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Extended quantumU(1)-liquid phase in a three-dimensional quantum dimer model

Olga Sikora and Nic ShannonH. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK.

Frank PollmannMax-Planck-Institut fur Physik komplexer Systeme, 01187Dresden, Germany

Karlo PencResearch Institute for Solid State Physics and Optics, H-1525 Budapest, P.O.B. 49, Hungary.

Peter FuldeMax-Planck-Institut fur Physik komplexer Systeme, 01187Dresden, Germany and

Asia Pacific Center for Theoretical Physics, Pohang, Korea

Recently, quantum dimer models have attracted a great deal of interest as a paradigm for the study of exoticquantum phases. Much of this excitement has centred on the claim that a certain class of quantum dimer modelcan support a quantumU(1) liquid phase with deconfined fractional excitations in three dimensions. Thesefractional monomer excitations are quantum analogues of the magnetic monopoles found in spin ice. In thisarticle we use extensive quantum Monte Carlo simulations toestablish the ground-state phase diagram of thequantum dimer model on the three-dimensional diamond lattice as a function of the ratioµ of the potentialto kinetic energy terms in the Hamiltonian. We find that, forµc = 0.75± 0.02, the model undergoes a first-order quantum phase transition from an ordered “R-state” into an extended quantumU(1) liquid phase, whichterminates in a quantum critical “RK point” forµ = 1. This confirms the published field-theoretical scenario.We present detailed evidence for the existence of theU(1) liquid phase, and indirect evidence for the existenceof its photon and monopole excitations. Simulations are benchmarked against a variety of exact and perturbativeresults, and a comparison is made of different variational wave functions. We also explore the ergodicity of thequantum dimer model on a diamond lattice within a given flux sector, identifying a new conserved quantityrelated to transition graphs of dimer configurations. Theseresults complete and extend the analysis previouslypublished in [O. Sikoraet al. Phys. Rev. Lett.103, 247001 (2009)].

PACS numbers: 75.10.Jm 75.10.Kt, 11.15.Ha, 71.10.Hf

I. INTRODUCTION

Dimer models, which describe the myriad possible con-figurations of hard-core objects on bonds, have long been atouch-stone of statistical mechanics1,2. Quantum dimer mod-els (QDM’s), in which dimers are allowed to resonate betweendifferent degenerate configurations, were first introducedtodescribe antiferromagnetic correlations in high-Tc supercon-ductors3. It has since been realized that QDM’s arise naturallyas effective models of many different condensed matter sys-tems, and provide a concrete realizations of several classesof lattice gauge theories. As such, they have become centralto the theoretical search for new quantum phases and excita-tions. A key question in this context is when, if ever, a QDMcan support a liquid ground state?

The answer to this question depends on lattice dimensionand topology. In two dimensions (2D), the situation is rela-tively well-understood4, and a liquid ground state is knownto exist in the QDM on thenon-bipartite triangular5 andkagome6 lattices. This liquid is gapped and has deconfinedfractional excitations, which are vortices of an underlyingZ2

(Ising) gauge theory. Meanwhile, for 2Dbipartite lattices,the underlying gauge theory has aU(1) character, akin toelectromagnetism7,8. In this case a liquid state is found onlyat a single, critical, “Rokhsar-Kivelson” (RK) point3. Awayfrom this, the system crystallizes into phases with broken lat-

FIG. 1: (Color online) Phase diagram of the quantum dimer modelon a diamond lattice, as a function of the ratioµ of potential to kineticenergy, following8,12,14.

tice symmetries. For the QDM on a square lattice, these arecolumnar dimer and resonating plaquette phases9,10. A verysimilar phase diagram is found for the QDM on a honeycomblattice11.

Much less is known about QDM’s in three dimensions (3D).However here field theoretical arguments suggest that a new,extendedU(1)-liquid phase with gapless photon-like excita-tions might “grow” out of the RK point for a 3D QDM ona bipartite lattice8,12 (see Fig.1). Similar claims have beenmade for the closely related quantum loop model in three di-mensions13.

In a recent Letter14, we presented numerical evidence for

2

the existence of an extended quantumU(1) liquid phasewith deconfined fractional excitations in a three-dimensionalQDM, taken directly from its microscopic Hamiltonian. Weconsidered the specific example of a QDM on a diamondlattice, which may serve as an effective model for the frus-tration found in various spinel compounds12,15. From zero-temperature GFMC simulations of clusters of up to2000 siteswe were able to demonstrate the existence of a quantumU(1)-liquid phase for a range of parameters0.77 ± 0.02 < µ < 1bordering on the RK point [cf. Fig.1]. These results comple-ment a recent finite-temperature quantum Monte Carlo studyof a microscopic model of interacting bosons which can bemapped onto an effective quantum loop model16.

