual issues  that need to  be confronted by
%any theory of the cuprates based  on fractionalization ideas. We argue
%that whether or  not the electron fractionalizes  in the cuprates is a
%sharp and well-posed question with a definite  answer. We elaborate on
%our recent proposal  for an  experiment  to unambiguously settle  this
%issue.

\bibitem{kitaev}
A. Kitaev, quant-ph/9707021 (unpublished).
%Fault-tolerant quantum computation by anyons

%A  two-dimensional quantum  system  with  anyonic excitations  can  be
%considered as a  quantum   computer. Unitary transformations  can   be
%performed by moving the  excitations around each  other.  Measurements
%can be  performed by  joining  excitations in pairyaks and  observing the
%result of fusion.  Such computation is  fault-tolerant by its physical
%nature.

\bibitem{ioffe02}
L. B. Ioffe, M.~V.~Feigel'man, A.~Ioselevich, D.~Ivanov, M.~Troyer and
G.~Blatter.
%Topologically protected quantum bits from Josephson junction arrays.
Nature {\bf 415}, 503 (2002).


%All physical implementations  of quantum  bits (qubits), carrying  the
%information and  computation in a  putative quantum  computer, have to
%meet the  conflicting  requirements of  environmental decoupling while
%remaining  manipulable  through designed  external signals.  Proposals
%based   on quantum optics naturally   emphasize  the aspect of optimal
%isolation,  while those  following the  solid  state route exploit the
%variability  and    scalability    of  modern   nanoscale  fabrication
%techniques. Recently, various designs using superconducting structures
%have been successfully tested for quantum coherent operation, however,
%the ultimate  goal of  reaching coherent  evolution over thousands  of
%elementary operations  remains a   formidable task.  Protecting qubits
%from decoherence by exploiting  topological stability, a qualitatively
%new proposal due  to Kitaev,  holds the  promise for  long decoherence
%times, but its practical physical  implementation has remained unclear
%so far. Here, we  show how strongly  correlated systems  developing an
%isolated two-fold degenerate  quantum dimer liquid groundstate  can be
%used  in the construction of topologically   stable qubits and discuss
%their implementation using Josephson junction arrays.


\bibitem{ns01}
C.~Nayak and K.~Shtengel,
Phys. Rev. B {\bf 64}, 064422 (2001).

%Microscopic   models  of two-dimensional  magnets  with fractionalized
%excitations

%We demonstrate that spin-charge separation can occur in two dimensions
%and    note its  confluence   with  superconductivity, topology, gauge
%theory,   and  fault-tolerant  quantum computation.   We  construct  a
%microscopic  Ising-like model and,   at  a special coupling   constant
%value, find its exact ground state as  well as neutral spin- (spinon),
%spinless   charge e  (holon),   and   Z2  vortex  (vison)  states  and
%energies. The fractionalized excitations reflect the topological order
%of the ground state which is evinced by its fourfold degeneracy on the
%torus-a degeneracy which  is unrelated to translational  or rotational
%symmetry-and is described   by a Z2   gauge theory. A  magnetic moment
%coexists with the topological order. Our model is a member of a family
%of topologically ordered   models, one  of   which is integrable   and
%realizes the toric quantum error correction code but does not conserve
%any component of the spin. We relate our  model to a dimer model which
%could  be a spin SU(2) symmetric  realization of topological order and
%its concomitant quantum number fractionalization.

\bibitem{bfg01}
L.~Balents, M.~P.~A.~Fisher,  S.~M.~Girvin,  Phys.  Rev.  B {\bf  65},
224412 (2002). Although dealing with a spin  model on the {\em kagome}
lattice, the  solvable model  discussed  by  these authors is  closely
related  to   that  of  Ref.~\cite{ms01}  with   dimers  on   the {\em
triangular} lattice.

