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All of the ground-state equal-time correlators of these theories are equal to correlation functions of a {\em local} two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as ${\mathbb Z}_2$ and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these {\em quantum Lifshitz points} is just the statistical (Gibbs) weight of the ordinary 2D free boson, the 2D Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously-varying critical exponents separating phases with long-range order from a ${\mathbb Z}_2$ deconfined topologically-ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-Abelian symmetry, the strong-coupling limit of $2+1$-dimensional Yang-Mills gauge theory with a Chern-Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \maketitle \section{Introduction} During the past decade and a half there has been an intense search for new kinds of theories describing quantum condensed-matter systems. Many experimental results have implied that strongly-correlated fermionic systems exhibit qualitatively new types of physical behavior. The now-classic example of this is the fractional quantum Hall effect, where one of the striking consequences of strong correlations is that the Laughlin quasiparticles have fractional charge and fractional statistics, even though though the microscopic degrees of freedom are electrons with integer charge and fermionic statistics [\onlinecite{kivelson01}]. Traditionally one classifies different phases in terms of order parameters which give a global characterization of the physical state. In turn, the local fluctuations of this order parameter field drive the phase transitions between ordered and disordered states of these systems. This viewpoint, pioneered by Landau and his school, has been extremely successful in condensed matter physics and in other areas of physics, such as particle physics, through the powerful underlying concept of spontaneous symmetry breaking. Much of the structure of modern theory of critical phenomena is based on this point of view [\onlinecite{lubensky95,cardy96}]. However, there are many different experimentally-realizable phases (and even more realizable theoretically!) in the fractional quantum Hall effect, but no {\em local} order parameter distinguishes between them. These phases are incompressible liquid states which have a fully gapped spectrum and do not break any symmetries of the Hamiltonian. The lack of a local order parameter led to many interesting discussions of the off-diagonal long-range order in the Hall effect [\onlinecite{prange:QHE}]. One particularly elegant way of characterizing the order in the fractional quantum Hall effect is as {\em topological order} [\onlinecite{wen90}]. The topological order parameters are non-local; they are expectation values of operators which are lines or loops. Because of this, they can (and do) depend on topology: their value depends on the genus of the two-dimensional surface on which the electrons live. One interesting characteristic of a {\em topological phase} is that the correlation functions in the ground state do not depend on the locations of the operators, but only on how the loops braid through each other. In addition, the degeneracies of these non-symmetry breaking ground states on topologically non-trivial manifolds are determined by the topology of these manifolds [\onlinecite{wen-niu90}]. Although so far the only unambiguous experimental realizations of topological phases are in the fractional quantum Hall effect, there has been considerable effort to find, both theoretically and experimentally, condensed matter systems whose phase diagrams may exhibit topological ground states. Much of the current work involves studying fractionalized phases in time-reversal invariant systems (see e.g.\ [\onlinecite{senthil00,moessner01a}]). One reason is that the ``normal state" of high-temperature superconductors lacks an electron-like quasiparticle state in its spectrum. There are reasons to believe that frustrated magnets may also exhibit fractionalized behavior as well. A particularly well-known and simple model with a topological phase is the {\em quantum dimer model}, which was invented as a way of modeling the short-range resonating-valence-bond theory of superconductivity [\onlinecite{kivelson87}]. The degrees of freedom of this two-dimensional model are classical dimers living on a two-dimensional lattice. With a special choice of Hamiltonian (called the RK point), the exact ground-state wave function can be found [\onlinecite{rokhsar88}]. When the dimers are on the square lattice, the result is a critical point. If one deforms this special Hamiltonian, one generically obtains ordered phases. However, a topological phase occurs in the quantum dimer model on the triangular lattice [\onlinecite{moessner01a}]. When the quantum dimer model is in a topological phase, an effect analogous to fractionalization occurs [\onlinecite{kivelson87}]. This is called spin-charge separation. One can view the dimers as being created by nearest-neighbor pairs of lattice electrons in a spin-singlet state. Even though the fundamental degrees of freedom (the electrons) have both spin and charge, one finds that the basic excitations have either charge (holons) or spin (spinons), but not both. To prove this occurs, one must show that if one breaks apart an electron pair (dimer) into two holons or two spinons, they are deconfined. For the triangular-lattice quantum dimer model, this was shown in Ref.\ [\onlinecite{moessner01a}]; the analogous statement in terms of holon-holon correlators was proven in Ref.\ [\onlinecite{fendley02}]. At a quantum critical phase transition between an ordered/confining phase and a disordered/deconfining phase (or between different confining states), confinement is lost: the RK point is deconfining [\onlinecite{fradkin90b,fradkin91,moessner02a}]. The notion of spin-charge separation is one of the basic assumptions behind the RVB theories of high-temperature superconductivity [\onlinecite{anderson87,kivelson87,baskaran88,affleck88,kotliar88,ioffe89,palee98,kalmeyer88,wen89}], which effectively can be regarded as strongly-coupled lattice gauge theories. In $2+1$-dimensional systems spin-charge separation can only take place if these gauge theories are in a deconfined phase [\onlinecite{read-sachdev90,fradkin91,mudry94}]. In $2+1$-dimensions this is only possible for discrete gauge symmetries. For a continuous gauge group, say $U(1)$ or $SU(2)$, $2+1$-dimensional gauge theories are always in a confining phase, unless the matter fields carry a charge higher than the fundamental charge so that the gauge symmetry is broken to a discrete subgroup [\onlinecite{fradkin-shenker79}]. Thus, the only consistent scenarios for spin-charge separation necessarily involve an effective discrete gauge symmetry, which in practice reduces to the simplest case ${\mathbb Z}_2$. Of particular interest is the fact that the low-energy sector of the deconfined phases of discrete gauge theories are the simplest topological field theories [\onlinecite{witten88,preskill90,dewild95}]. Many of these ideas have their origin in the conceptual description of confined phases of gauge theories as monopole condensates, and of their deconfined states as ``string condensates" [\onlinecite{polyakov-book}]. In gauge theories it has long been known that their phases cannot characterized by a local order parameter, since local symmetries cannot be spontaneously broken. The phases of gauge theories are understood instead in terms of the behavior of generally non-local operators such as Wilson loops and disorder operators [\onlinecite{thooft78,fradkin78}], a concept borrowed from the theory of the two-dimensional Ising magnet [\onlinecite{kadanoff-ceva71}]. Interesting as they are, the applicability of these ideas to the problem of high-temperature superconductivity and other strongly correlated systems is still very much an open problem. Topological fractionalized ground states are not the only possible explanation of the unusual physics of the cuprates. In fact, when constructing local microscopic models of strongly-correlated systems which are suspected to have fractionalized phases, many theorists have found that instead these models have a strong tendency to exhibit spatially-ordered states, {\it a.k.a.\/} ``valence bond crystals", which appear to compete with possible deconfined states. It is now clear that the regimes of strongly-correlated systems which may favor fractionalized phases also favor, and perhaps more strongly, non-magnetic spatially ordered states of different types, including staggered flux states [\onlinecite{affleck88}], or $d$-density wave states [\onlinecite{ddw}], and electronic analogs of liquid crystalline phases [\onlinecite{nature,subir-rmp}]. By now there are a number of examples of models with short-range interactions whose phase diagrams contain both fractionalized and spatially-ordered phases [\onlinecite{moessner01a,moessner01b,balents02,senthil02a}]. It has recently been proposed that deconfined critical points may describe the quantum phase transitions between ordered N\'eel states and valence bond crystals [\onlinecite{vishwanath03b}]. For several reasons, most of the studies of topological order and quantum critical points have focused on examples with two spatial dimensions. The experimental reason is that the Hall effect is two-dimensional, and typical strongly-correlated systems, such as the cuprate high-temperature superconductors, are often effectively two-dimensional. Theoretically, it is because in two dimensions particles can have exotic statistics interpolating between bosonic and fermionic [\onlinecite{wilczekbook}]. A common characteristic of topological phases in two dimensions is the presence of exotic statistics, which occur in the fractional quantum Hall effect [\onlinecite{prange:QHE}]. The statistics can even be non-abelian: in some cases, the change in the wave function depends on the order in which particles are exchanged [\onlinecite{moore-read91}]. Systems with non-abelian statistics are particularly interesting because they are useful for error correction in quantum computers [\onlinecite{kitaev97,freedman01a,freedman01b,preskill02b,ioffe03}]. In this paper we will discuss models with topological phases and ordered phases, as well as quantum phase transitions separating them. There has also been a great deal of interest in quantum critical points in and of themselves [\onlinecite{sachdev-book}]. At a quantum critical point, the physics is of course scale invariant, but it need not be Lorentz invariant. The quantum critical points discussed in this paper have dynamical critical exponent $z=2$, instead of the usual $z=1$ of a Lorentz-invariant theory. This allows for some striking new physics. The action of these $z=2$ quantum critical points is invariant under time-independent conformal transformations of the two-dimensional space. A remarkable consequence is that the {\em ground-state wave functionals} of the field theories discussed here are conformally invariant in space. This means that the ground state wave functional is invariant under any angle-preserving coordinate transformations of space. % (but not time, since $z\neq 1$). For two-dimensional space, there is an infinite set of such transformations, as is familiar from studies of two-dimensional conformal field theory[\onlinecite{yellow}]. This sort of behavior is not common at all: the action of a field theory at a critical point is often scale invariant (and also conformally invariant), but the ground-state wave functional itself in general is not. We dub critical points with this behavior {\em conformal quantum critical points}. One of the consequences of the conformal {\em invariance} of the ground state wave function is that all the equal-time correlators of the quantum theory are equal to suitable correlation functions of observables of a two-dimensional Euclidean conformal field theory. We will exploit this connection in this paper quite extensively. However, just as important, conformal invariance of the wave function implies that the ground state of this $2+1$-dimensional theory at a conformal quantum critical point must have zero resistance to shear stress in the two-dimensional plane. This can be seen as follows. Consider an infinitesimal local distortion of the geometry of the two-dimensional plane represented by an infinitesimal change $\delta g_{ij}(x)$of the two-dimensional metric, as is conventional in the theory of elasticity[\onlinecite{lubensky95}]\footnote{Recall that the change in the metric, given by the strain tensor, is quadratic in the local deformation of the system.