In this article we document the evidence for these claims,taken from explicit calculations of (i) the order parameterofthe competing (dimer)-ordered phase [cf. Fig.2], (ii) the“string tension” associated with fractional monomer excita-tions and (iii) the finite-size scaling of a characteristic set ofenergy gaps associated with the U(1) liquid phase. We alsoexplore the internal consistency of these calculations, cross-checking simulations against exact diagonalization of smallclusters and perturbative expansions about exactly solublelimits of the model.

We pay particular attention to the thorny question of the er-godicity of GFMC simulations in the presence of hidden quan-tum numbers. In the 2D QDM’s studied previously, the quan-tum numbers are well understood, and it is clear which dimerconfigurations are connected by the QDM Hamiltonian. Thisis not, however, the case for the 3D QDM considered in thisarticle. Despite the fact that we can define a “magnetic” flux~φ which is a good quantum number — in a manner similar tothe square-lattice QDM in 2D — the QDM Hamiltonian ona diamond lattice connects only sub-sets of dimer configura-tions within of each flux sector. This means that the QDM ondiamond lattice isnot ergodic within a given flux sector, andgreat care must be taken to ensure that simulations accuratelyrepresent the true ground state for each value of flux~φ. Weshed some light on this issue by identifying a new, hidden,quantum number conserved by the QDM on diamond lattice.

The paper is structured as follows :In SectionII we introduce the model and summarize the

properties of the proposedU(1)-liquid phase. We demon-strate a way of constructing dimer configurations with differ-ent values of magnetic flux, and define an order parameter forthe ordered “R-state” found for negative values ofµ. In Sec-tion III we use Green’s function Monte Carlo simulations toobtain explicit predictions for theµ dependence of the orderparameter, monomer string tension and ground state energyas a function of magnetic flux for a range of different clustersizes and geometries. We confirm the conjectured form of thephase diagram Fig.1, and use careful finite size scaling to ob-tain an improved estimate of the location of the phase transi-tion from ordered to liquid phases in the thermodynamic limitasµc = 0.75 ± 0.02. In SectionIV, we explore the ergodic-ity of our simulations within a given flux sector and confirmthat the techniques used give reliable estimates of ground stateproperties. We also identify an additionalZ2 quantum numberwhich divides each flux sector into two distinct parts. In Sec-

tion V we summarise our results and briefly discuss remainingopen issues.

The paper concludes with some technical appendices. InAppendix A we discuss the generalization to a fermionicQDM and show that it is equivalent to the bosonic case in low-est order. In AppendixB we exhibit a Landau theory for theordered,R-state. In AppendixC we compare different vari-ational wave functions, and use these to investigate the tran-sition from ordered to liquid ground states. In AppendixDwe benchmark our Green’s function Monte Carlo simulationsagainst perturbation theory about exactly soluble limits of themodel.

II. THE QUANTUM DIMER MODEL AND WHAT YOUNEED TO KNOW ABOUT IT

A. The model and its derivation

In the spirit of Ref. 3, we study the quantum dimer model(QDM)

HQDM = −gKf + µNf (1)

where the matrix elements of

Kf =∑

(

∣

∣

∣

⟩⟨ ∣

∣

∣+∣

∣

∣

⟩⟨ ∣

∣

∣

)

(2)

connect different, degenerate dimer configurations by cycli-cally permuting dimers on six-bond “flippable” plaquettes ofalternating filled and empty links within the diamond lattice[cf. Fig. 2], while

Nf =∑

(

∣

∣

∣

⟩⟨ ∣

∣

∣+∣

∣

∣

⟩⟨ ∣

∣

∣

)

(3)

counts the number of such flippable plaquettes. In what fol-lows we will considerg > 0, with −∞ < µ ≤ g.

This QDM Hamiltonian acts on the≈ 1.3N/2 differentdimer configurations which cover the links theN -bond di-amond lattice17. This Hilbert space breaks up into differ-ent subsectorsλc, defined by the sets of dimer configurationswhich are connected by the matrix elements ofKf . The num-ber and nature of these subsectors remains an open problem,which will be discussed at length in SectionIV of this article.

Crucially, however,all off-diagonal matrix elements ofEq. (1) are zero or negative. By the Frobenius-Perron the-orem, the lowest energy state of the QDM withinany givensubsector of the Hilbert spaceλc must then be a nodeless su-perposition of dimer configurations

| ψ0〉λc=∑

c∈λc

ac|c〉 (4)

with real,positivecoefficientsac. As a result, Quantum MonteCarlo simulations of Eq. (1) arenotsubject to a sign problem.

