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\begin{document}

\title{Resonating-valence-bond physics and topological order in two
dimensions: from dimer models to high-temperature superconductivity}

%%%%%%%
\classification{74.20.Mn, 75.10.Jm, 03.75.Lm, 71.10.Fd}

\keywords{resonating-valence-bond state, quantum dimer models,
high-temperature superconductivity, topological order}
%%%%%% 

\author{Dmitri A.~Ivanov}{
  address={Institute of Theoretical Physics, Ecole Polytechnique F\'ed\'erale
de Lausanne (EPFL), CH-1015 Lausanne, Switzerland}
}

\begin{abstract}
The notion of $Z_2$ topological order in
two-dimensional resonating-valence-bond systems is reviewed. 
Using quantum dimer models as an example, I discuss general properties
of the topological order and the resulting vortex-like excitations
(visons). Some connections to theories of frustrated magnetism
and of high-temperature superconductivity are pointed out.
\end{abstract}

\maketitle

\section{Resonating-valence-bond state}

The idea of a quantum spin liquid, i.e., a spin system which does not
order in any way down to zero temperature (in two or three dimensions), 
has been an important
topic of research in condensed-matter physics since many years. Under
certain conditions, such a state would imply a novel strongly correlated
phase of matter with unusual low-temperature properties \cite{Lhuillier2},
and it has been sought for both theoretically and experimentally.
Obvious candidates for a quantum liquid phase are so-called frustrated
spin systems, where the signs of interactions between the spins disfavor
any sort of magnetic ordering. 
On the other hand, spin-liquid physics has also been proposed to be relevant
for high-temperature superconductivity, where magnetic order is
destroyed by the doped holes \cite{PWA}. By now, many theories and models
have been constructed for spin liquids 
\cite{Lhuillier,Moessner-Sondhi,Balents-Fisher-Girvin,Senthil-Motrunich,%
Misguich-Serban-Pasquier,Ioffe-Feigelman}
and some experiments (e.g., on the herbertsmithite ZnCu$_3$(OH)$_6$Cl$_2$
\cite{kagome-experiment}) seem to indicate a possible spin-liquid state. 
Nevertheless, up to date, there has been no rigorous argument for a
stable spin-liquid state in a realistic spin model.

The most popular spin-liquid scenario is the so-called 
resonating-valence-bond (RVB) state, which has been proposed for
spin-1/2 systems in the two-dimensional geometry \cite{Fazekas},
and which has soon found its application also in the theory of 
high-temperature superconductors \cite{PWA}. 
Since then, the development of the RVB theory
continued in two parallel lines (with mutual
influence): RVB physics in spin systems and in dimer models on one side,
and the RVB scenario of high-temperature superconductivity on the other
side.

\subsection{RVB state in spin and dimer systems}

In analyzing  low-temperature phases of frustrated magnets, 
it has soon been realized that spins usually
tend to order either magnetically or as a crystal of singlets. The RVB-liquid
phase of frustrated magnets has mostly been suggested numerically, without
a solid analytic support
(see, e.g., Ref.~\cite{Misguich-Lhuillier} for a review). A more rigorous
and controllable way to realize the RVB phase is to consider quantum
dimer models or other models with an explicit $Z_2$ gauge 
symmetry \cite{MSF}. One of the simplest and most convenient systems
for studying the RVB phase is the Rokhsar--Kivelson (RK) dimer model \cite{RK}.
If formulated on the square lattice, it only has the RVB phase at
one point in the space of parameters (corresponding to a transition
between two crystal phases). The excitation spectrum is
gapless at this point, which is a consequence of a conserved ``winding
number'' for dimer models on bipartite lattices 
\cite{bipartite-Stephenson,Henley}. 
On the triangular lattice, however, there is a finite region of
parameters where the RVB phase is stable with a finite energy gap 
\cite{Moessner01}.

While properties of the RVB state in dimer models are quite well
understood (see the sections below), finding a realistic spin model
with a confirmed RVB state remains a challenging theoretical problem.