}. Let $\ket{\Psi}$ be the ground state wave function for the undistorted plane and $\ket{\Psi(g)}$ be the ground state wave function in the distorted plane with two-dimensional metric $g_{ij}(x)=\delta_{ij}+\delta g_{ij}(x)$. Under this distortion the Hamiltonian of the system changes by an amount \begin{equation} \delta H(g)=\int d^2 x \; \frac{\delta H}{\delta g_{ij}(x)} \; \delta g_{ij}(x)+\ldots \label{deltaH} \end{equation} To first order in perturbation theory in $\delta H$, the change of the ground state wave energy is \begin{equation} \delta E_0=\frac{\me{\Psi}{\; \delta H(g)\; }{\Psi}}{\ipr{\Psi}{\Psi}} \equiv \langle \delta H(g) \rangle= \int d^2x \; \displaystyle{\Big\langle {\frac{\delta H }{\delta g_{ij}(x)}\Big\rangle}} \delta g_{ij}(x)+\ldots \label{change} \end{equation} where $E_0$ is the exact ground state energy of the distorted system. On the other hand, the change of the norm of the ground state wave function $\|\Psi\|$ is, to all orders in perturbation theory, given by[\onlinecite{Baym90}] \begin{equation} \|\Psi\|^2=\frac{\partial E_0}{\partial\varepsilon_0} \end{equation} where $\varepsilon_0$ is the ground state energy of the undistorted system. Thus, the change of the norm $\|\Psi\|$ is determined by the ($ 2 \times 2$) {\em stress tensor} $T_{ij}$ of the $2+1$-dimensional theory \begin{equation} T_{ij}(x)=\Big\langle \frac{\delta H\; }{\delta g_{ij}(x)}\Big\rangle\ . \label{stress} \end{equation} On the other hand, we can regard $\| \Psi\|^2$ as the partition function $Z$ of a two-dimensional Euclidean conformal field theory. This theory has an Euclidean {\em stress-energy} tensor, $T^{\rm cft}_{ij}$, defined by[\onlinecite{yellow}] \begin{equation} T^{\rm cft}_{ij}=-\Big\langle \frac{\delta S_{\rm cft}}{\delta g_{ij}(x)}\Big\rangle=\frac{\delta \ln Z}{\delta g_{ij}(x)}\ , \label{2Dcft} \end{equation} which essentially coincides with the {\em stress} tensor of the $2+1$-dimensional quantum field theory defined above. Scale invariance, rotational invariance and conservation require that $T_{ij}$ be a conserved (divergence free) symmetric traceless tensor. Consequently, the effective Hamiltonian (as well as the action) at this quantum critical point can depend on the spatial gradients of the field only through the ``spatial curvature", {\it e.g.\/} $(\nabla^2\varphi)^2$ in a scalar field theory. In other words, at a conformal quantum critical point for a scalar theory, the stiffness vanishes: the usual $(\nabla\phi)^2$ term is not possible. This means that the dynamical critical exponent of this quantum critical theory must be $z=2$. We call such theories quantum Lifshitz theories; we will discuss such critical points in detail. In this paper we discuss both lattice models which exhibit both ordered/confined phases and disordered/deconfined phases. We will also discuss the field-theory description of these phases and of the phase transitions. To simplify matters, and to be able to obtain exact results, we will introduce models whose ground-state wave function will be known exactly and whose properties we will be able to determine quite explicitly. In this sense, these models are a generalization of the quantum dimer model at the RK point. The basis of the Hilbert space of these models is the configuration space of a two-dimensional classical statistical-mechanical system or Euclidean field theory. Each of these basis states is defined to be orthogonal with respect to the others. An arbitrary state in this Hilbert space can therefore be described as some linear combination of these basis elements. Describing the Hilbert space in such a fashion is not particularly novel. The unusual feature of the models we will discuss is that the ground-state wave function can be expressed as in terms of the action or Boltzmann weights of a {\em local} two-dimensional classical theory. The normalization of the wave function will then be the partition function or functional integral of the classical two-dimensional model. This special property is why the wave functionals at the critical points are have a time-independent conformal invariance at their critical points. The field theory of these conformal quantum critical points can be extended to describe nearby ordered and disordered phases, including their confinement properties. We will study this quite explicitly in a quantum generalization of the eight-vertex model. However, much of the physics we discuss should apply to topological phases and ($z=2$) quantum critical points in general. We will also study theories with a continuous non-abelian symmetry. We show that, interestingly enough, there seems to be no way to construct a non-trivial conformal quantum critical point. We do find a Hamiltonian whose ground state is the doubled Chern-Simons theory of Ref.\ [\onlinecite{freedman03,shivajinew}]. This is a time-reversal invariant theory of interest in topological quantum computation and in (ordinary) supercondutivity; it is in a gapped topological phase. In section \ref{sec:scale-wf} we discuss the simplest model with a scale-invariant critical wave function, the quantum dimer model at the RK point. Here we also introduce the quantum Lifshitz model, the effective field theory of these new quantum critical points. In section \ref{sec:2dwavefn}, we generalize the relation between the quantum dimer model and the scalar field theory discussed in section \ref{sec:scale-wf} to include perturbations which drive the system in to a quantum disordered/deconfined phase or to a ordered/confined phase. In section \ref{sec:q8v}, we define the quantum eight-vertex model by finding a Hamiltonian whose ground-state wave function is related to the classical eight-vertex model. This will allow us to find quantum critical lines with variable critical exponents separating a ${\mathbb Z}_2$-ordered phase from a topologically-ordered phase. It will also allow us to place a number of previously-known models, in particular that of Ref.\ [\onlinecite{kitaev97}], in a more general setting. We show in detail how to use the known results from the Baxter solution of the classical model to map out the critical behavior of the quantum theory. In particular we analyze in detail the confinement and deconfinement properties of the different phases and at criticality. In section \ref{sec:YMCS}, we study the non-abelian case, and see that the strongly-coupled limit of Yang-Mills theory with a Chern-Simons term has a wave functional local in two-dimensional classical fields [\onlinecite{witten92}]. This theory is in a phase with topological order. In three appendices we give details of the correlators of the quantum Lifshitz field theory (Appendix \ref{app:gaussian}), and of the gauge-theory construction of the quantum six-vertex (Appendix \ref{app:gauge}) and eight-vertex (Appendix \ref{app:z2gauge}) models. \section{Scale-Invariant Wave Functions and Quantum Criticality} \label{sec:scale-wf} The simplest lattice model we discuss is the quantum dimer model; the simplest field theory we dub the quantum Lifshitz theory. They both provide very nice illustrations of the properties discussed in the introduction. In the quantum dimer model, the space of states consists of close-packed hard-core dimers on a two-dimensional lattice. A quantum Hamiltonian therefore is an operator acting on this space of dimers, taking any dimer configuration to some linear combination of configurations. In every configuration exactly one dimer must touch every site, so any off-diagonal term in the Hamiltonian must necessarily move more than one dimer. The simplest such operator is called a ``plaquette flip'': if one has two dimers on opposite sites of one plaquette, one can rotate the dimers around the plaquette without effecting any other dimers. For example, for the $i$th plaquette on the square lattice one has \begin{equation} \begin{picture}(350,30) \thicklines \put(-40,6){$\hat{F}_i:$} \put(0,0){\line(0,1){20}} \put(0.3,0){\circle*{4}} \put(0.3,20){\circle*{4}} \put(20,0){\line(0,1){20}} \put(20.3,20){\circle*{4}} \put(20.3,0){\circle*{4}} \put(40,8){$\longrightarrow$} \put(70,0){\line(1,0){20}} \put(70,20){\line(1,0){20}} \put(70.3,0){\circle*{4}} \put(70.3,20){\circle*{4}} \put(90.3,0){\circle*{4}} \put(90.3,20){\circle*{4}} \put(170,6){and} \put(240,0){\line(1,0){20}} \put(240,20){\line(1,0){20}} \put(240.3,0){\circle*{4}} \put(240.3,20){\circle*{4}} \put(260.3,0){\circle*{4}} \put(260.3,20){\circle*{4}} \put(280,8){$\longrightarrow$} \put(310.3,0){\circle*{4}} \put(310.3,20){\circle*{4}} \put(310,0){\line(0,1){20}} \put(330,0){\line(0,1){20}} \put(330.3,0){\circle*{4}} \put(330.3,20){\circle*{4}} \end{picture} \end{equation} The operator $\hat{F}_i$ is defined as zero on any other dimer configuration around a plaquette (i.e.\ if the $i$th plaquette is not flippable). We define the operator $\hat{V}_i$ as the identity if the plaquette is flippable, and zero otherwise. The Rokhsar-Kivelson Hamiltonian for the quantum dimer model [\onlinecite{rokhsar88}] \begin{equation} H_{RK} = \sum_i (\hat{V}_i - \hat{F}_i) \label{HRK} \end{equation} has the remarkable property that one can find its ground states exactly. They have energy zero, and every state (in a given sub-sector labeled by global conserved quantities) appears with equal amplitude in its ground-state wave function. These properties follow from the facts that $H_{RK}$ is self-adjoint, and $(\hat{V}_i - \hat{F}_i)^2 = 2(\hat{V}_i - \hat{F}_i)$. Hamiltonians of the form $H = \sum_i Q^\dagger_i Q_i$ necessarily have eigenvalues $E$ obeying $E\ge 0$. Moreover, if one can find a state annihilated by all the $Q_i$, then it is necessarily a ground state. The equal-amplitude sum over all states is indeed such a state. In the Schr\"odinger picture, the wave function for this state is easy to write down. Define $Z$ as the number of all dimer configurations in some finite volume. This is precisely the classical partition function of two-dimensional dimers with all configurations weighted equally. Then the properly-normalized ground-state wave function for any basis state $|C\rangle$ in the Hilbert space (\ie any classical dimer configuration $C$) is \begin{equation} |\Psi_0\rangle=\frac{1}{\sqrt{Z}} \sum_C |C\rangle \Rightarrow \Psi_0(C) = \frac{1}{\sqrt{Z}} \end{equation} The wave function of the quantum system is indeed related to the classical system. One can extend this sort of analysis to compute equal-time correlators in the ground state. One finds simply that these correlators are given by the correlation functions of the two-dimensional classical theory. Thus for $H_{RK}$ for dimers on the square lattice, one finds algebraic decay of the correlation