3

Exactly at the RK pointµ = g, the QDM Hamiltonian canbe written as a sum of projectors

HRK = g∑

[

∣

∣

∣

⟩

−∣

∣

∣

⟩

] [

⟨ ∣

∣

∣−⟨ ∣

∣

∣

]

(5)

In this case the ground state inany subsector of the Hilbertspaceλc is the zero-energy eigenstate given by the equallyweighted sum overall dimer configurations within that sub-sector.

|RK〉λc∝∑

c∈λc

|c〉 (6)

FIG. 2: (Color online) (a) Cubic unit cell of the diamond lattice, withbonds occupied by dimers shown by thick red links. The dimer con-figuration shown is the ordered “R-state” found for negativeµ. Thiscontains a “flippable” hexagonal plaquette, which is shadedblue. (b)Equivalent cubic unit cell of the pyrochlore lattice, with sites occu-pied by hard-core bosons shown in red. The boson configurationshown is the same ordered “R-state”.

The QDM Eq. (1) can be derived from various microscopicmodels. One example is a hard-core boson model on a py-rochlore lattice formed of corner sharing tetrahedra

HtV = −t∑

〈ij〉

(b†i bj +H.c.) + V∑

〈ij〉

ninj (7)

at quarter filling, in the limit of large nearest-neighbor inter-actionsV ≫ t. In this case, the low energy configurations arethose without bosons on neighbouring sites. This imposes theconstraint of placingexactlyone boson in each tetrahedron.

The diamond lattice is the medial lattice of the pyrochlorelattice, formed by connecting the centres of all tetrahedra(seeFig. 2), and and these constrained states are in exact one-to-one correspondence with the dimer coverings of a diamondlattice. The lowest-order process in degenerate perturbationtheory which connects these states is a cyclic exchange ofbosons around the smallest, hexagonal plaquette of the py-rochlore lattice. This occurs at third order in the hopping ofbosons, givingg = 12t3/V 2.

Another physically motivated model which reduces to thequantum dimer model on the diamond lattice is the easy-axisquantum antiferromagnet on a pyrochlore lattice

HXXZ = Jxy∑

〈ij〉

(Sxi S

xj + Sy

i Syj ) + Jz

∑

〈ij〉

Szi S

zj

−h∑

i

Szi (8)

at half-magnetization〈Sz〉 ≡ S/2. ForS = 1/2 this is ex-actly equivalent to the hard-core boson problem consideredabove, with each occupied site corresponding to a “up” spin,and each empty site to a “down” spin, with the caveat thatthe kinetic energy termJxy now has the opposite sign, and sog = −12J3

xy/J2z .

At first sight this might seem to imply that theXXZ modelEq. (8) with g < 0 has a different ground state from the at-Vmodel withg > 0 Eq. (7). However the global sign ofg canbe changed by the simple unitary transformation

|c〉 → exp[iπNΛ/2]|c〉 (9)

where|c〉 is an arbitrary dimer configuration andNΛ countsthe number of dimers which occupy the subset of diamondlattice bondsΛ shown in Fig.3. The cyclic permutation ofdimers on any six-bond flippable plaquette changesNΛ bytwo, and so the unitary transformation Eq. (9) mapsg → −g.Moreover, exactly the same quantum dimer model Eq. (1) canbe derived from a fermionict-V model, i.e. Eq. (7) with Fermioperators substituted for the bosonic ones. This mapping isdescribed in AppendixA.

FIG. 3: (Color online) The subset of diamond lattice bondsΛ (thick,blue, shaded bonds) used in the unitary transformation Eq. (9) whichmapsg → −g in Eq. (1).

An effective Hamiltonian of the QDM form Eq. (1) can alsobe derived from Eq. (8) for larger values of spin. ForS = 3/2,

4

where the model might be of relevance to Cr spinels, this wasaccomplished by Bergmanet al.18, who foundµ/g ≈ −7. Atwo-dimensional analogue of Eq. (1) might also be realizedusing cold atoms19. Finally, quantum dimer models also ariseas effective models of spin–orbital models20.

Since the goal of this paper is to determine the ground statephases of Eq. (1), and not to relate it to any particular physicalsystem, in what follows we setg = 1, and study the model asa function of the single adjustable parameterµ. Forµ→ −∞the potential energy dominates, and the ground state of Eq. (1)is the set of dimer configurations which maximise the numberof “flippable” hexagons, with one out of four hexagons beingflippable. This is the so-calledR-state18, illustrated in Fig.2,which has a cubic unit cell and is eight-fold degenerate.