%Fractionalization in an Easy-axis Kagome Antiferromagnet

%We study an antiferromagnetic     spin-1/2  model with up  to    third
%nearest-neighbor couplings on the Kagome    lattice in the   easy-axis
%limit,  and show that its low-energy  dynamics are governed  by a four
%site XY ring exchange Hamiltonian. Simple ``vortex pairing'' arguments
%suggest that the model sustains a novel fractionalized phase, which we
%confirm by exactly solving a modification of the Hamiltonian including
%a further four-site   interaction. In this   limit,  the system is   a
%featureless   ``spin  liquid'', with gaps    to   all excitations,  in
%particular: deconfined S^z=1/2  bosonic ``spinons'' and Ising vortices
%or ``visons''. We use an Ising duality transformation to express vison
%correlators as non-local strings  in terms of  the spin operators, and
%calculate the string  correlators using the  ground state wavefunction
%of the modified Hamiltonian.  Remarkably, this wavefunction is exactly
%given  by      a   kind  of   Gutzwiller      projection    of  an  XY
%ferromagnet. Finally, we show  that  the deconfined spin liquid  state
%persists over a finite  range as the additional four-spin  interaction
%is reduced, and study the effect of this reduction  on the dynamics of
%spinons and visons.

\bibitem{s88}
B. Sutherland, Phys. Rev. B {\bf 37}, 3786 (1988).
%Systems with resonating-valence-bond  ground  states: Correlations and
%excitations

%We consider a resonating-valence-bond  state as a variational estimate
%for the  ground state of   a Heisenberg antiferromagnet. We  derive an
%expression  for the normalization of  this  wave function, which is  a
%superposition of a very large number  of singlet pair states, which we
%then use to show  that the singularities, correlations, and  low-lying
%excitations of    this  quantum system  can  be   calculated  from the
%partition function  of   a  related  dimerlike   classical statistical
%mechanics problem. Possible calculational schemes are discussed.

\bibitem{rc89k89}
N. Read and B.~Chakraborty, Phys. Rev. B 40, 7133 (1989).
S. Kivelson,     Phys. Rev. B 39, 259 (1989).

\bibitem{msf02}
R. Moessner, S.  L.  Sondhi, and  E.~Fradkin.
%Short-ranged  resonating valence bond  physics, quantum  dimer models,
%and Ising gauge theories
Phys. Rev. B {\bf 65}, 024504 (2002).

%Quantum dimer models are believed  to capture the essential physics of
%antiferromagnetic   phases dominated   by  short-ranged   valence bond
%configurations.  We show that these models  arise as particular limits
%of Ising (Z2)   gauge theories, but that in   these limits the  system
%develops   a  larger  local    U(1)  invariance   that  has  different
%consequences  on  different  lattices. Conversely,  we   note that the
%standard Z2  gauge  theory is  a  generalized quantum dimer  model, in
%which the  particular  relaxation of the   hardcore constraint for the
%dimers breaks the U(1)  down to Z2.   These mappings indicate that  at
%least   one      realization  of the     Senthil-Fisher   proposal for
%fractionalization is exactly the  short ranged resonating valence bond
%(RVB) scenario of  Anderson and of  Kivelson, Rokhsar and Sethna. They
%also suggest  that other realizations  will require the identification
%of   a   local low    energy, Ising  link    variable and    a natural
%constraint. We also  discuss  the notion of   topological order in  Z2
%gauge theories and its connection  to earlier ideas  in RVB theory. We
%note that  this notion is not   central to the experiment  proposed by
%Senthil  and Fisher to detect   vortices in the  conjectured Z2  gauge
%field.

%URL: http://link.aps.org/abstract/PRB/v65/e024504 

\bibitem{hw88elser89}
A.~J.~Phares and F.~J.~Wunderlich, Nuovo Cimento B {\bf 101}, 653 (1988).
V.~Elser.
%Nuclear antiferromagnetism in a registered $^3$He solid.
Phys. Rev. Lett. {\bf 62}, 2405 (1989).

\bibitem{ez93}
V. Elser and C. Zeng.
%kagome spin-1/2 antiferromagnets in the hyperbolic plane.
Phys. Rev. B {\bf 48}, 13647 (1993). 