functions [\onlinecite{rokhsar88}]. This model is then interpreted as a critical point between two ordered phases of the dimers [\onlinecite{read-sachdev90,fradkin91}]. However, for the analogous Hamiltonian on the triangular lattice, the classical two-dimensional correlators are exponentially decaying [\onlinecite{moessner01a}]. One can also show that spinon-type excitations (sites without a dimer) are deconfined on the triangular lattice [\onlinecite{fendley02}]. This means the quantum dimer model with $H_{RK}$ on the triangular lattice is interpreted as being in a ``liquid'' phase, which has a mass gap and exponential decay of interactions, but which has no non-zero local order parameter. Such a relation between two-dimensional quantum theories is not limited to lattice models, nor are the ground-state wave functions required to be equal-amplitude sums over all configurations. We will construct now a simple (non-Lorentz invariant) two-dimensional quantum critical field theory \ie a theory whose ground state wave function represents a two-dimensional conformal theory. Consider a free boson $\varphi(x,t)$ in two spatial dimensions and one time dimension. Instead of the usual Hamiltonian quadratic in derivatives, we use one which has been conjectured by Henley [\onlinecite{henley97a}] to belong to the same universality class as the square-lattice quantum-dimer model. It is \begin{equation} H = \int d^2x \; \left[\frac{\Pi^2}{2} + \frac{\kappa^2}{2}(\nabla^2\varphi)^2\right] \label{eq:Hboson} \end{equation} where $\Pi=\dot\varphi$ as usual. The associated Euclidean action for the field $\varphi$ is \begin{equation} S=\int d^3x \left[\frac{1}{2} \left(\partial_\tau \varphi\right)^2+\frac{\kappa^2}{2}\left(\nabla^2 \varphi\right)^2\right] \label{3D-Euclidean} \end{equation} This system, Eq.\ (\ref{3D-Euclidean}), also arises in three-dimensional classical statistical mechanics in the field-theory description of Lifshitz points [\onlinecite{lubensky95}], for example in (smectic) liquid crystals. For this reason we will call the system with Hamiltonian (\ref{eq:Hboson}) the {\em quantum Lifshitz model}. Of particular relevance to our discussion is the long-ago observation by Grinstein [\onlinecite{grinstein81a}] that this system is analogous to the two-dimensional Euclidean free boson in that it represents a line of fixed points parametrized by $\kappa$. Let us rederive this result by quantizing the Hamiltonian (\ref{eq:Hboson}). We impose the canonical commutation relations \begin{equation} [\varphi(\vec x),\Pi(\vec x^{\prime})]=i\delta(\vec x -\vec x^{\prime}) \label{eq:ccr} \end{equation} so in the Schr{\"o}dinger picture the canonical momentum is the functional derivative $\Pi(\vec x)= -i{\delta}/{\delta\varphi(\vec x)} $. The Schr{\"o}dinger equation for the wave functional $\Psi[\varphi]$ is then \begin{equation} \int d^2 x\, \left[-\12 \left(\frac{\delta}{\delta\varphi}\right)^2 + \frac{\kappa^2}{2}(\nabla^2\varphi)^2 \right]\Psi[\varphi] = E\Psi[\varphi] . \label{eq:SE} \end{equation} We can find the ground-state wave function in the same fashion as we did for $H_{RK}$. Indeed, if we define \begin{equation} Q(x)\equiv \frac{1}{\sqrt{2}} \left(\frac{\delta}{\delta\varphi} + \kappa \; \nabla^2\varphi \right)\ , \qquad\quad Q^\dagger(x)\equiv \frac{1}{\sqrt{2}} \left(-\frac{\delta}{\delta\varphi} + \kappa\; \nabla^2\varphi \right) , \label{eq:QAbelian} \end{equation} the (normal-ordered) quantum Hamiltonian is then \begin{equation} H= \12 \int d^2x \; \left\{Q^\dagger(\vec x), Q(\vec x)\right\} -\varepsilon_{\rm vac} V \equiv \int d^2x \; Q^\dagger(\vec x) Q(\vec x) \label{eq:HQQ} \end{equation} which is Hermitian and positive. Here $V$ is the spatial volume (area) of the system, and we have normal-ordered the Hamiltonian by subtracting off the (UV divergent) zero-point energy density \begin{equation} \varepsilon_{\rm vac}=-\frac{\kappa}{2} \lim_{\vec y \to \vec x} \nabla_x^2 \delta(\vec x-\vec y)>0 \ . \end{equation} Any state annihilated by $Q(x)$ for all $x$ must be a zero-energy ground state. The corresponding ground-state wave functional $\ipr{[\varphi]}{{\rm vac}}=\Psi_0[\varphi]$ satisfies $Q\Psi_0 [\varphi] =0 $, where $Q$ is defined in Eq.\ (\ref{eq:QAbelian}). This is simply a first-order functional differential equation, and is easily solved, giving \begin{equation} \Psi_0[\varphi] = \frac{1}{\sqrt{{\mathcal Z}}}e^{\displaystyle{-\frac{\kappa}{2} \int d^2x \left(\nabla \varphi(x)\right)^2}} \label{eq:Psi0} \end{equation} where ${\mathcal Z}$ is the normalization \begin{equation} {\mathcal Z}=\int [{\mathcal D} \varphi] \; e^{\displaystyle{-\kappa \int d^2x \; \left(\nabla \varphi\right)^2}}\ . \label{eq:Zboson} \end{equation} The probability of finding the ground state in the configuration $\ket{[\varphi]}$ is therefore \begin{equation} \big\vert \Psi_0[\varphi]\big\vert^2=\frac{1}{{\mathcal Z}} e^{\displaystyle{-\kappa \int d^2x \; \left(\nabla \varphi\right)^2}}\ . \label{eq:probab} \end{equation} Consequently, the ground state expectation value of products of Hermitian local operators ${\mathcal O}[\varphi(\vec x)]$ reduces to expressions of the form \begin{equation} \me{{\rm vac}}{ {\mathcal O} [\varphi( {\vec x}_1)] \ldots {\mathcal O}[\varphi( {\vec x}_n)]}{{\rm vac}} =\frac{1}{\mathcal Z} \int [{\mathcal D} \varphi] \; {\mathcal O}[\varphi(\vec x_1)] \ldots {\mathcal O}[\varphi(\vec x_n)] \; e^{\displaystyle{-\kappa \int d^2x \; \left(\nabla \varphi\right)^2}} \ . \label{eq:mapping} \end{equation} This two-dimensional quantum theory has a deep relation with a two-dimensional classical theory: the ground-state expectation value of all local observables are mapped one-to-one to correlators of a two-dimensional massless Euclidean free boson. The latter is a well-known conformal field theory, and its correlation functions are easily determined (for convenience we give them explicitly in Appendix \ref{app:gaussian}). This two-dimensional critical field theory is conformally-invariant, so the equal-time correlators of the quantum theory must reflect this. This scalar field theory is therefore not only quantum critical but it also has a time-independent conformal invariance. The equal-time expectation values of the ``charge operators" ${\mathcal O}[\varphi]$, as well as the correlation functions of the dual vortex (or ``magnetic") operators discussed in Appendix \ref{app:gaussian}, exhibit a power-law behavior as a function of distance, as expected at a quantum critical point. As shown also in Appendix \ref{app:gaussian}, their autocorrelation functions also exhibit scale invariance albeit with a dynamic critical exponent $z=2$. This behavior of the equal-time correlator was shown earlier to occur in the quantum dimer model on the square lattice at the RK point: there is a massless ``resonon'' excitation, and the equal-time correlation functions for two static holons has a power law behavior equal to that of the monomer correlation function in the classical 2D dimer model on the square lattice. However, not all theories whose ground state can be found in this fashion need be critical with power-law behavior. As noted above, the quantum dimer model on the triangular lattice is not. As we will discuss in section \ref{sec:2dwavefn}, adding say a mass-like term to $Q(x)$ in the scalar field theory gives a theory with exponentially-decaying correlations in the ground state. Thus one can understand phase transitions in such theories as well, and we will explore several of these in this paper. Nevertheless, we expect that the quantum critical points, that is provided the quantum phase transition is continuous, of generic theories of this type to have the basic structure of the quantum Lifshitz model. As discussed in the introduction, only quantum Lifshitz points can be conformal quantum critical points.\footnote{The role of conformal invariance in 2D classical dynamics with $z=2$ and anisotropic 3D classical Lifshitz points was considered recently in Refs.\ [\onlinecite{henkel02,henkel03}].} In the next sections we will show that generalizations of this $z=2$ quantum Lifshitz Hamiltonian also describe the quantum phase transitions between generalizations of the valence bond crystal states and quantum disordered states which describe deconfined topological fluid phases (provided these quantum phase transitions are continuous). Notice that phase transitions from deconfined to confined, uniform and translationally-invariant states are described by the standard Lorentz invariant $z=1$ critical point of gauge theories [\onlinecite{fradkin78,fradkin-shenker79,kogut79}]. These examples show that one can obtain precise information about some 2d quantum systems in terms of known properties of 2d classical systems. The trick of doing so is in finding a set of $Q_i$ (or $Q(x)$ in the continuum) which annihilates the equal-amplitude state (or some other desired state), and then defining $H = \sum_i Q^\dagger_i Q_i$ [\onlinecite{arovas91}]. This seems like it should be possible to do for any classical 2d theory, and indeed, there are many known examples of this sort. However, it is not clear for a given 2d classical theory one can always find a Hamiltonian which is both local and ergodic (in this context, ergodic means that the Hamiltonian will eventually take the system through all of phase space with a given set of conserved quantum numbers). It is also not clear that even if such a Hamiltonian exists, whether it will have any physical relevance. Moreover, this simple relation of the ground-state wave function of a 2d quantum system to a 2d classical system is not at all generic: the quantum dimer model with $H_{RK}$ and this quantum Lifshitz field theory are quite special. To illustrate this, let us discuss briefly the ground-state wave functions of standard quantum field theories at a (quantum) critical point. Consider first the most common case, the Lorentz invariant $\varphi^4$ field theory at criticality. Below $D=4$ space-time dimensions this critical theory is controlled by its non-trivial Wilson-Fisher fixed point. The resulting theory is massless and in general it has an anomalous dimension $\eta \neq 0$. Scale and Lorentz invariance fully dictate the behavior of {\it all} the correlation functions at this fixed point. General fixed point theories are scale invariant and, in addition, they exhibit an enhanced, generally finite-dimensional, conformal symmetry. It is a very special feature of $D=1+1$-dimensional Lorentz-invariant fixed point theories that they exhibit a much larger, infinite-dimensional, conformal invariance. This enhanced symmetry leads to a plethora of critical behaviors in $1+1$ dimensions. In contrast, there are relatively few known distinct critical points in higher dimensions for Lorentz-invariant field theories. It is well known that the knowledge of all the equal-time correlation functions determines completely the form of the ground state wave function, {\it i.e.