At the RK pointµ = 1, the ground state is the equallyweighted sum of all possible dimer configurations3, while forµ > 1, ground states are the many “isolated states” whichcontainno flippable hexagons, and so have no kinetic energy.The interest of the QDM therefore rests in the parameter range−1 . µ ≤ 1, for which kinetic and potential energy competeon an equal footing. It is this range of parameters which weconsider below.

B. Effective electrodynamics

For purpose of comparison with numerics, we now brieflyreview the field theory arguments underlying the proposedU(1) liquid phase. The defining property of a dimer modelis that every lattice site is touched byexactlyone dimer. Thediamond lattice is bipartite — we can divide it in two sublat-tices,A andB, such that every every link (bond) of the latticeconnects anA sublattice site with aB sublattice site. We canuse this property to associate a magnetic flux vectorB witheach bond of the lattice. Bonds not occupied by a dimer areassignedone(arbitrary) unit of magnetic flux, directed fromB to A. Where a dimer is present, we assignthreeunits offlux to that bond, directed fromA toB43.

In this language, the constraint that every lattice site istouched by one dimer becomes the condition

∇ ·B = 0. (10)

We resolve this constraint by writing

B = ∇×A, (11)

where the gauge fieldA is defined on thesitesof the orig-inal diamond lattice while the fictitious magnetic fieldB isdefined on itsbonds. We chose to work in the Coulomb gauge∇ · A = 0. In the absence of quantum effects, the statisticaldescription of the dimer model reduces to a relatively straight-forward problem in magnetostatics, and the ground state of thedimer model is found to be classicalU(1)-liquid. Correlationsbetween dimers have a dipolar form in real space, and the con-straint∇ ·B = 0 manifests itself as a set of “pinch points” inthe structure factor in k-space7,21–24. Exactly this situation isrealized in the rare-earth magnet spin ice25.

The purpose of this paper is to treat the quantum effectswhich arise from the tunnelling of the system from one dimer

configuration to another, as described by Eq. (1). This tun-nelling introduces fluctuations in time of the gauge fieldA,which in turn give rise to an effective electric field

E = −∂tA. (12)

None the less, the total magnetic fluxφ =∫

dS · B throughany plane in the lattice remains a conserved quantity, sincethedynamics present in the quantum dimer model only permit areversal in the sense of flux around a closed loop.

This representation clearly has a lot in common with con-ventional electromagnetism and, following Ref. 8, we can usethis analogy to write down a plausible long-wavelength actionfor the QDM on a diamond lattice

S =

∫

d3xdt[

E2 − c2B2

]

, (13)

wherec2 > 0. This is the Maxwell action of conventionalelectromagnetism, and the system must posses linearly dis-persing “photons” (transverse excitations of the gauge fieldA) with speed of lightc. These conditions were argued tobe realized in a quantumU(1)-liquid state bordering the RKpoint in three-dimensional quantum dimer models on a bipar-tite lattice forµ . 1 [8], leading to the conjectured phasediagram Fig.1. An exactly parallel story can be told about thequantum loop model in three dimensions13.

These “emergent” photons are the signature feature of theproposedU(1)-liquid state, and offer a beautiful realizationof Maxwell’s laws in a condensed matter system. However,for the purposes of this study, we wish to emphasize thatEq. (13) also contains information about thefinite sizescal-ing of the energy spectra. A fluxφ through a cluster of vol-umeL3 corresponds to an average magnetic fieldB = φ/L2.In theU(1)-liquid state, this magnetic field is uniformly dis-tributed on the “coarse-grained” scale of the effective actionEq. (13). It then follows from Eq. (13) that the energy differ-ence∆φ = Eφ−E0 between the ground state of the zero-fluxsector, and the lowest energy state of the sector with fluxφscales as

∆φ = Eφ − E0 = c2φ2

L. (14)

In the thermodynamic limit∆φ → 0 and different flux sectorsshould be treated as degenerate ground states. A second finite-size prediction of Eq. (13) is that the ground state energy oftheU(1)-liquid state should have a finite size correction∆Efrom the zero-point energy of photons which in leading orderscales as

∆E(L)

N∼

[

∫ Λ′

2π

L

dk −

∫ Λ′

0

dk

]

k2ck ∼ −c

L4(15)

whereΛ′ is a size-independent momentum cutoff. In this pa-per we make use of both Eq. (14) and Eq. (15) to study finitesize properties of Eq. (1).