%Spin-dimerized      states  are   useful   in    the   construction of
%spin-disordered wave functions but  difficult to deal with  because of
%nonorthogonality.   For the  spin-1/2   kagome antiferromagnet, a
%systematic expansion of  matrix elements of these nonorthogonal states
%is  made possible  by considering  generalizations  of the kagome
%structure in  the hyperbolic plane. The  first nontrivial term in this
%expansion  is an effective  spin Hamiltonian which describes resonance
%among  dimerized  states.  Minimum-energy   states  of  the  effective
%Hamiltonian correspond  to  a high degree of  resonance  among a small
%fraction of the dimers.

\bibitem{ze95}
C.~Zeng    and   V.~Elser.
%Quantum     dimer  calculations   on  the
%spin-$\frac{1}{2}$ kagome Heisenberg  antiferromagnet.
Phys.  Rev. B {\bf 51}, 8318 (1995).
%In analogy to the quantum dimer model on the square lattice introduced
%by Rokhsar  and Kivelson, we identify the  low-energy Hilbert space of
%the  spin-1/2  kagome Heisenberg   antiferromagnet  with  the set   of
%dimerized states (nearest-neighbor  singlets) and derive  an effective
%Hamiltonian which describes  resonance and  includes fluctuations into
%next-nearest-neighbor   singlets. Complete   diagonalization   of this
%effective Hamiltonian for systems as  large as 36 spins makes possible
%the   study of  the     low-energy thermodynamics   of this    quantum
%antiferromagnet  and strongly supports  the double peak feature in the
%heat capacity.



\bibitem{sg01}
R. Siddharthan and A. Georges.
%Square kagome quantum antiferromagnet and the eight-vertex model.
Phys. Rev. B {\bf 65}, 014417 (2002).

%We  introduce  a two-dimensional  network  of corner-sharing triangles
%with square-lattice  symmetry.  Properties  of magnetic  systems  here
%should be similar  to those  on  a  kagome  lattice. Focusing  on  the
%spin-1/2  Heisenberg quantum antiferromagnet,  we  generalize the spin
%symmetry group from SU(2)  to SU(N). In  the large-N limit, we map the
%model exactly to the  eight-vertex model solved  by Baxter. We predict
%an exponential number of low-lying  singlet states, a triplet gap, and
%a  two-peak specific heat. In  addition, the  large-N limit suggests a
%finite   temperature phase   transition   into a  phase  with  ordered
%"resonance loops" and broken translational symmetry.

% URL: http://link.aps.org/abstract/PRB/v65/e014417 

\bibitem{waldtmann}

Ch.  Waldtmann, H.~Kreutzmann, U.~Schollw\"ock, K.~Maisinger, and
H.-U.~Everts, Phys. Rev. B {\bf 62} 9472 (2000).

%Ground states   and   excitations  of a    one-dimensional kagom�-like
%antiferromagnet

%We study a Heisenberg antiferromagnet  (AF) on a one-dimensional strip
%composed   of two rows of  corner   sharing triangular plaquettes. The
%geometry  of  this  ladder-like  lattice   naturally allows  for   two
%different exchange couplings, one between  pairs of spins on the outer
%legs (J) and a different  one between the spins on  the axis and their
%nearest    neighbors  on the  legs  (J).  For   J/J0.5 the  model is a
%ferrimagnet. Our main   interest is in  the region  J/J0.5, where  the
%classical ground   state of  this system   shows the same  macroscopic
%degeneracy as the classical ground state of the antiferromagnet on the
%kagom� lattice. To explore to which extent this similarity between the
%classical ground  states of these   two models carries  over to  their
%quantum states we have   applied exact diagonalization techniques  and
%density-matrix    renormalization group    (DMRG)  methods   to    our
%model. Exact diagonalization is restricted to system  sizes of up to N
%=  30  sites. For J/J   = 1, the results   obtained by this technique,
%low-energy spectra, correlation functions and the specific heat, agree
%qualitatively with the  results  obtained  for finite samples  of  the
%kagom� AF. As in the case of the  kagom� AF, these finite size results
%suggest that our model is a spin liquid  with a gap between the ground
%state  and  the lowest spin  excitation.  However, extrapolations from
%DMRG data for strips of up to N =  120 sites point towards a vanishing
%spin gap in a wide range of couplings, 0.5J/J1.25, so that contrary to
%the  above conjecture,    our one-dimensional model  may  in   fact be
%critical in this  parameter  range. Moreover, our DMRG   data indicate
%that the model  undergoes a transition to a  gapped state as the ratio
%J/J  increases  through the value   J/J1.25. In  the vicinity  of  the
%transition point, a high density of low-lying singlets develops in the
%spin gap.