\/} in the Schr\"odinger representation of the field theory [\onlinecite{symanzik81,symanzik83,fradkin93a}]. For a general theory, the ground-state wave function is a non-local and non-analytic functional of the field configuration. Thus, at the Wilson-Fisher fixed point, which describes theories with only a global conformal (scale) invariance, the structure of the ground state wave function is quite complicated. For instance, the probability of a {\rm constant} field configuration $\varphi(\vec x)=\varphi$ at a critical point has the universal form [\onlinecite{fradkin93a}] \begin{equation} \left\vert \Psi_{\rm vac}(\varphi)\right\vert^2=A \; e^{\displaystyle{-B \; \vert \varphi \vert^{1+\delta}}} \label{eq:constant} \end{equation} For a Lorentz-invariant $\varphi^4$ theory the universal critical exponent is given by $\delta=(d+2-\eta)/(d-2+\eta)$, and $A$ and $B$ are two non-universal constants. Therefore, at criticality the wave function in general is a non-analytic non-local functional of the field configuration. %This is to be expected since {\em local} conformal invariance is natural only in $1+1$ dimensions. In contrast, the ground-state wave function of a $1+1$-dimensional relativistic interacting fermions (a Luttinger liquid), which is a conformal field theory, has a universal non-local non-analytic Jastrow-like power law factorized form [\onlinecite{fradkin93b}] consistent with the form found by the Bethe-Ansatz solution of the Calogero-Sutherland model [\onlinecite{sutherland71,haldane88,shastri88}]. This structure is a consequence of the (local) conformal invariance of the $1+1$-dimensional theory. Notice that, even in $1+1$ dimensions the wave function is non-local. \section{Dimers, fermions and the quantum Lifshitz field theory} \label{sec:2dwavefn} In Section \ref{sec:scale-wf}, we showed how to find the exact ground-state wave functions of the quantum dimer model with Rokhsar-Kivelson Hamiltonian $H_{RK}$, and the quantum Lifshitz scalar field theory. In this section, we will describe their properties in more detail, and some simple generalizations. \subsection{From the square to the triangular lattice} \label{sec:from} The ground states of $H_{RK}$ are the sum over all classical dimer configurations in a sector with equal amplitudes [\onlinecite{rokhsar88}]. There are a number of useful generalizations to models where the ground state is still a sum over all states in a sector, but not necessarily with equal amplitudes. One interesting case is a quantum dimer model which interpolates between the square and triangular lattices. A triangular lattice can be made from a square lattice by adding bonds across all the diagonals in one direction. The classical dimer model on the square lattice can be deformed continuously into the triangular-lattice model by assigning a variable weight $w$ for allowing dimers along these diagonals: for $w=0$ we have the original square-lattice model, while $w=1$ gives the triangular-lattice one [\onlinecite{fendley02}]. There is a two-dimensional quantum Hamiltonian which has the $w$-dependent classical dimer model as its ground state. For dimers on opposite sides of a plaquette of the original square lattice, the Hamiltonian remains $H_{RK}$. In addition, however, parallel dimers on adjacent diagonals can also be flipped [\onlinecite{moessner01a}], see figure \ref{triflips}. %%%%%%%%%%%%%%%%%% \begin{figure}[h] \begin{center} \psfrag{a}{$\longleftrightarrow$} \psfrag{h}{\hspace{-.5cm} horiz} \psfrag{v}{\hspace{-.4cm} vert} \psfrag{s}{\hspace{-.6cm} square} \includegraphics[width= .7\textwidth]{triflips.eps} \caption{Dimer flips on the triangular lattice} \label{triflips} \end{center} \end{figure} %%%%%%%%%%%%%%%%%% Like the flip on the square lattice, this flip can be done without violating the close-packing and hard-core constraints. When the off-diagonal terms in the Hamiltonian $H_w$ consist solely of these flips, $H_w$ breaks up into two-by-two blocks like $H_{RK}$ does. In the ground state dimers along diagonals should get a weight $w$, so the Hamiltonian $H_{w}$ must explicitly depend on $w$. To construct a Hamiltonian with the desired ground state, we find a set of operators $Q_i$, each of which annihilates this state. For example, denote a configuration with dimers on adjacent diagonals as $|1\rangle$, and $|2\rangle$ as the configuration to which it is flipped, as shown in fig.\ \ref{triflips}. The $Q_i$ acting on these two configurations is then \begin{equation} Q_i = \frac{1}{w^2+w^{-2}} \begin{pmatrix} w^{-2}&-1\\ -1&w^2 \end{pmatrix} \label{Qtriang} \end{equation} where the first row and column correspond to state $|1\rangle$, while the second corresponds to state $|2\rangle$. There are three types of $Q_i$: $Q^{(\hbox{square})}_i$ acts on dimers on opposite sides of a plaquette of the original square lattice, while $Q^{(\hbox{horiz})}_i$ and $Q^{(\hbox{vert})}_i$ are associated to flips involving dimers on the `diagonal' links; see figure \ref{triflips}. The operator $Q^{(\hbox{square})}_i$ is given by (\ref{Qtriang}) with $w=1$. We then take \begin{equation} H_w = \sum_i \left[Q^{(\hbox{square})}_i + w Q^{(\hbox{vert})}_i + wQ^{(\hbox{horiz})}_i \right] \end{equation} where the sum is over all plaquettes $i$. Each $Q_i$ is a projection operator, so $Q_i=Q^\dagger_i =(Q_i)^2$. We have defined the operator $Q_i$ so that it annihilates the state $w^2|1\rangle\ +\ |2\rangle$. The ground state of $H_w$ is then the sum over classical dimer model states with each state weighted by $w^{\mathcal D}$, where ${\mathcal D}$ is the number of dimers along diagonals in that state. More precisely, one must find the conserved quantities for a given value of $w$ and boundary conditions; the sum over all states with weight $w^{\mathcal D}$ in that sector is an eigenstate of $H_{RK}$ with zero energy. The ground-state wave function for a configuration with ${\mathcal D}$ diagonal dimers is then \begin{equation} \Psi_0[{\mathcal D}] = \frac{w^{\mathcal D}}{\sqrt{Z(w^2)}} \label{psiD} \end{equation} where $Z(w)$ is the partition function for the classical dimer model with diagonal dimers receiving weight $w$. Note the $w^2$ in the argument of $Z$ in the denominator: this is because in quantum mechanics probabilities are given by $|\Psi_0|^2$. For the square or triangular lattice, this is unimportant, because $w=0$ or $w=1$, both of which have $w^2=w$. For $w=1$, we have the equal-amplitude sum over all dimer states of the triangular lattice, the model discussed in Ref.\ [\onlinecite{moessner01a}]. For $w = 0$, we recover a slight generalization of the original square-lattice quantum-dimer model of Rokhsar and Kivelson [\onlinecite{rokhsar88}]. In this limit, this Hamiltonian reduces to $H_{RK}$ plus a potential term forbidding dimers on adjacent diagonal links. Isolated diagonal dimers are still allowed for $w\to 0$, but since none of them can be flipped, the Hamiltonian does not affect them at all. Thus they can be viewed as fixed zero-energy defects in the square-lattice quantum-dimer model. The ground-state wave function for a given set of defects is the equal-amplitude over all configurations of dimers on the sites without defects. Since we know the exact ground-state wave function for any $w$, one would like to compute the correlation functions in this model. In most two-dimensional lattice models, even those solvable by the Bethe ansatz, this is extraordinarily difficult or impossible. However, the classical dimer model is special in that one can do such computations, because like the two-dimensional Ising model, it is essentially free-fermionic. Precisely, its partition function and correlators in the classical dimer model can be written in terms of the Pfaffian (the square root of the determinant) of known matrices [\onlinecite{fishersteph}]. One can rewrite the Pfaffians in terms of a functional integral over Grassmann variables at every site on the lattice [\onlinecite{samuel}]. The action in the case of equal Boltzmann weights is quadratic in the Grassmann variables, so one can compute easily any ground-state correlation function using the dimers, because the dimers can be written in terms of the fermions. This was discussed for the triangular lattice in [\onlinecite{moessner01a}]. The correlators of spinon-like or holon-like excitations are much more complicated, but the computation was done for $w=0$ in [\onlinecite{fishersteph}], and for arbitrary $w$ in [\onlinecite{fendley02}]. On the lattice, the holon is a defect or monomer, a site without a dimer. The holon-creation operators are not local in terms of the fermionic variables, in a manner reminiscent of how the spin and fermion operators are non-local with respect to each other in the two-dimensional Ising model. The holon two-point function is valuable in that it gives an order parameter for the phase with topological order: if it is non-vanishing as two holons are taken far apart, the holons are deconfined and we are indeed in a topological phase. The existence of topological order was previously established for the triangular lattice $w=1$ [\onlinecite{moessner01a,moessner02a}]; in [\onlinecite{fendley02}] the explicit correlator was computed, and indicates the topologically-ordered phase exists for any non-zero $w$. The ground state of the quantum dimer model with $H_{w}$ is therefore well understood for any $w$. There are no exact results for the excited states, however. In fact, since $H_w$ does not have any action upon empty sites, one can give the holon any gap desired without changing the ground states. It is therefore useful to find a continuum limit and study the field theory describing this model. In other words, we would like to understand a field theory with partition function equal to the continuum limit of partition function $Z(w)$ in Eq.\ (\ref{psiD}). Since ground-state correlators for the square lattice are algebraically decaying, the $w=0$ model is critical and should have a sensible continuum limit. Indeed, when $w=0$, the Grassmann variables turn into a single free massless Dirac fermion field; its action is the usual rotationally-invariant kinetic term. The dimer correlation length $\xi$ was computed exactly as a function of $w$; for the triangular lattice it is about one lattice spacing, while $\xi$ diverges as $1/w$ for $w$ small [\onlinecite{fendley02}]. Thus there is a field-theory description of the continuum limit of the classical dimer model valid as long as $w$ is scaled to zero with the lattice spacing $a$ such that $w/a$ remains finite. Since the action is still quadratic in the Grassmann variables, the resulting fermionic field theory remains free. However, the Dirac fermion receives a mass proportional to $w/a$ [\onlinecite{fendley02}]. There are several useful aspects of taking the continuum limit, apart from finding the excited-state spectrum. Correlators are easier to compute: for $w=0$ one can use conformal field theory [\onlinecite{yellow}], while in the scaling limit $w\to 0$ one can use form-factor techniques [\onlinecite{formfac}]. Another useful fact is that (ignoring boundary conditions), a Dirac fermion can be described in terms of two decoupled Ising field theories. In the continuum, the holon can be written in terms of the product of the spin field in one Ising model with the disorder field in the other Ising model. Taking $w$ away from zero amounts to giving one Ising order field an expectation value, and the other disorder field an expectation value. [\onlinecite{fendley02}]. Thus one can see directly in continuum that the holon order parameter is non-vanishing for $w\ne 0$. We will see in the next section that one can also understand the physics of the quantum Lifshitz critical line for all $\kappa$ in terms of two (coupled) Ising models. \subsection{The critical field theory} \label{critft} We would therefore like to find a natural-looking quantum field-theory Hamiltonian which has as its ground-state wave functional \begin{equation} \Psi_0[\psi] = \frac{e^{-S_{Dirac}[\psi]}}{\sqrt{Z_{Dirac}}} \label{psidirac} \end{equation} where $S_{Dirac}$ is the usual action for a rotationally-invariant action for a free Dirac fermion in two Euclidean dimensions. Since this wave functional involves Grassman numbers, the easiest way to think of $|\Psi_0|^2$ as a weight in the path integral defining all correlators. While it is possible to find a Hamiltonian acting on this fermionic basis, it is more convenient and more intuitive to instead use bosonic variables. It is more convenient because correlators in a massless Dirac fermion theory (including those involving the product of spin fields) can be bosonized, meaning that they can be written in terms of correlators of free scalar fields [\onlinecite{kadanoff79,yellow}]. It is more intuitive because the classical dimer model on the square lattice has a simple description in terms of a ``height'' variable [\onlinecite{henley97a}]. A height is an integer-valued variable, which typically in the continuum limit turns into a scalar field. This description will allow us also to make contact with the quantum Lifshitz model. Recently, Moessner {\it et al.