5

TABLE I: Cluster geometries used in simulations. The first columngives the short notation for cluster geometry used in the text; thesecond column, the number of diamond lattice bonds; the lastthreecolumns show the vectorsgi which define the translation vectors ofthe cluster. The integerl defines the size of the cluster, within a givenfamily. The [100] clusters have side of length ofL = 2l, and containN = 2L3 bonds.

number of bonds g1 g2 g3

[100] 16 l3 2l (1, 0, 0) 2l (0, 1, 0) 2l (0, 0, 1)

[110] 32 l3 2l (1, 1, 0) 2l (0, 1, 1) 2l (1, 0, 1)

[111] 64 l3 2l (1, 1,−1) 2l (1,−1, 1) 2l (−1, 1, 1)

C. Clusters used in simulation

Our evidence for the existence of a quantumU(1)-liquidphase is taken from simulation of the quantum dimer modelEq. (1) on finite-size clusters of the diamond lattice with pe-riodic boundary conditions. For the size of a cluster use thenotation in whichN is the number of pyrochlore lattice sites,corresponding to the diamond lattice bonds. The periodicboundary conditions permit us to associate a set of (integer)topological quantum numbers~φ = (φ1, φ2, φ3) with the fluxthrough a set of orthogonal planes. This reduces the Hilbertspace of the problem from the≈ 1.3N/2 different dimer con-figurations to a set of discrete sectors with given~φ.

An important lesson from the exact diagonalization of smallclusters is that the matrix elements of Eq. (1) do not connectall dimer configurations within a given value of~φ. None theless, the size of the largest blocks is still too large to permit theexact diagonalization of clusters with more that 128 diamondlattice bonds. These clusters are very small by the standardsof the (implicitly) course-grained field theory Eq. (13), andquantum Monte Carlo simulations must therefore be used todetermine the ground state of the model.

The quantum Monte Carlo method we chose is GFMC(Green’s function Monte Carlo), which constructs MonteCarlo updates using only the Hamiltonian dynamics of Eq. (1).This has the advantage that all quantum numbers (includingmagnetic flux) are preserved in simulation. However since notall of these quantum numbers are knowna priori, great caremust be taken to simulate in the appropriate subsectors of theHilbert space. This is an issue which we discuss at lengthin SectionIV. Moreover, because an exponential number ofdimer configurations contribute to the ground state wave func-tion of theU(1)-liquid, quantum Monte Carlo simulations arethemselves limited to clusters of up to 2000 diamond latticebonds.

In order to gain the most from finite size scaling, we there-fore consider a range of different cluster geometries. Werestrict ourselves to clusters which are compatible with the16-bond unit cell of theR-state, and group them into fami-lies of clusters with the same shape. These are summarizedin Table I. The main family of clusters used in our simu-lations consists of clusters with2L3 diamond lattice bonds(pyrochlore lattice sites) with side of lengthL = 4, 6, ..., 16,

which have the full (cubic) symmetry of the diamond lattice.We denote this family as [100], since the edges of the clusterare parallel to cubic lattice axes. The two other families ofclusters considered are denoted [110] and [111]. All resultsare quoted for [100] clusters, unless specified otherwise.

D. String defects and flux sectors

FIG. 4: (Color online) Construction of the “string” defect (one mag-netic flux) configuration from the maximally flippableR-state con-figuration (assuming periodic boundary conditions): (a) two adjacent16-bond unit cells of the diamond lattice within theR-state, with flip-pable hexagons denoted by blue shading; (b) a pair of monomerex-citations created by removing a dimer; (c), (d) and (e) monomers areseparated. The dashed line shows the “string” of displaced dimerswhich connects them. (f) one of the monomers has traversed thecluster and the dimer is re-introduced. The resulting configurationhas a single string excitation and a nonzero magnetic flux.

6

As discussed in SectionII B and SectionII C above, anygiven dimer configuration on the diamond lattice can be clas-sified according to its total “magnetic” flux

~φ = (φ1, φ2, φ3) (16)

where each component of~φ is a (topological) quantum num-ber conserved under the dynamics of the quantum dimermodel Eq. (1).

In a finite size cluster, each component of fluxφi servesas a winding number which takes on an extensive number of(discrete) values. These are bounded by the size of the sys-tem, and can be expressed as combinations of the four occu-pation numbers(n1, n2, n3, n4) which count the number ofdimers on each of the four fcc sublattices of diamond-latticebonds (equivalently, pyrochlore lattice sites). These occu-pation numbers are themselves invariant under the dynamicsof the QDM. However the mapping between the occupationnumber representation (cf. Ref. 12), and the flux represen-tation of these conserved quantities depends on the specificcluster geometry.