\subsection{RVB state in high-temperature superconductors}

In application to high-$T_c$ superconductivity, the RVB physics
appears in the Gutzwiller-projected construction of variational wave
functions \cite{PWA}. After the first systematic application of this method
\cite{Gros}, a variety of works have appeared
analyzing low-temperature properties of the $t$--$J$ model
(a generic model for high-$T_c$ superconductors) with the use
of variational wave functions \cite{Yokoyama,Himeda,Ivanov-sf}. 
Those predictions, even for the simplest version of the model with
only nearest-neighbor hopping and with the only parameter $t/J=3$
appear to be qualitatively consistent with the commonly observed properties of
cuprates \cite{A-Z}. The low-temperature phases include a
$d$-wave superconductor \cite{Gros} with an antiferromagnetic instability at 
very low doping \cite{Himeda}, in agreement with the experimental phase
diagram. Recently, more detailed studies have appeared
with computations of such quantities as condensation energy,
superfluid density, coherence length, quasiparticle spectral weight,
Fermi velocity, etc \cite{Paramekanti,Yang-Zhang,Yang-Li,Ho}.


\section{$Z_2$ topological order and vortexlike excitations (visons)}

The RVB state is usually defined in the following way: 
the spins are grouped into singlet pairs, and the
ground state is a linear combination of various singlet 
configurations (Fig.~\ref{figure:RVB}).

\begin{figure}
\includegraphics[height=.15\textheight]{rvb3.eps}
\caption{A schematic view of a RVB state. The
electrons form short-range singlets. The RVB state is a linear
combination of various singlet configurations that preserves
all the lattice symmetries.}
\label{figure:RVB}
\end{figure}

Such a state is conjectured to have a $Z_2$ topological 
order \cite{Read-Chakraborty}, and
its elementary excitations are spin-1/2 spinons and $Z_2$ vortices
(visons) \cite{Kivelson,Senthil-Fisher}. The topological order and
the vortexlike excitations remain at the level of conjecture for spin 
systems, but can be rigorously proven and studied for a simpler
type of systems: dimer models. 

\subsection{Topological order}

In dimer models, the spin singlets
are replaced by unbreakable dimers (Fig.~\ref{figure:dimer-models}),
and different dimer configurations are orthogonal by definition.
Then a new type of conservation law emerges: for any closed contour,
any \emph{local} rearrangement of dimers does not change the \emph{parity}
of the number of intersecting dimers ($Z_2$ index of intersection)
(Fig.~\ref{figure:dimer-models}) \cite{Kivelson,Senthil-Fisher}.
Note that this conservation law does not depend neither on 
the geometry of the lattice nor on the type of local rearrangement.
For bipartite lattices, an additional conservation law appears, which
extends the symmetry from $Z_2$ to $U(1)$ \cite{bipartite-Stephenson,Henley}.
In this paper, I assume that the lattice of available dimer positions
is not bipartite, and therefore I will not consider the $U(1)$ case.

\begin{figure}
\includegraphics[height=.25\textheight]{dimer-model.eps}
\caption{Dimer models. {\bf (a)}: Replacing singlets by dimers. The
overlap between different dimer configurations is zero by definition,
unlike for singlets. {\bf (b)}: An example of a dimer configuration.
For any (closed) contour, the parity of the number of intersections with dimers
is invariant with respect to local dimer rearrangement. {\bf (c)}: An
example of a local dimer rearrangement. This elementary rearrangement 
of two dimers constitutes the kinetic term in the Rokhsar--Kivelson
dimer model (\ref{RK-hamiltonian}).}
\label{figure:dimer-models}
\end{figure}

This conservation law leads to the existence of \emph{topological sectors} 
for systems defined on multiply connected domains (cylinder, torus, 
plane with holes, etc.): the Hilbert space splits into several disconnected
components. Assuming the \emph{absence of dimer crystallization}
(all correlations are exponentially decaying), this further leads to
a topological degeneracy in the thermodynamic limit (as the system
size $L$ tends to infinity) (Fig.~\ref{figure:topological-sectors}).

\begin{figure}
\includegraphics[height=.10\textheight]{dimer-cylinder.eps}
\caption{For a cylinder, one chooses a contour connecting the two
edges. The parity of dimer intersections with this contour distinguishes
the two topological sectors: even and odd.}
\label{figure:topological-sectors}
\end{figure}

In general, the notion of $Z_2$ topological order in RVB-type systems
may be formulated in the form of 
\emph{two conditions} \cite{Ivanov-top,Ivanov-soup}: on a multiply
connected domain, the degenerate ground states $\left| A \right\rangle$
and $\left| B \right\rangle$ should obey:
\begin{enumerate}
\item
identical local properties: $\left\langle A \right| X \left| A \right\rangle
= \left\langle B \right| X \left| B \right\rangle $ in the limit
$L\to\infty$ for any local operator $X$;
\item
orthogonality: $\left\langle A \right| X \left| B \right\rangle$,
again, in the thermodynamic limit $L\to\infty$ for any local operator $X$.
\end{enumerate}

Note that the second condition is automatically satisfied for dimer models,
while the first condition amounts to the absence of dimer crystallization. 
In spin systems, the second condition becomes also nontrivial and can be 
approximately related to the absence of spin ordering \cite{Ivanov-soup}.