\bibitem{fs78}
E.~Fradkin and L. Susskind, Phys. Rev. D {\bf 17}, 2637 (1978).
%Order and disorder in gauge systems and magnets
%We show  how  phase transitions  in  Abelian two-dimensional  spin and
%four-dimensional     gauge systems can    be  understood  in  terms of
%condensation of topological objects. In the spin systems these objects
%are  kinks   and in  the  gauge systems   either magnetic monopoles or
%fluxoids (quantized lines of  magnetic flux). Four models are studied:
%two-dimensional Ising and  XY models and  four-dimensional Z2 and U(1)
%gauge

\bibitem{kogut}
%An introduction to lattice gauge theory and spin systems
J. B. Kogut, Rev. Mod. Phys. {\bf 51}, 659 (1979).
%This  article is an  interdisciplinary review  of lattice gauge theory
%and  spin systems. It discusses  the  fundamentals,  both physics  and
%formalism,  of these   related subjects. Spin  systems   are models of
%magnetism and  phase  transitions. Lattice gauge theories   are cutoff
%formulations  of     gauge  theories     of     strongly   interacting
%particles. Statistical mechanics  and field theory are closely related
%subjects, and the connections between them are developed here by using
%the transfer matrix. Phase diagrams  and critical points of continuous
%transitions are stressed as   the keys to understanding the  character
%and continuum limits of  lattice theories.  Concepts such as  duality,
%kink condensation,  and the existence of  a  local, relativistic field
%theory at a critical  point of a  lattice theory are illustrated  in a
%thorough discussion of the two-dimensional Ising model.  Theories with
%exact local (gauge) symmetries are introduced following Wegner's Ising
%lattice gauge theory.  Its gauge-invariant "loop" correlation function
%is   discussed  in detail.  Three-dimensional   Ising  gauge theory is
%studied thoroughly. The  renormalization group of  the two dimensional
%planar model is presented  as  an illustration  of a phase  transition
%driven by  the condensation of  topological excitations. Parallels are
%drawn to Abelian lattice gauge theory  in four dimensions. Non-Abelian
%gauge theories are introduced and the possibility of quark confinement
%is discussed. Asymptotic  freedom  of O(n) Heisenberg spin  systems in
%two  dimensions  is  verified for n  >=   and is explained   in simple
%terms. The direction of present-day research is briefly reviewed.

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

%Duality in generalized Ising    models and phase  transitions  without
%local order parameters

%It is shown that  any Ising model with  positive coupling constants is
%related to another Ising model by a duality transformation. A class of
%Ising models M/sub dn/ on  d-dimensional  lattices characterized by  a
%number n=1,2,...,d(n=1 corresponds to   the Ising model  with two-spin
%interaction)  is defined.  These  models are   related by  two duality
%transformations.  The models with   1(n(d  exhibit a phase  transition
%without local order parameter.  A nonanalyticity in the  specific heat
%and  a  different  qualitative behavior of   certain  spin correlation
%functions  in the  low and the  high temperature  phases indicate  the
%existence of  a phase transition. The  Hamiltonian of the simple cubic
%dual model contains products of  four Ising spin operators. Applying a
%star square transformation, one obtains  an Ising model with competing
%interactions exhibiting a   singularity  in the specific  heat  but no
%long-range order of the spins in the low temperature phase


\bibitem{sm02}
%Hidden long range order in Heisenberg Kagome antiferromagnets
A.~V.~Syromyatnikov and S.~V.~Maleyev, cond-mat/0202117 (unpublished).


\end{thebibliography}