For the (100) cluster series we obtain the following expres-sion:

φx =1

2L(n1 − n2 − n3 + n4)

φy =1

2L(n1 − n2 + n3 − n4)

φz =1

2L(n1 + n2 − n3 − n4), (17)

where the corresponding positions of the sites in the 4-siteelementary unit cell of the pyrochlore lattice are numbered

r1 = (1, 1, 1)/4

r2 = (−1,−1, 1)/4

r3 = (−1, 1,−1)/4

r4 = (1,−1,−1)/4

It follows that−L2/4 ≤ φi ≤ L2/4 for a cluster withN =2L3 bonds in the diamond lattice (i.e. the number of sitesin the pyrochlore lattice). In the highly symmetricR–state(cf. Fig. 2) n1 = n2 = n3 = n4. This state therefore belongsto the zero-flux sector~φ = (0, 0, 0).

In order to test the predictions of the effective field the-ory, notably Eq. (14), we need to be able to construct otherstates in neighbouringflux sectors. The flux quantum numbers(φ1, φ2, φ3) are conserved underall local operations whichconnect configurations satisfying the dimer constraint. How-ever, we can construct states with finite magnetic flux, startingfrom theR–state, by creating non-local “string” defects whichtraverse the periodic boundaries of the cluster.

We do this following the procedure illustrated in Fig.4.When a dimer is removed from the lattice, two monomers(sites not touched by a dimer) are created. Subsequently, wecan separate these monomers, creating a “string” of dimerswhich have been translated by exactly one bond. Moving adimer by one bond reverses the sense of the magnetic fieldB associated with that dimer. We can therefore construct a

state with a different flux by moving one of the monomersacross the periodic boundary of the cluster in such a way thatthe two monomers meet and the dimer can be re-introduced.The resulting configuration fulfills again the zero-divergencecondition onB, but the magnetic flux through the periodicboundary traversed by the monomer has changed by exactlyone unit. For a cubic cluster this is the smallest magneticflux possible in the system, and we choose convention whereφ = 1 in the new configuration. States with higher flux can beconstructed by repeating the procedure above.

The monomers created by removing a dimer are excitationswith fractional quantum numbers which depend on the origi-nal microscopic model, e.g. they might be fractional chargese/2 [15], or “spinons” carrying spins = 1/2 [18]. In field-theoretical terms, they are the magnetic monopoles of aU(1)gauge theory, and the central question addressed in this paperis whether the QDM on a diamond lattice can support a quan-tumU(1) liquid phase in which these monopoles are decon-fined. In a finite size system, deconfined monopoles must befree to traverse the periodic boundaries of the cluster, andsothe string defects described above also provide a direct test ofmonopole confinement — a necessary condition for monopoledeconfinement in a periodic cluster is that the ground statesinflux sectors connected by strings should be degenerate [c.f.Eq. (14)]. We return to this idea below.

E. Competing ordered phase

(b)(a)14

3

62

1216

98

45

1

13

89

16 12

2

11

7

11

3

15101510

6

14

13

15

4

7

FIG. 5: (Color online) Numbering convention in the definition ofthe order parametermR for theR-state (Eq. 18): (left) cubic cell ofpyrochlore lattice with 16 sites, (right) corresponding diamond latticecell with 16 bonds.

In order to determine the phase diagram of the quantumdimer model on a diamond lattice (cf. Fig.1), we need also tocharacterize the competing ordered “R-state” [Refs. 12,18].The R-state (illustrated in Fig.2) occupies the 8-site (16-bond) cubic unit cell of the diamond lattice (16-site on the py-rochlore lattice), and is 8-fold degenerate. TheR-state breaksthe inversion symmetry of the diamond lattice, its 8-fold de-generacy is best thought of as2 × 4 states with opposite chi-rality. The4 states with equal chirality can be related to eachother with rotations or translations. In terms of the gauge field,this 8-fold degeneracy can be thought of as a 4-fold choice of111 axis (for a given tetrahedron), and a two fold choice ofchirality about that axis. However for purposes of simula-

7

TABLE II: Coefficients of bond occupation numbers within thesix-dimensional irreducible representation used to define an order pa-rameter for the orderedR-state, following Eq. (18).

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

f1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1

f2 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1

f3 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1

f4 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1

f5 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1

f6 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1

tion it is easier to keep track of an order parameter definedin terms of the occupation numbers for dimers on diamondlattice bonds.

We define this through a 6-dimensional irreducible repre-sentation of the diamond lattice point group as follows

mR =

√

√

√

√

6∑

η=1

m2R,η, (18)

where

mR,η =

16∑

ξ=1

f ξη nξ. (19)

Here mR,η measures the occupancy of diamond latticebonds/pyrochlore lattice sites, numbered as in Fig.5,weighted by the factorsf ξ

η given in TableII .Naively, one might have expected the order parameter to

have been given by the sum of projectors

8∑

ζ=1

〈Rζ |ψ〉2

where |Rζ〉 are the 8 possible orderedR-states(ζ = 1, 2, ..., 8). However,

8∑

ζ=1

〈Rζ |ψ〉 =N

4

measures the total number of dimers (equal to one quarter ofthe number of pyrochlore sites), and the only linearly inde-pendent combinations of|Rζ〉 are those given in Table.II .