\subsection{Topological order in the RK dimer model}

\begin{figure}
\includegraphics[height=.22\textheight]{vison-transition.eps}
\caption{A numerical evidence for the second-order crystallization
transition in the RK dimer model. Solid squares and diamonds represent
the dimer Bragg peaks at two different points in the Brillouin zone. 
Solid circles represent the vison Bragg peaks (multiplied by 0.01, to
be shown on the same scale). Open circles represent the vison gap, and
open squares and diamonds represent the dimer gaps at the two points
in the Brillouin zone. The horizontal axis is the ratio $v/t$ in the
Hamiltonian (\ref{RK-hamiltonian}). From Ref.~\cite{Ralko-3}.}
\label{figure:phase-transition}
\end{figure}

A typical example of the system with the $Z_2$ topological order described
above is the Rokhsar--Kivelson dimer model on the triangular 
lattice \cite{Moessner-Sondhi,Fendley}. For dimers on the triangular
lattice, the quantum dynamics is defined by the Hamiltonian:
\def\horparallel{
\lower.5ex\hbox{
\includegraphics[height=2ex]{fig-hor.eps}}\,\, }
\def\vertparallel{
\lower.5ex\hbox{
\includegraphics[height=2ex]{fig-vert.eps}}\,\, }
\begin{equation}
H_{\rm RK}= \sum \Big( -t 
\left| \horparallel  \right\rangle\left\langle \vertparallel \right|
+ v 
\left| \horparallel  \right\rangle\left\langle \horparallel \right|
\Big)
\label{RK-hamiltonian}
\end{equation}
where the sum is performed over all rhombi containing two parallel
dimers. The kinetic term with the coefficient $t$ flips the pair of
dimers as shown in the bottom panel of Fig.~\ref{figure:dimer-models},
while the potential term with the coefficient $v$ assigns a potential
energy to such pairs of dimers.

For the special case $v=t$ (the ``Rokhsar--Kivelson point''), the RVB
phase is rigorously proven, with the exponential decay of correlation
functions and with the two conditions of the topological order 
(formulated above) satisfied \cite{Ioselevich,Fendley}. An extensive
numerical study confirms a finite region of the RVB phase for the
range of parameters $0.83(2) < v/t \le 1$ 
\cite{Moessner-Sondhi,Ralko-1,Ralko-2,Ralko-3}. At $v/t = 0.83(2)$,
the RVB phase undergoes a transition to a crystal phase, with the
transition apparently of the second order \cite{moessner2,Ralko-3}
(Fig.~\ref{figure:phase-transition}).  At $v/t = 1$, a transition
between the RVB phase and another crystal phase is of the first order.


\subsection{Vortex-like excitations (visons)}

\begin{figure}
\includegraphics[height=.25\textheight]{dimer-vison-frustrated.eps}
\caption{{\bf (a)}: The point-like visons $V_i$ are defined on the dual lattice
with frustration. {\bf (b)}: For the triangular lattice, the dual lattice
is hexagonal. The frustration may be represented by the magnetic field
of half flux quantum per plaquette}
\label{figure:vison-frustration}
\end{figure}

The existence of the topological order on
multiply-connected domains also implies a new type of vortex-like excitations.
In the definition of the topological order, the contour is chosen to be 
either closed or connecting the edges of the domain. If now one takes 
a contour $\Gamma$ connecting two points \emph{inside} the lattice 
(connecting two plaquettes), then
the operator of the parity of dimer intersections with such a contour
no longer commutes with the Hamiltonian. In fact, one can show that this
parity operator can be represented as a product of two point-like objects:
\begin{equation}
V_1 \cdot V_2 = (-1)^{N_\Gamma}
\end{equation}
where $N_\Gamma$ is the number of the dimer intersections with the contour
$\Gamma$. The operators $V_1$ and $V_2$ are associated with the end points
of the contour, but they are nonlocal in terms of dimer operators. They
produce a certain type of vortexlike excitations (dubbed visons)
\cite{Kivelson,Senthil-Fisher}. While the operators $V_1$ and $V_2$ applied
to the ground state do not produce an eigenstate, they may be dressed
by local dimer operators and translated with a given wave vector to
produce eigenstates of the Hamiltonian of the quantum dimer model. The
operators $V_i$ have the structure of $Z_2$ vortices: they can only be created
in pairs in a finite system, and a product of two such operators can
be expressed locally in terms of dimer operators. So the true vison
eigenstates are the lowest-energy states equivalent to the states $V_i$
\cite{Ivanov-vison}.