\/} [\onlinecite{moessner02a}] generalized Henley's argument of Ref. [\onlinecite{henley97a}], and used the connection between quantum dimer models and their dual quantum roughening (height) models [\onlinecite{fradkin90a,fradkin90b,fradkin91,read-sachdev89,read-sachdev90,zheng-sachdev89,levitov90,elser89}] to argue that the Hamiltonian of Eq.\ (\ref{eq:Hboson}) actually defines the universality class of quantum critical points between valence bond crystal phases. The nature of the phase transition between valence bond crystal states is the focus of much current research. Quite recent results by Vishwanath {\it et al.\/} [\onlinecite{vishwanath03}] \footnote{We became aware of Ref. [\onlinecite{vishwanath03}] as this paper was being completed.}, and by Fradkin {\it et al.\/} [\onlinecite{fradkin03}], show that the transition between valence bond crystals is generically first order, as expected from a simple Landau argument. Nevertheless, when the transition is continuous, it is described by the Hamiltonian of Eq.\ (\ref{eq:Hboson}) which must be regarded as a (rather rich) multicritical point. To see how the height description arises, let us go back to the square-lattice quantum dimer model, where $H_w$ reduces to $H_{RK}$. To map the square-lattice classical dimer model onto a height model, one first assigns a height variable to each plaquette. In going around a vertex on the even sub-lattice clockwise, the height changes by $+3$ if a dimer is present on the link between the plaquettes, and by $-1$ if no dimer is present on that link. On the odd sub-lattice, the heights change by $-3$ and $+1$ respectively. The flip operator $\hat{F}_i$ on a plaquette $i$ changes the height on that plaquette by either $\pm 4$. To take the continuum limit, is convenient to turn this into a model with heights on the sites. We define $h$ on each site to be the average value of the four plaquette heights around that site\footnote{This construction has been employed extensively in statistical mechanics models of classical dimers and loops; see for instance [\onlinecite{kondev-henley94}].}; see figure \ref{heights}. \begin{figure}[ht] \begin{center} \setlength{\unitlength}{1 mm} \begin{picture}(80,20)(0,-8) \multiput(0,5)(25,0){4}{\line(1,0){10}} \multiput(5,0)(25,0){4}{\line(0,1){10}} \linethickness{1 mm} \put(5,5){\line(0,1){5}} \put(30,5){\line(1,0){5}} \put(55,5){\line(0,-1){5}} \put(80,5){\line(-1,0){5}} \multiput(2,7)(25,0){4}{$0$} \put(7,7){$3$} \put(32,7){$-1$} \put(57,7){$-1$} \put(82,7){$-1$} \put(7,1){$2$} \put(32,1){$2$} \put(57,1){$-2$} \put(82,1){$-2$} \put(2,1){$1$} \put(27,1){$1$} \put(52,1){$1$} \put(74,1){$-3$} \put(1,-8){$h=3/2$} \put(26,-8){$h=1/2$} \put(50,-8){$h=-1/2$} \put(76,-8){$h=-3/2$} \end{picture} \caption{4 height configurations} \label{heights} \end{center} \end{figure} To avoid overcounting configurations, we identify the height $h$ with $h+4$. The flip operator $\hat F$ corresponds to changing $h\to h \pm 1$ on all four sites around a plaquette (the $\pm$ depending on the sub-lattice). Columnar order for dimers corresponds to an expectation value for $h$, while staggered order for dimers corresponds to an expectation value for $\partial h$. One can obtain a Hamiltonian with ordered ground states by allowing the coefficients of the two terms in Eq.\ (\ref{HRK}) to be different. If the coefficient of $\hat F_i$ is larger, this favors columnar order; if the coefficient of $\hat V_i$ is larger, staggered order is favored. It is widely assumed that $H_{RK}$, where the coefficients are equal, describes a phase transition between the two kinds of order [\onlinecite{fradkin90a,fradkin90b,read-sachdev90,sachdev-park02}]. However, this has never been proven, and there exists the possibility of an intermediate ``plaquette" phase[\onlinecite{sachdev-jalabert90}]. We would now like to take the continuum limit of the quantum dimer model in its height description. In this limit, we identify the height $h$ with a scalar field $4\varphi(x)$. Like the height, the scalar field must be periodic, so we identify $\varphi$ with $\varphi +1$. We have already noted that the continuum correlators of the square-lattice quantum dimer model are those of a massless Dirac fermion. The correlators in the ground state of the quantum Hamiltonian in terms of $\varphi$ must be identical. The quantum Lifshitz Hamiltonian Eq.\ (\ref{eq:Hboson}) and Eq.\ (\ref{eq:HQQ}) has ground-state correlators of the form given in Eq. (\ref{eq:mapping}). When $\kappa^{-1}=2\pi$, these correlators are precisely those of a Dirac fermion; this is a result of the widely-known procedure known as bosonization [\onlinecite{yellow}]. In fact, correlators of many two-dimensional critical classical statistical mechanical systems, not just free fermions, can be written in terms of exponentials of a free boson [\onlinecite{kadanoff79}]; we collect some of these results in Appendix \ref{app:gaussian}. In the next section we will display quantum lattice models whose continuum limit corresponds to all values of $\kappa$. The Hamiltonian $H_{RK}$ on the square lattice in the continuum limit is therefore identified with the quantum Lifshitz Hamiltonian with $\kappa^{-1}=2\pi$. This also allows a qualitative understanding of the physics away from the RK point [\onlinecite{henley97a}]. A phase with staggered order should have an expectation value of $\partial_x \varphi$ or $\partial_y \varphi$ in the continuum theory. Adding a term $(\nabla\varphi)^2$ to the Hamiltonian with negative coefficient will drive the system into such an ordered phase. Adding this term with positive coefficient will favor a constant value of $\varphi$, driving the system into columnar order. Adding terms like $\cos(2\pi\varphi)$ will also drive the system into a phase with columnar order. Thus one expects that at a critical point like that described by $H_{RK}$ on the square lattice, the coefficients of $(\nabla\varphi)^2$ and $\cos(2n\pi\varphi)$ will vanish. This leaves Eq.\ (\ref{eq:Hboson}) as the simplest non-trivial Hamiltonian with the desired properties. The requirement that the ground state be equivalent to a free fermion then fixes the coefficient $\kappa$; note that if desired $\kappa$ can scaled out of the Hamiltonian by redefining the compactification relation to be $\varphi\sim \varphi + \sqrt{\kappa}$. It is not at all clear whether critical $2+1$-dimensional field theory and the continuum limit of $H_{RK}$ are identical for excited states, although the above heuristic argument is very suggestive. The Hamiltonian for the scalar field theory Eq.\ (\ref{eq:HQQ}) is purely quadratic in the field $\varphi$, so one can obtain essentially any desired information exactly. $H_{RK}$ is not so simple on the lattice, but one may hope the two models are in the same universality class. In any event, we can easily extract all the excited-state energies for the quantum Lifshitz field theory. The operators $Q(x)$ and $Q^\dagger(x)$ are essentially harmonic-oscillator creation and annihilation operators: the equal-time commutation relation Eq.\ (\ref{eq:ccr}) implies that \begin{equation} \left[ Q(\vec x), Q^\dagger(\vec y) \right] = \kappa \; \nabla^2\delta^{(2)} (\vec{x} - \vec{y}) \label{QQccr} \end{equation} The ground state is indeed annihilated by all $Q(x)$, so we can create excited states by acting with $Q^\dagger(x)$. The commutation relations Eq.\ (\ref{QQccr}) mean that the dispersion relation is $E = \kappa p^2$. This theory is gapless but not Lorentz-invariant: the dynamical critical exponent is $z=2$. The exponent $z=2$ can also be seen by looking at the classical action associated with this Hamiltonian. This will also allow us to make contact with the earlier three-dimensional statistical-mechanical results. The action consistent with the Hamiltonian of Eq.\ (\ref{eq:Hboson}) and the canonical commutation relations of Eq.\ (\ref{eq:ccr}) is \begin{equation} {\mathcal S}=\int d^3x \left[\12 \left(\partial_t \varphi\right)^2-\frac{\kappa^2}{2}(\nabla^2\varphi)^2\right] \label{eq:Sboson} \end{equation} Clearly, this action is not Lorentz invariant and has $z=2$. It is rotationally invariant only in the XY plane. Defining the imaginary time $\tau=it$ gives the Euclidean action (\ref{3D-Euclidean}). %\begin{equation} %{\mathcal S}_{\rm euclidean}=\int d^3x \left[ \12 \left(\partial_{\tau} %\varphi\right)^2+\frac{\kappa^2}{2}(\nabla^2\varphi)^2\right] %\label{eq:Sboson-euclidean} %\end{equation} The imaginary time axis $\tau$ can be regarded as the $z$-coordinate of a three-dimensional classical system in which $\varphi(\vec x,\tau)$ is an angle-like variable and the action represents the spin-wave approximation of an anisotropic classical $XY$ model. In general one would have expected a term proportional to the operator $\left(\nabla \varphi\right)^2$ with a finite positive stiffness in the plane. This is so in the $XY$ ferromagnetic phase. On the other hand, if the stiffness becomes negative, there is an instability to a modulated helical phase. The action of Eq.\ (\ref{3D-Euclidean}) represents the Lifshitz point, the critical point of this phase transition [\onlinecite{grinstein81a,grinstein82a}] where the stiffness vanishes. This effective action also plays a central role in the smectic A-C transition [\onlinecite{grinstein81b}] and in other classical liquid crystal phase transitions associated with the spontaneous partial breaking of translation and/or rotational invariance [\onlinecite{lubensky95,degennes98}]. The compactified version of the problem (\ie the identification $\varphi\sim \varphi + 1$) has also been considered in this context, for example in Ref.\ [\onlinecite{grinstein81a}]. The choice of period in general depends on the physical context of the problem. The square-lattice quantum dimer model and the scalar field theory with Hamiltonian Eq.\ (\ref{eq:HQQ}) are both at critical points, in that the correlators in the ground state are algebraically decaying. The quantum dynamics implied by this Hamiltonian must be {\em compatible} with the 2D time-independent conformal invariance. In particular, the spectrum of the quantum theory must be gapless and, as we learned from this example, the dynamic critical exponent must be $z=2$. Notice however, that $z=2$ alone does not guarantee a gapless (or even critical) theory. Indeed, instructive counter-examples to this statement are well known in the theory of (the absence of) quantum roughening [\onlinecite{fisher83,fradkin83}] where quantum fluctuations destroy the critical behavior and lead to an ordered state through an order-from-disorder mechanism. \subsection{The off-critical field theory} \label{offcritft} We have thus shown that the continuum limit of the square-lattice quantum dimer model is described by the quantum Lifshitz model at a special point $\kappa^{-1}=2\pi$. We also argued that at least some deformations of the two models result in ordered phases. However, we saw at the beginning of this section that not all deformations of the square-lattice quantum dimer model result in an ordered phase. Allowing dimers across the diagonals with Hamiltonian $H_w$ results in a topologically-ordered phase, where the order parameter is not local. In this subsection, we find a bosonic field theory describing the topological phase in the continuum limit. We showed above that in the scaling limit $w\to 0$ with $w/a$ finite, the ground-state wave function Eq.