III. EVIDENCE FOR A QUANTUM U(1)-LIQUID PHASE

A. Comments on simulations

In this section of the paper, we calculate the ground stateproperties of the quantum dimer model Eq. (1) using theGreen’s Function Monte Carlo (GFMC)26,27 method. Thismethod has previously been applied very successfully to

QDM’s in two dimensions28–30. We benchmark our GFMCsimulations against results from exact diagonalization forsmall clusters, against a perturbation theory about the RKpoint forµ→ 1 and, in AppendixD, against expansions abouta perfectly orderedR–state forµ < 0. We chose to use GFMCbecause it automatically conserves the flux quantum numberswhich are central to our study of the QDM, and because itcan provide numerically exact answers for the ground state(T = 0) properties of reasonably large clusters.

GFMC works by calculating a set of running averages on arandom walk through configuration space, using on Hamilto-nian matrix elements to move from one configuration to an-other. For large clusters and/or complicated problems, thisrandom walk must be guided using a trial wave function, pre-viously optimized by a suitable variational calculation. In thissense GFMC can perhaps best be understood as a systematicway of improving upon a known variational wave function.

In AppendixC we discuss a range of different variationalwave functions which can be used to explore the ground stateof the QDM on a diamond lattice. The most useful guide func-tion we have found to date is :

|ψVarαβγ〉φ,c = exp[αNf + βmR +

∑

〈ij〉

γijτiτj ]|ψRK〉φ,c (20)

Here |ψRK〉φ,c is the equally weighted superposition of alldimer configurations within the maximally-connected subsec-tor with flux ~φ = (φ1, φ2, φ3), i.e. those which are connectedto anR-state by string excitations and/or matrix elements ofEq. (1). The sumij runs over pairs of hexagonal plaquetteswithin the diamond lattice, andτi = 0,1 is an Ising-like vari-able which takes on the valueτi = 1 when the hexagonal pla-quettei is “flippable”. Nf =

∑

i τi counts the total numberof flippable plaquettes andmR is the order parameter associ-ated with theR–state, as defined in Eq. (18). The variationalparametersα, β andγij are minimized using a stochastic re-configuration algorithm31. For large lattices there are a manydifferent pairs of hexagonal plaquettes. In practice we restrictourselves to a set of up to 40 inequivalentγij .

Our GFMC algorithm follows the prescription of [27]. Sim-ulations are typically performed using up to a few thousandsof “walkers” in parallel. The transition probabilities betweendimer configurations are calculated using the matrix elementsfrom Eq. (1). If the guide wave function is exact (for exampleat the RK point, with all variational parameters set to zero)GFMC converges to the exact answer in a single step. Awayfrom the RK point simulations typically converge after a fewhundreds of reconfiguration steps. A “forward walker” tech-nique is used to calculate the average value of the order pa-rametermR [27].

For ~φ = 0, all simulations are started from a perfectlyorderedR-state, and explore only dimer configurations con-nected with this. For more general~φ, starting configura-tions are constructed using the prescription described in Sec-tion II D. In neither case is the GFMC technique ergodic, inthe sense of exploring all dimer configurations which might,in principle, contribute to the ground state. However GFMCperformed in this way does produce results which are repre-

8

sentative of the entire configuration space. This question isdiscussed at length in SectionIV.

B. Death of an ordered state

We begin by studying how the orderedR-state evolves as afunction ofµ. In Fig.6 we present both VMC and GFMC sim-ulation results for the order parametermR, for−0.2 ≤ µ ≤ 1,in a series of cubic clusters of the form [100] which havethe full symmetry of diamond lattice and have2L3 bonds[cf. TableI]. The linear dimension of the clusters ranges fromL = 4 (128 bonds) toL = 10 (2000 bonds).

0 0.05 0.1

L-3/2

0

0.1

0.2

0.3

0.4

orde

r pa

ram

eter

-0.2 0 0.2 0.4 0.6 0.8 1µ

0

0.2

0.4

0.6

0.8

1

orde

r pa

ram

eter

128, VMC

128, GFMC

128, ED

432, VMC

432, GFMC

1024, VMC

1024, GFMC

2000, gfmc

FIG. 6: (Color online) Collapse of the ordered state: the order pa-rametermR in the sector connected to theR-state. The results ofVMC (dashed lines) and GFMC (solid lines) simulations are shownfor series of clusters with cubic symmetry, and compared to exactdiagonalization results for the smallest, 128 bond cluster. The insetshows finite size scaling of the order parameter at the RK point.