Note that the operators $V_i$ are defined on the dual lattice (i.e.,
on the lattice formed by the plaquettes of the original lattice) with
a frustration (so that the operator $V_i$ acquires the sign $-1$ if moved
around one site of the original lattice), Fig.~\ref{figure:vison-frustration}. 
Equivalently, one may think of
a particle on the dual lattice in the magnetic field of half flux quantum
per plaquette of the dual lattice. This determines the symmetry of the
vison excitations and the structure of its Brillouin zone.

\subsection{Visons in the RK dimer model}

Vison excitations have been studied numerically in the Rokhsar--Kivelson
dimer model. The numerics is especially simple at the Rokhsar--Kivelson
point (at $v=t$): in that case, one can use the equivalence between the
quantum system in imaginary time and a classical stochastic process
\cite{Henley} and model the quantum dynamics with a classical Monte Carlo
simulation. This approach has been implemented in the work \cite{Ivanov-vison},
where the full dispersion of vison excitations has been obtained (except for 
the high-energy part of the branch, where the visons are unstable with respect 
to decay into triples of visons). The lowest energy of the vison excitations
equals 0.089(1) (in the units of $v=t$), which is
smaller than the lowest energy of excitations generated by dimer
operators. It has been thus shown that visons constitute the elementary
excitations in this RVB state.

The analysis of the vison spectrum has been later extended
away from the RK point, to $v/t<1$ (with a more complicated numerical
technique) \cite{Ralko-3}. It has been found that the visons continue
to be the lowest-energy excitations throughout the RVB phase, and that
their energy decreases (and apparently goes to zero) at the crystallization
phase transition around $v/t=0.83(2)$, see Fig.~\ref{figure:phase-transition}.
Thus the crystallization transition
may be described as a second-order vison condensation, in agreement with
the conjecture of Ref.~\cite{moessner2}.

\section{RVB physics in Gutzwiller-projected wave functions}

An interesting and not yet fully resolved question is to what extent
the physics described above (topological order and vison excitations)
survives if one goes back from dimer models to spin systems. One of the
obstacles is the overlap between various dimer configurations and
non-uniqueness of the decomposition of a spin state in terms of products
of singlets. A promising way around this difficulty 
is to consider Gutzwiller-projected wave functions as prototype states 
with RVB structure. 

The idea of the Gutzwiller-projected construction is the 
following \cite{Gutzwiller}:
First, one considers a mean-field BCS-type quadratic Hamiltonian
$H_{\rm BCS}(\chi,\Delta)$ (where $\chi$ and $\Delta$ are the sets of hopping
and pairing amplitudes, typically restricted to a certain symmetry
class and involving only one or several nearest-neighbor types of
bonds). Then the ground state $\Psi_{\rm BCS}$ of this
Hamiltonian is projected onto the configurations with no double occupancy of
sites. In addition, the total electron density is fixed either by
projecting onto a fixed-electron-number sector or by adjusting
a chemical-potential term. This procedure is called the
Gutzwiller projection $P_G$. The resulting wave function
$\Psi_{\rm GP}=P_G \Psi_{\rm BCS}$ is considered as a variational
wave function for the physical Hamiltonian $H_{\rm phys}$. The
variational parameters $\chi$ and $\Delta$ are then adjusted
to minimize the variational energy 
$\langle \Psi_{\rm GP} | H_{\rm phys} | \Psi_{\rm GP} \rangle$.

This variational approach to strongly-correlated Hamiltonians
$H_{\rm phys}$ has been widely used both for a variety of
spin systems \cite{Dmitriev,Sorella-J1J2,Patrick-kagome} and
for the $t$--$J$ model in the context of high-temperature
superconductivity \cite{Gros,Yokoyama,A-Z}. Not only provides it
a good variational ansatz for such systems, but it also implies
a RVB-type structure \cite{PWA}. One is then tempted to ask whether
the argument in Ref.~\cite{PWA} can be promoted to the topological
order and visons, by analogy with the dimer models. Several developments
have been made in this direction.