\ (\ref{psidirac}) can be written in terms of a free massive Dirac fermion of mass proportional to $w/a$. In two dimensions, the bosonic version of a massive Dirac fermion is the sine-Gordon model at a particular coupling. Precisely, the two-dimensional fermion action is equivalent to \begin{equation} S_{2d} = \int d^2 x \left[ {\kappa}(\nabla\varphi)^2 - \lambda \cos(2\pi\varphi) \right] . \label{sgaction} \end{equation} For the free-fermion case, we have $\kappa=1/2\pi$; we will discuss the more general case in section \ref{sec:q8v}. In the fermion language, different values of $\kappa$ correspond to adding a four-fermion coupling to $S_{2d}$. To find a Hamiltonian with this two-dimensional action describing the ground state, we again find an operator $Q(x)$ and define the Hamiltonian via Eq.\ (\ref{eq:HQQ}). The operator \begin{equation} Q (x) \equiv \frac{1}{\sqrt{2}} \left(\frac{\delta}{\delta\varphi} + \kappa \; \nabla^2\varphi + \frac{\lambda}{2\pi} \sin(2\pi\varphi)\right) \label{Qsg} \end{equation} annihilates the wave functional $\Psi \propto e^{-S_{2d}}$. Because of the extra term in $Q$, the commutator $\left[ Q(\vec x), Q^\dagger(\vec y) \right]$ is not a simple c-number, but in fact depends on the field configuration $\varphi$, \begin{equation} \left[ Q(\vec x), Q^\dagger(\vec y) \right] = \kappa \nabla^2\left(\delta^{(2)} (\vec{x} - \vec{y}) \right) + \lambda \sin(2\pi \varphi(\vec{x})) \delta^{(2)}(\vec{x}-\vec{y}) \ . \label{QQccrpot} \end{equation} Thus, normal-ordering the Hamiltonian is not just an innocent ground state energy shift: the two parts of (\ref{eq:HQQ}) are not the same here. To obtain the desired ground-state wave functional, we must define $H$ of the form $\int Q^{\dagger} Q$. The three-dimensional version of this model was discussed in Ref.\ [\onlinecite{grinstein81a}]. This Hamiltonian is not quadratic in the field $\varphi$ except at the critical point $m=0$, so that even with this fine tuning this model cannot be solved simply. Since $Q$ and $Q^\dagger$ do not have simple commutation relations, we cannot simply find the spectrum of this theory. Of course, one can compute properties in the fermionic picture, but as noted before, computations involving spin fields are non-trivial in this basis as well. However, since there are dimensionful parameters in the Hamiltonian and no spontaneous breaking of a continuous symmetry, it seems likely that the Hamiltonian is gapped. Moreover, in the limit %$\kappa\to \infty$, $\lambda\to\infty$ with $\lambda/\kappa$ finite, the action Eq.\ (\ref{sgaction}) reduces to that of a free massive boson. Then one can solve the model explicitly, and the quantum Hamiltonian indeed has a gap. When $\lambda$ is reduced to a finite value, the gap should remain\footnote{The equal-time correlators of the vertex operators defined and computed in Appendix \ref{app:gaussian} can also be computed away from the critical point by either by a naive semi-classical argument, which predicts the simple exponential decay we just discussed, or in a more sophisticated way by means of form factors [\onlinecite{formfac}]. Apart from a subtle bound state structure in the spectral functions, the more sophisticated approach confirms the essence of the naive semi-classical result. On the other hand, the $z=2$ character of the critical theory suggests that time-dependent correlation functions must be consistent with this fact, and that the correlators must be functions of $x^2$ or $t$, and that the time dependent Euclidean auto-correlation functions may obtained from the equal-time correlator by replacing $x^2 \leftrightarrow |t|$. These arguments are consistent with the renormalization-group results of Ref.\ [\onlinecite{grinstein81a}]}. There are many terms in the Hamiltonian of this field theory, and their coefficients must be fine-tuned to enable us to compute the ground-state wave function explicitly. There are terms like $\cos(4\pi\varphi)$ and $(\nabla\varphi)^2$ which, as noted above, tend to order the system. However, the exact lattice results for $H_w$ from [\onlinecite{fendley02}] show that the ground-state correlators for $w$ small are those of the bosonic Hamiltonian with these special couplings. Thus this model is not ordered but rather topologically ordered. Moreover, since the model is gapped, we expect that its physical properties are robust and persist even when the coefficients are tuned away from this special point. Thus there must exist a topological phase, not just an isolated point. An interesting open problem is to understand how large (in coupling constant space) the topological phase is, as compared to the ordered phases. \section{A quantum eight-vertex model} \label{sec:q8v} In the last section, we discussed a $2+1$-dimensional theory whose ground-state wave function is simply described in terms of a classical two-dimensional bosonic field. With vanishing potential and a particular value of the coupling $\kappa$, it is believed to describe the continuum limit of the quantum dimer model on the square lattice. In this section, we will study lattice models which in the continuum limit allow arbitrary values of $\kappa$. These models also have a quantum critical line separating an ordered phase from a topologically-ordered phase. The degrees of freedom in our model are those of the classical two-dimensional eight-vertex model. These are arrows placed on the links of a square lattice, with the restriction that the number of arrows pointing in at each vertex is even. This means that there are eight possible configurations at each vertex, which we display in figure (\ref{fig8v}). %%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] \begin{center} \includegraphics[width= .9\textwidth]{8v.eps} \caption{The eight vertices and their Boltzmann weights } \label{fig8v} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The classical Boltzmann weights for a given vertex in the zero-field eight-vertex model are usually denoted by $a$, $b$, $c$ and $d$, as shown in the figure. Since we are interested in rotationally-invariant theories, we set $a=b$ in the following; moreover, since we can rescale all the weights by a constant, we set $a=b=1$. A typical configuration is displayed in figure (\ref{fig8vtypical}); %%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[ht] \begin{center} \includegraphics[width= .4\textwidth]{8vtypical.eps} \caption{A typical configuration in the eight-vertex model} \label{fig8vtypical} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%% the Boltzmann weight of such a configuration is given by the product of Boltzmann weights of the vertices. The classical eight-vertex model is integrable, and many of its properties can be derived exactly [\onlinecite{baxbook}]. For $a=b=1$, it has ordered phases for $c>d+2$ and $d>c+2$. In these phases the ${\mathbb Z}_2$ symmetry of flipping all the arrows is spontaneously broken. Critical lines with continuously varying exponents at $c=d+2$ and $d=c+2$ separate the ordered phases from the disordered one $|c-d|<2$. The correlation length diverges as [\onlinecite{baxbook}] \begin{equation} \xi \sim \big| |c-d| -2 \big|^{-\pi/(2\mu)}\ , \qquad \mu \equiv 2 \tan^{-1}(\sqrt{cd})\ , \label{critexp} \end{equation} near these critical lines. (For $\pi/\mu$ an even integer this is multiplied by $\log | |c-d| -2 |$ .) When $c=0,d\leq 2$ or $d=0,c\leq 2$, the model is also critical; in fact the partition function on this line can be mapped onto that for the order-disorder critical line. For $d=0$, the exponent in Eq.\ (\ref{critexp}) diverges: there is a Kosterlitz-Thouless transition as one brings $c$ through $2$. Another useful result for the classical correlation length is that it is zero on the line $c=d$; this is the state of maximal disorder. An order parameter which will be useful later comes by rewriting the model in terms of an Ising spin at the center of each plaquette. This description is best thought of as two Ising models, with spins $\tau(A)$ on one sublattice, and $\tau(B)$ on the other. Then the Boltzmann weights can be written in terms of two Ising couplings between nearest sites on the $A$ lattice and on the $B$ lattice, and a four-spin coupling between the two $A$ and two $B$ spins around a site of the original lattice. The polarization operator of the eight-vertex model becomes $\tau(A)\tau(B)$. One finds that its expectation value is non-vanishing in the ordered phase, and vanishes in the disordered phase $|c-d|<2$ [\onlinecite{baxbook}]. One also can define Ne\'el-like staggered order parameters, in terms of $\tau(A)$ and $\tau(B)$ individually, which do not vanish in the ordered phase. Along the line $cd=1$, the four-spin coupling vanishes, so the eight-vertex model turns into two decoupled Ising models; the model here in this case be solved by using Pfaffian techniques [\onlinecite{FanWu}]. Along the line $c=d$, the two Ising couplings vanish, leaving only the four-spin coupling. Thus this line in the classical model has an extra ${\mathbb Z}_2$ gauge symmetry. The classical eight-vertex model has a number of useful dualities [\onlinecite{baxbook}]. They can be described by defining the combinations $W_1=(a+b)/2$, $W_2=(a-b)/2$, $W_3=(c+d)/2$ and $W_4=(c-d)/2$. The partition function is invariant under the exchange of any two of the $W_j$ and under the $W_j\to -W_j$ for any $i$. These dualities, for example, map the critical line $c=d+2$ to the critical line $d=0,c\le 2$ by exchanging $W_1$ with $W_4$. In Ising language this amounts to performing Kramers-Wannier duality on one of the two types of Ising spins. The line $c=d+2$ is invariant under the exchange $W_1\leftrightarrow W_3$. In Ising language, this duality amounts to taking the Kramers-Wannier dual of both types of Ising spins. Denoting the dual spins as $\mu(A)$ and $\mu(B)$, duality means therefore that in the disordered phases, the expectation value $\langle \tau(A) \tau(B)\rangle$ is non-vanishing. \subsection{Construction of the quantum eight-vertex Hamiltonian} \label{sec:8vqh} We now define a quantum Hamiltonian acting on a Hilbert space whose basis elements are the states of this classical eight-vertex model. To define such a Hamiltonian, we first need the analog of the flip operator in the quantum dimer model. A flip operator needs to be ergodic: by flips on various plaquettes one should be able to reach all the states with the same global conserved quantities. The simplest such operator for the eight-vertex model is the operator which reverses all the arrows around a given plaquette. We write this flip operator $\hat{\mathcal F}_i$ explicitly in gauge-theory language in appendix \ref{app:z2gauge}. Note that as opposed to the quantum dimer model on the square or triangular lattice, all configurations in the quantum eight-vertex model are flippable\footnote{However, the quantum dimer models on the Kagome [\onlinecite{misguich}] and Fisher [\onlinecite{moessner02d}] lattices have Hamiltonians where all plaquettes are flippable.}: $\hat{\mathcal F}_i$ preserves the restriction that an even number of arrows be pointing in or out at each vertex. The simplest Hamiltonian has no potential energy, just a flip term. It is convenient to write this in terms of a projection operator: for $I$ the identity matrix, we have $(I-\hat{\mathcal F}_i)^2 =2(I-\hat{\mathcal F}_i)$. Then the Hamiltonian \begin{equation} H_{c=d=1}= \sum_i (I-\hat{\mathcal F}_i) \label{hkitaev} \end{equation} has a ground state corresponding to the equal-amplitude sum over all eight-vertex model states. In terms of the Boltzmann weights introduced above, this is the state with $a=b=c=d=1$. A Hamiltonian with the same ground state was introduced by Kitaev [\onlinecite{kitaev97}]. There the eight-vertex-model restriction of having an even number of arrows in and out at each vertex was not required a priori, but instead a term was introduced giving a positive energy to vertices not obeying the restriction. The zero-energy ground state can therefore include no such vertices, so the ground state for the model of [\onlinecite{kitaev97}] is indeed the sum over the eight-vertex-model configurations with equal weights. Because every plaquette is flippable and all configurations have equal weights, different $\hat{\mathcal F}_i$ commute. Therefore all the terms in Eq.\ (\ref{hkitaev}) commute with each other, so they can be simultaneously diagonalized and their eigenstates can easily be found, as for the quantum Lifshitz field theory. The model is gapped, and is in a topologically-ordered phase [\onlinecite{kitaev97}]. This follows as well from the results for the classical eight-vertex model discussed above: for $c=d=1$ all Ising couplings vanish resulting in two decoupled Ising models at infinite temperature. At this point the order parameter vanishes, but the non-local order parameter does not. We can thus interpret the product of dual Ising variables $\mu(A)\mu(B)$ as a topological order parameter. This ground state can also be mapped onto the ground state of a ${\mathbb Z}_2$ gauge theory deep in its deconfined phase. In Appendix \ref{app:z2gauge}, we give a detailed derivation of the $ {\mathbb Z}_2$ gauge theory of the full quantum eight-vertex model we define below, as well as the its dual theory which we will use to characterize some of the phases. In Appendix \ref{app:gauge} we discuss the $U(1)$ gauge theory description of the quantum six-vertex model, which describes the limit $d=0$, and by duality, the lines $c=0$, $c^2=d^2+2$ and $d^2=c^2+2$. We now find a two-parameter quantum Hamiltonian whose ground state is a sum over the states of the eight-vertex model with amplitudes given by the classical Boltzmann weights with arbitrary $c$ and $d$. We are still keeping $a=b=1$ to preserve two-dimensional rotational invariance, but this restriction can be relaxed if desired. Our model is neither the simplest nor the most natural extension of the classical eight-vertex model: a simpler Hamiltonian was proposed by Chakravarty [\onlinecite{chakravarty}] in the context of $d$-density waves. Although we believe that our Hamiltonian and Chakravarty's describe the same physics, we are not aware of any simple mapping between these models. Another lattice model related to the quantum eight-vertex model discussed here was introducted in Ref.\ [\onlinecite{ioffe02}]. The main virtue of the construction that we use here is the structure of the ground-state wave function. Finding a quantum Hamiltonian with a known ground state is straightforward to do by using the trick discussed above (and in Ref.\ [\onlinecite{arovas91}]). Namely, we find a Hamiltonian of the form \begin{equation} \label{q8vham} H_{q8v} = \sum_i w_i {\mathcal Q}_i \end{equation} where ${\mathcal Q}_i={\mathcal Q}_i^\dagger \propto {\mathcal Q}_i^2$. To yield the desired ground state, each operator ${\mathcal Q}_i$ must annihilate the sum over states with each state weighted by $c^{N_c} d^{N_d}$, where $N_c$ and $N_d$ are the number of $c$ and $d$ type vertices in that state. In particular, we look for a ${\mathcal Q}_i$ of the form \begin{equation} {\mathcal Q}_i = \sum_i \left[\hat{\mathcal V}_i - \hat{\mathcal F}_i\right] \label{q8v} \end{equation} where $\hat{\mathcal V}_i$ is diagonal and depends on the Boltzmann weights for the four vertices at the corners of the plaquette $i$. Since $(\hat{\mathcal F}_i)^2=I$, this Hamiltonian breaks into $2$ by $2$ blocks like $H_{RK}$ for the quantum dimer model. If we choose the potential ${\mathcal V}_i$ so that the blocks are of the form \begin{equation} \begin{pmatrix} v & -1\\ -1 & v^{-1} \end{pmatrix} \ , \end{equation} ${\mathcal Q}_i$ will have the desired properties. To find $\hat{\mathcal V}_i$, let $n_c$ be the number of $c$-vertices at the corners of the plaquette $i$, and let $\widetilde{n}_c$ be the number of $c$-vertices around the plaquette after it is flipped by $\hat{\mathcal F}_i$. Likewise, let $n_d$ be the number of $d$-vertices around the plaquette, while $\widetilde{n}_d$ is the number of $d$ vertices in the flipped configuration. Note that $\hat{\mathcal F}_i$ always flips a $c$ or $d$ vertex to an $a$ or $b$ vertex, and vice versa. Consequently, we take the operators $\hat {\mathcal V}_i$ to be of the form \begin{equation} \hat{\mathcal V}_i = c^{\widetilde{n}_c - n_c} d^{\widetilde{n}_d - n_d} \label{Vpot} \ . \end{equation} Explicitly, we can write the projectors as follows \begin{equation} \mathcal{Q}_i = \begin{pmatrix} c^{\widetilde{n}_c - n_c} d^{\widetilde{n}_d - n_d} & -1 \\ -1 & c^{\widetilde{n}_c - n_c} d^{\widetilde{n}_d - n_d} \\ \end{pmatrix} \ . \end{equation} This ${\cal Q}_i$ is indeed proportional to a projection operator: the Hamiltonian (\ref{q8vham}) has the classical eight-vertex model as an eigenstate. This holds for any choice of the $w_i$, but in the ${\mathbb Z}_2$ gauge-theory language of appendix \ref{app:z2gauge}, it is natural to set all $w_i=1$. The amplitude of the ground-state wave function of this $H_{q8v}$ on a state with $N_c$ $c$-vertices and $N_d$ $d$-vertices is \begin{equation} \Psi_0[N_c,N_d] = \frac{c^{N_c} d^{N_d}}{\sqrt{Z(c^2,d^2)}} \ , \end{equation} where $Z(c,d)$ is the partition function of the classical two-dimensional eight-vertex model with weights $a=1$, $b=1$, $c$ and $d$. The arguments in the denominator are $c^2$ and $d^2$ because averages in the quantum model are calculated with respect to $|\Psi_0|^2$. In section \ref{sec:2dwavefn}, we discussed how by taking the limit $w\to 0$ in $H_w$, one can recover a slightly-generalized square-lattice quantum dimer model which allows defects with no dynamics. Similarly, here one can find a quantum six-vertex model with some defects allowed by taking the limit $d \rightarrow 0$.\footnote{This Hamiltonian in the special case $c=1$, $d=0$ was discussed in Ref.\ [\onlinecite{balents03b}]. Moreover, the quantum six-vertex model should be in the same universality class as the``supersymmetric'' $XY$ model introduced some time ago [\onlinecite{girvin93}]; this model defines a similar Hamiltonian acting on the 2D classical XY model, and presumably can also be understood in terms of a mapping to the quantum Lifshitz model. Another quantum six-vertex model has been proposed as a model of a planar pyrochlore lattice in Ref.\ [\onlinecite{moessner01c}]. This six-vertex model is not of the Rokhsar-Kivelson type; it is the six-vertex limit of the simpler quantum eight-vertex model we discuss in section \ref{another}.} Configurations with $n_d>\widetilde{n}_d$ on any plaquette will receive infinite potential energy and so is disallowed. Some useful facts are that $n_c+n_d + \widetilde{n}_c +\widetilde{n}_d =4$, and $n_c-\widetilde{n}_c =0$ mod 2 and $n_d-\widetilde{n}_d =0$ mod 2. When $d\to 0$, configurations with $n_d=\widetilde{n}_d=0$ are the flippable plaquettes of the six-vertex model, and are obviously included in the ground-state wave function. Configurations with $n_d=\widetilde{n}_d=1$ or $n_d=\widetilde{n}_d=2$ are not suppressed, but the flip therefore preserves the number of $d$ vertices around each plaquette. Thus in the $d\to 0$ limit we can view these $d$ vertices as defects in the quantum six-vertex model. The ground-state wave function is a sum over all allowed states with a given set of plaquettes with defects. The amplitude of each configuration in this sum is proportional to $c^{N_c}$. We discuss the relation between the quantum six- and eight-vertex models in more detail in the Appendices \ref{app:gauge} and \ref{app:z2gauge}. \subsection{The phase diagram} \label{sec:8vphases} Now that we have the exact ground-state wave function for the quantum eight-vertex Hamiltonian of \eqref{q8vham}, we can use it to determine the phase diagram. Given the fact that the probability of a configuration of arrows in the ground state wave function is equal to the Boltzmann weight of a classical eight-vertex model, we can deduce much of the physics of the quantum theory (at least its equal-time properties) directly from the Baxter solution of the classical 2D eight-vertex model [\onlinecite{baxbook}], as well as from Kadanoff's classic work on its critical behavior [\onlinecite{kadanoff77,kadanoff79,kadanoff-brown79}]. The only change is that the weights must be squared here, since in quantum mechanics we weigh configurations with $|\Psi|^2$. The phase diagram is displayed in figure \ref{eight}; note that the axes are labeled by $c^2$ and $d^2$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure} \begin{center} \psfrag{bz2}{Broken ${\mathbb Z}_2$} \psfrag{ubz2}{Unbroken ${\mathbb Z}_2$} \psfrag{d}{$d^2$} \psfrag{c}{$c^2$} \psfrag{confining}{Confining} \psfrag{deconfining}{Deconfining} \psfrag{kitaev}{Kitaev} \psfrag{kt}{KT} \psfrag{2}{$2$} \psfrag{6-vertex}{$6$-vertex} \psfrag{ising}{Dual $6$-vertex} \psfrag{I}{$I$} \psfrag{II}{$II$} \psfrag{III}{$III$} \psfrag{o}{Ordered} \psfrag{qd}{Quantum Disordered} \includegraphics[width=0.6 \textwidth]{eight.eps} \caption{Phase diagram of the quantum eight-vertex model: phases $I$ and $II$ are separated by a dual $6$-vertex transition (same with $I$ and $III$); $6$-vertex denotes the $6$-vertex model critical lines and KT are $2D$ Kosterlitz-Thouless transitions; the dotted line shows that the Kitaev point is smoothly connected to the critical regime of the eight-vertex model. } \label{eight} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We will now use this knowledge, as well a simple perturbative arguments in the quantum theory, to determine the phase diagram, the behavior of physical observables in the different phases, and (much of) their critical behavior. A useful fact is that for $cd=1$, the classical model with partition function $Z(c^2,1/c^2)$ is equivalent to two decoupled Ising models. This decoupling is a property only of the wave function, not of the Hamiltonian of the full $2+1$-dimensional quantum theory. Since this line $cd=1$ goes through both ordered and disordered phases, much of the physics of the quantum eight-vertex model can be described at least qualitatively in terms of decoupled Ising models. In particular, for any values of $c$ and $d$ except those on the critical line, correlators decay exponentially fast with distance with a correlation length $\xi$ which diverges as the phase boundary is approached, in a manner given in Eq.\ \eqref{critexp}. This exponential decay occurs in general, not just on the decoupling curve. As with the dimer models discussed in section \ref{sec:2dwavefn}, the partition function for $cd=1$ can be expressed in terms of Grassmann variables with only quadratic terms, i.e.\ free fermions. Duality means that $Z(c^2,d^2)$ is free-fermionic for $c^4+d^4=2$ as well. The correlators in the continuum limit of the (critical) square-lattice quantum dimer model are therefore identical to those obtained for $c=\sqrt{2}$, $d=0$. The special point $c=d=1$ discussed above and in Ref.\ [\onlinecite{kitaev97}] (labeled in fig.