\subsection{Topological order in projected wave functions}

To test for topological order, one needs to verify the two conditions
formulated above for the ground states in different
topological sectors. Such states are constructed variationally
by projecting fermionic states with either periodic or antiperiodic
boundary conditions on multiply connected domains \cite{Ivanov-top}.
If one considers an undoped system (a spin model), such a projection
produces the same spin boundary conditions,  regardless of the 
choice of fermionic boundary conditions. One can then check 
whether the two criteria
for the topological order are satisfied or not (both criteria become
nontrivial in this case).

In Ref.~\cite{Ivanov-top}, we have performed such
a test for several types of wave functions similar to those used
for the $t$--$J$ model and for the $J_1$--$J_2$ spin model. We have
found that some of them do not have a topological order, while
some others do. Note that since the mean-field states $\Psi_{\rm BCS}$
used in our study have nodes in the spectrum, the conditions
for the topological order, even if they are satisfied, hold not
with an exponential, but only with a power-law accuracy for large
systems. For example, for the nearest-neighbor spin correlations we find
\begin{equation}
\langle {+} | S_i S_j | {+} \rangle -
\langle {-} | S_i S_j | {-} \rangle \propto L^{-\alpha}\, , 
\quad \alpha \approx 2\, ,
\end{equation}
where $ | {+} \rangle$ and  $ | {-} \rangle$ are Gutzwiller-projected
states with different fermionic boundary conditions, and $L$ is the
linear system size.

A similar numerical study has been performed later by another group
\cite{TLi-top} on a different set of wave functions. They have also observed
that some of the projected wave functions possess a topological order
while others don't. However the problem of finding
the exact criteria on the wave function for the appearance of the 
topological order is not solved yet: in Ref.~\cite{Ivanov-top}, we
argue that it should be determined by whether the mean-field ansatz is
of the $Z_2$ or of the $U(1)$ type, while in Ref.~\cite{TLi-top} they
claim that it is the bipartite geometry of the lattice which is
responsible for the presence or absence of topological order in the
projected wave function. Neither of the two conjectures is solidly
proven, and the collection of available numerical evidence collected in 
the two papers appear to be inconsistent with either of the two conjectures.
Therefore it seems necessary to undertake either a more detailed and systematic
numerical study of this problem or to present a rigorous analytic argument
to resolve the issue of topological order in Gutzwiller-projected
wave functions.

\section{Generalization of the RVB construction: loop path integral}

An interesting complication arising in the study 
of topological order in Gutzwiller-projected wave function is the presence
of nodes in the mean-field state. Those nodes result in the
\emph{algebraic} topological order. Moreover, the original argument
of Anderson about the RVB nature of Gutzwiller-projected wave functions
\cite{PWA} is not applicable to BCS states with nodes (since the
resulting singlet amplitudes become long-ranged in this case).
This issue has been addressed in Ref.~\cite{Ivanov-soup}, where the
RVB structure of wave functions has been interpreted in terms of the
loop path integral, instead of singlets.

\begin{figure}
\includegraphics[height=.13\textheight]{topological-soup.eps}
\caption{{\bf (a)}: In the conventional RVB construction, the loops
are given by the overlap of two singlet (dimer) configurations.
{\bf (b)}: The topological sectors may be accessed by changing the
sign of the amplitude for loops having a global winding.}
\label{figure:soup}
\end{figure}

The idea is to consider the ``partition function'' 
$\langle \Psi | \Psi \rangle$ instead of the wave function $| \Psi \rangle$.
For all known examples of RVB states, this normalization of the
wave function has the form
\begin{equation}
\langle \Psi | \Psi \rangle = \sum_{\{ C_n \} } \prod_n A(C_n)\, ,
\label{loop-path-integral}
\end{equation}
where the sum is taken over all possible fully packed coverings of
the lattice by the loops $C_n$ (so that every site belongs to exactly
one loop), and $A(C_n)$ is some function of the loop. This construction
obviously generalizes the expression for the overlap of the conventional
RVB states (built as products of singlets \cite{PWA}):
\begin{equation}
A(C)= -2 a_{12} a_{23} \dots a_{k1}
\label{loop-singlets}
\end{equation}
where $a_{ij}$ are the singlet amplitudes. The minus sign in the 
expression for $A(C)$ comes from the fermionic structure of 
the RVB construction (a positive sign would also be possible \cite{Liang}),
and the factor of 2 reflects the spin multiplicity. The loops
are formed by the overlap of dimer singlets in the bra and ket vectors
$\langle \Psi |$ and $| \Psi \rangle$ (Fig.~\ref{figure:soup}). 