\ \ref{eight} as ``Kitaev'') is also free fermionic. At this point one does not need the Pfaffian techniques to compute correlators exactly, and one finds that the model is in a disordered phase in the Ising-spin language. However, the expectation value $\langle\mu(A)\mu(B)\rangle$ is non-vanishing, so there is topological order at this point. Let us now discuss the different phases of this system. \begin{enumerate} \item {\em The Ordered (Confined) Phase}:\\ {}From the known phase diagram of the classical eight-vertex model [\onlinecite{baxbook}], we conclude that the ground state of the {\em quantum} model with the Hamiltonian of Eq.\ \eqref{q8vham} has an {\em ordered} phase for $c^2>d^2+2$ (and also for $d^2>c^2+2$). That this is an ordered phase can be seen easily by considering the limit $c \to \infty$ (with $d$ fixed). In this limit the ground state is dominated by just two configurations, related to each other by a lattice translation of one lattice spacing, which have a $c$ vertex on every site, as shown in fig. \ref{ddw}. In this phase expectation value of the staggered polarization operator $\langle \tau(A)\tau(B)\rangle$ has a non-vanishing expectation value. This result can also be obtained directly from the Hamiltonian of the ${\mathbb Z}_2$ gauge theory, Eq.\ \eqref{q8vham2}, since for $c$ large the potential energy term $H_V$ dominates and in it the piece associated with the $c$ projection operators. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure} \begin{center} \includegraphics[width=0.2 \textwidth]{ddw.eps} \caption{The ordered phase ``antiferroelectric" or ``staggered flux" phase of the eight-vertex model for $c \gg 1$. In this limit there are only $c$ vertices in the ground state.} \label{ddw} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The ordered phase is also {\em confining}. Below we will discuss the behavior of the Wilson loop operator and show that in this regime it obeys an area law, the hallmark of confinement [\onlinecite{fradkin78,kogut79}]. We will also show that the energy of a state with two static sources grows linearly with their separation. We should also note here that the equal-time fermion correlation function has an exponential decay in this phase, suggesting that this phase may support massive fermionic excitations. \item {\em The Disordered (Deconfined) Phase}:\\ {}From the exact solution of the classical model, we know that there is, a {\em disordered} phase for $c^2\widetilde{n}_d$ will be suppressed as they receive infinite energy. The flip term has to be modified, because we need to have a flip term which commutes with the constraint, which has become a stronger statement in the six-vertex case, namely (c.f.\ Eq.\ \eqref{con6v}) \begin{equation} \label{scon6v} \sigma_1^1 (\vec x) - \sigma_{1}^1 (\vec x-\vec e_1) + \sigma_2^1 (\vec x) - \sigma_{2}^1 (\vec x-\vec e_2) = 0 \qquad \forall \vec x \ . \end{equation} The flip term which preserves the six-vertex constraints is \begin{equation} \label{sflip6v} H_{\rm flip,6v} = - \sum_{\vec x} \bigl(\sigma^-_1 (\vec x) \; \sigma^-_2 (\vec x + e_1) \; \sigma^+_1 (\vec x + e_2) \; \sigma^+_2 (\vec x) + {\rm h.c.} \bigr) \ , \end{equation} where the raising and lowering operators (in the representation we use) are given by \begin{equation} \sigma^\pm = \tfrac{1}{2} (\sigma^3 \mp i\sigma^2) \end{equation} To make contact with the flip term for the eight-vertex model, we rewrite Eq.\ \eqref{sflip} as \begin{equation} \begin{split} H_{\rm flip,q8v} = & - \sum_{\vec x} \bigl(\sigma^+_1 (\vec x) + \sigma^-_1 (\vec x)\bigr) \bigl(\sigma^+_2 (\vec x + e_1) + \sigma^-_2 (\vec x + e_1)\bigr) \times \\ &\bigl(\sigma^+_1 (\vec x + e_2) + \sigma^-_1 (\vec x + e_2)\bigr) \bigl(\sigma^+_2 (\vec x) + \sigma^-_2 (\vec x)\bigr) \ . \end{split} \end{equation} The flip term for the six-vertex model is therefore precisely the flip term for the eight-vertex model minus the terms which cause the six-vertex constraint to be violated. It is easily checked that the six-vertex flip term Eq.\ \eqref{sflip6v} commutes with the constraint \eqref{scon6v}. We thus find that on the level of the wave function, the limit $d \rightarrow 0$ is smooth, as the amplitude of the configurations which contain $d$ vertices goes to zero. In addition, for $d \neq 0$, the flip term commutes with constraint \eqref{scon8v2}, while for $d=0$, the flip term commutes with the $U(1)$ constraint \eqref{scon6v}. Thus, the symmetry is enhanced from $\mathbb{Z}_2$ for $d\neq 0$ to $U(1)$ for $d=0$, as was to be expected. \subsection{The dual of the gauge theory} \label{app:duality} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[h] \begin{center} \psfrag{x}{$\vec x$} \psfrag{R1}{$\vec r$} \psfrag{R2}{$\vec r-\vec e_1$} \psfrag{R3}{$\!\!\!\!\!\!\!\!\!\vec r-\vec e_1-\vec e_2$} \psfrag{R4}{$\vec r-\vec e_2$} \includegraphics[width=0.23 \textwidth]{dual.eps} \caption{The dual lattice sites are labeled by $\vec r$.} \label{fig:dual} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We now have a Rokhsar-Kivelson generalization of the eight-vertex model, in a gauge-theory language. We can use this representation of our model to study the various phases. However, this is more easily done in a dualized version, as the dual takes the form of an Ising model. In the dual picture, the spin degrees of freedom will live on the sites of dual square lattice, {\it i.e.\/} the centers of the plaquettes of the direct lattice. Thus, we will label by $\vec r$ the site of the dual lattice on the center of the plaquette labeled by $\vec x$ (its SW corner). Of course, the potential term in the dual language will still be quite formidable. We will denote the dual Pauli operators by $\tau^1$ and $\tau^3$. To start with the flip term, the product of $\sigma^3$'s around a plaquette becomes $\tau^1$ on the plaquette [\onlinecite{fradkin78,kogut79}] \begin{equation} \label{tx} \tau^1 (\vec r) = \sigma^3_1(\vec x) \; \sigma^3_2(\vec x+\vec e_1) \; \sigma^3_1(\vec x+ \vec e_2) \; \sigma^3_2(\vec x) \end{equation} To see what happens with the constraint and the projector operators defined by Eq.\ (\ref{vms}) and Eq.\ (\ref{P-S}) we need the dual form of the $\sigma^1$\;'s living on the links. In term of the dual variables $\tau^3$, and using the notation of Fig. \ref{fig:dual}, the $\sigma^1$\;'s are given by \begin{equation} \label{tz} \sigma^1_1(\vec x)= \tau^3(\vec r) \; \tau^3(\vec r-\vec e_2), \qquad \sigma^1_2(\vec x)= \tau^3(\vec r) \; \tau^3(\vec r-\vec e_1) \end{equation} We thus easily find that the constraint is automatically satisfied (again, going to the dual picture amounts to solving the constraint). Also, it is trivial to show [\onlinecite{fradkin78,kogut79}] that the inverse relation, {\it i.e.\/} to express the dual lattice $\tau^3$ operators in terms of the $\sigma^1$ operators of the original lattice is \begin{equation} \tau^3(\vec r)=\prod_{\ell \in \Gamma(\vec r)} \sigma^1(\ell) \label{tau3} \end{equation} where $\{\ell\}$ is a set of links of the direct lattice pierced by a path $\Gamma(\vec r)$ on the {\em dual} lattice ending at the dual site $\vec r$ (but which is otherwise arbitrary); see Fig. \ref{dual-path}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[h] \begin{center} \psfrag{R}{$\vec r$} \psfrag{G}{$\Gamma$} \includegraphics[width=0.27 \textwidth]{disorder-op.eps} \caption{The dual path $\Gamma$.} \label{dual-path} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Note that we need to choose the spin on one of the plaquettes; all the others are subsequently determined by the $\sigma^x$'s on the links. In terms of the dual variables, the projection operators for site $\vec x$, defined by Eq.\ \eqref{P-S}, take the form \begin{equation} \label{vms-dual} \begin{split} \mathcal{P}_a(\vec x) &= \frac{1}{4} \left( 1 + \mathcal{A}(\vec r) + \mathcal{B}(\vec r) + \mathcal{C}(\vec r) \right) \\ \mathcal{P}_b(\vec x) &= \frac{1}{4} \left( 1 - \mathcal{A}(\vec r) - \mathcal{B}(\vec r) + \mathcal{C}(\vec r) \right) \\ \mathcal{P}_c(\vec x) &= \frac{1}{4} \left( 1 - \mathcal{A}(\vec r) + \mathcal{B}(\vec r) - \mathcal{C}(\vec r) \right) \\ \mathcal{P}_d(\vec x) &= \frac{1}{4} \left( 1 + \mathcal{A}(\vec r) - \mathcal{B}(\vec r) - \mathcal{C}(\vec r) \right) \ , \\ \end{split} \end{equation} where $\mathcal{A,B}$ and $\mathcal{C}$ are given by \begin{equation} \begin{split} \mathcal{A}(\vec r) &= \tau^3(\vec r-\vec e_1) \; \tau^3(\vec r-\vec e_2) \\ \mathcal{B}(\vec r) &= \tau^3(\vec r) \; \tau^3(\vec r-\vec e_1-\vec e_2) \\ \mathcal{C}(\vec r) &= \tau^3(\vec r) \; \tau^3(\vec r-\vec e_1) \; \tau^3(\vec r-\vec e_1) \; \tau^3(\vec r - \vec e_1-\vec e_2) \ . \end{split} \end{equation} We see that we the interaction in these projectors has the same structure as in the spin representation of the classical eight-vertex model: it consists of two-body terms on interpenetrating sublattices, and a four-body term, which couples the two sublattices. In the total interaction term, all the two-body interaction terms will only couple spins on the same sublattice. The four and six body interaction terms (of which there are many!), couple the sublattices. The same holds for the eight-body term, naturally. Let us state the dual form of the theory \begin{equation} H_{\rm q8v, dual} = H_{\rm V,dual} - \sum_{\vec r} \tau^1(\vec r) \ , \label{q8v-dual} \end{equation} where $H_{\rm V,dual}$ is given by \eqref{q8vpotsig}, but now with the projectors given in Eq.\ \eqref{vms-dual}. Thus, formally this theory takes the form of a (multi-spin) Ising model in a transverse field. However, the two-body interactions only couple spins on the same sublattices, together with the multi-spin terms conspire to change the quantum critical behavior from the conventional $z=1$ Lorentz-invariant criticality of the standard Ising model in a transverse field to the $z=2$ quantum critical behavior discussed in the rest of this paper. Now that we found the dual version of our gauge theory, we would like to discuss the limits $a=b=c=d=1$ and $d=0$. Again, the first limit brings us back to the Kitaev point, because the potential term becomes the identity operator again, and we are left with the very simple spin flip term of Eq.\ \eqref{q8v-dual}, $H_{\rm f}=-\sum_{\vec r} \tau^1(\vec r)$. The limit $d \rightarrow 0$ is however more complicated in this dual gauge theory. First of all, we now do need a constraint, which was not present for $d \neq 0$. Moreover, the flip term now only can act, depending on the surrounding spins. Let us start by dualizing the constraint Eq.\ \eqref{scon6v}, which results in \begin{equation} \label{tcon6v} \bigl( \tau^3(\vec r) - \tau^3(\vec r-\vec e_1-\vec e_2) \bigr) \bigl( \tau^3(\vec r-\vec e_1) + \tau^3(\vec r-\vec e_2) \bigr) = 0 \qquad \forall \vec x \ . \end{equation} Obviously, the eight-vertex flip term $\tau^1 (\vec r)$ does not commute with this constraint. To find a flip term which does commute with the constraint, we dualize the six-vertex flip term \eqref{sflip6v}, which results in \begin{equation} \label{tflip6v} H_{\rm flip,q6v} = -\frac{1}{8}\sum_{\vec r} \tau^1 (\vec r) \bigl( 1-\tau^3(\vec r+\vec e_1) \tau^3(\vec r+\vec e_2) \bigr) \bigl( 1+\tau^3(\vec r-\vec e_1) \tau^3(\vec r+\vec e_2) \bigr) \bigl( 1+\tau^3(\vec r+\vec e_1) \tau^3(\vec r-\vec e_2) \bigr) \ . \end{equation} The factors in bracket can be seen to give a non zero result only on plaquettes which are flippable. Hence, this flip term commutes with the constraint \eqref{tcon6v}. Of course, this can also be checked explicitly. Apart from a factor $\tau^2$, there are factors depending on the $\tau^3$'s coming form both the flip term and the constraint. The signs in this product conspire in such a way to render the commutator zero. \bibliography{loops-alt} \end{document}