However, the loop-path-integral expression (\ref{loop-path-integral})
has a wider range of validity: in fact, all known RVB-like states
can be interpreted in this way. For example,
\begin{enumerate}
\item
As mentioned above, the RVB state composed of products of singlets
corresponds to the choice $A(C)= -2 a_{12} a_{23} \dots a_{k1}$.
\item
The ground state of the Rokhsar--Kivelson dimer model at the RK point
($t=v$) corresponds to
\begin{equation}
A(C)=\cases{
1 & for length-two loops \cr
0 & otherwise }
\end{equation}
\item
For Gutzwiller-projected wave functions \cite{Ivanov-soup},
\begin{equation}
A(C)=-{\rm Tr}\, G_{12} G_{23} \dots G_{k1}\, ,
\end{equation}
where $G_{ij}$ are the BCS equal-time Green functions (2$\times$2 matrices).
\end{enumerate}

The last example is interesting from the point of view of the algebraic
topological order. Indeed, the topological order may be formulated
in terms of the loop behavior in the path-integral theory 
(\ref{loop-path-integral}). Different topological sectors may be accessed by
assigning a minus sign for globally winding loops (Fig.~\ref{figure:soup}).
The two conditions of topological order may be qualitatively 
formulated as (1) loops are short ranged and (2) loops do not 
crystallize \cite{Ivanov-soup}. The short-rangedness of loops is however
only algebraic in the example of projected BCS states with nodes.
In such a case, since the Green functions $G_{ij}$
decay algebraically (as $L^{-2}$), the loop size cannot decay faster than
that (since there always exist ``direct-flight'' loops with long-range
links), but can only decay equally fast or slower than $G_{ij}$. This
imposes an upper bound on the convergence of the topological degeneracy
for such projected states in the thermodynamic limit.

A more accurate analytic description of algebraic topological order in
Gutzwiller-projected wave functions is still to be completed. Numerical
studies indicate that the close-packing constraint strongly renormalizes 
the algebraic topological order, as compared to the power-law decay
of the bare Green functions $G_{ij}$  \cite{Ivanov-soup}.


\section{RVB state in doped systems}

Finally, it is worth commenting on the fate of topological order
and of vortex-like excitations in RVB systems doped with
mobile charges (with high-temperature superconductivity in mind).
A general theory of doped RVB states still needs to be completed,
and a great part of research in high-temperature superconductivity
is devoted to this direction. 

One obvious scenario (supported by the study of Gutzwiller-projected
wave functions) is that upon doping the system immediately becomes
superconducting. In that case the vison excitations get promoted
to superconducting vortices \cite{Ivanov-top}. It has been even
conjectured that vison excitations may survive above the superconducting
transition temperature \cite{Senthil-Fisher}, but so far this conjecture 
has not found any experimental confirmation \cite{Bonn}.

Another interesting question is the spin-charge separation in doped RVB
systems. While the na\"ive picture predicts that the spin and charge
excitations should be decoupled in a liquid of RVB dimers (see, e.g.,
a monomer-monomer correlation function acquiring a finite expectation
value in the RK dimer model \cite{Fendley}), 
the Gutzwiller-projected construction of 
quasiparticles \cite{Paramekanti,Yang-Zhang,Yang-Li,Ho,Nave,Bieri-weight}
suggests that, to the contrary, the spin and charge are bound to
form a renormalized BCS-like quasiparticle. A resolution of this apparent
inconsistency may lie in the observation that the spectral weight of the
quasiparticle goes to zero in the limit of zero doping \cite{Paramekanti}.
Therefore, the spin and charge separated at half filling may
gradually recombine upon doping. It may be interesting to find a model
(possibly an extension of the RK model) where the problem of spin-charge
recombination may be treated analytically, to give an insight in the
physics of doped RVB systems.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% BACKMATTER
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{theacknowledgments}
I am grateful to P.~A.~Lee, X.-G.~Wen, T.~Senthil, M.~Feigelman, A.~Ioselevich,
F.~Mila, A.~Ralko, F.~Becca, and M.~Ferrero for collaboration on
different topics mentioned in this review. The work was supported
by the Swiss National Foundation.
\end{theacknowledgments}

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\end{document}