For the smallest 128-bond cluster we find very close agree-ment between GFMC and VMC simulations, and perfect nu-merical agreement between GFMC results and exact diago-nalization calculations. This cluster is too small to be repre-sentative, but a point of inflection inmR for µ ≈ 0 hints at apossible phase transition out of theR-state. For larger clus-ters, a first-order phase transition out of theR-state is clearlyvisible as a jump in the order parametermR at a critical valueof µ = µc(L). This jump is most pronounced in VMC cal-culations, where it occurs for smaller values ofµ, i.e. deeperinside the ordered state. The jump in themR shifts steadily tolarger values ofµ as the size of the system is increased. For theL = 6, 432-bond cluster, a jump inmR is observed in VMCfor µc ≈ 0.25, and in GFMC forµc ≈ 0.45. Meanwhile for

0 0.0005 0.001 0.0015 0.002

1/L4

0

0.2

0.4

0.6

0.8

1

µ c

100110111

FIG. 7: (Color online) Estimation of the critical value ofµc(∞)for the transition from the orderedR-state to the quantumU(1)-liquid, taken from GFMC simulation of three different families ofclusters. Linear fits toµc(L) as a function of1/L4 are made subjectto the constraint that all data sets must converge to the samevalueµc(∞) for 1/L → 0, regardless of the cluster geometry. We findµc(∞) = 0.75 ± 0.02.

the largestL = 10, 2000-bond cluster, the GFMC simulationssuggest that this transition takes place forµc ≈ 0.7.

For µc(L) < µ < 1,mR(µ) takes on a smaller, roughlyconstant value. However the finite size effects are large, withthe value ofmR decreasing as the system size is increased.Much larger systems can be simulated at the RK pointµ ≡ 1,where the form of the ground state wave function is knownexactly. Various methods of simulating at the RK point arediscussed in SectionIV. In the inset to Fig.6, we show re-sults obtained using local updates within the configurationsexplored by GFMC simulations. We find thatmR vanishes inthe thermodynamic limit asmR(µ ≡ 1) ∼ L−3/2.

Given the strong finite size effects visible in these data, itis important to make an estimate ofµc(L → ∞), to ensurethat the competing phase forµ > µc(L) does not collapse toa single point in the thermodynamic limit. The properties ofthe system forL→ ∞ mustbe independent of the cluster ge-ometry, and soµc(L) should interpolate to the same value ofµc(∞) for all different families of clusters. In Fig.7 we plotµc(L) as a function of1/L4 for clusters of the type [100],[110] and [111], as defined in TableI. All data collapse to asingle valueµc(∞) = 0.75 ± 0.02. We note that the slightlydifferent method of estimation used in Ref. 14 (fitting a sepa-rate line to each family of clusters) gave a very similar resultµc(∞) = 0.77± 0.02.

The fact thatµc(∞)− µc(L) ∼ 1/L4 provides strong, al-beit indirect, evidence for the existence of the gapless photonexcitations which are a signal feature of the quantumU(1)-liquid state. We can argue as follows : since the jump inmR implies a first-order, zero-temperature phase transition,the finite size correction toµc must follow from the finite sizecorrections to the ground state energy of the two competingphases. All excitations about theR-state are gapped, and finitesize corrections to its ground state energy should therefore be

9

FIG. 8: (Color online) Monopole string tensionκ1 [Eq. 21] fora range ofµ spanning the orderedR-state and proposed quantumU(1)-liquid phase. Results are taken from VMC (dashed lines)and GFMC (solid lines) simulations for a series of clusters with cu-bic symmetry, and compared to exact diagonalization results for thesmallest, 128 bond cluster.

small. Assuming that the competing phase is a quantumU(1)-liquid with gapless photon excitations, the finite size correc-tions to its ground state energy will scale as∼ 1/L4, as de-scribed by Eq. (15). Provided that the ground state energiesof both states evolve smoothly nearµc(L) (which is known tobe true from both exact diagonalization and quantum MonteCarlo studies), finite size corrections toµc should also thenscale as∼ 1/L4.

On the strength of the simulation results formR presentedabove, it seems very reasonable to conclude that the phasediagram proposed in Fig.1 is correct, with a single first-orderphase transition from an orderedR-state to a quantumU(1)-liquid occurring forµc = 0.75 ± 0.02. In what follows, wepresent a variety of other evidence that this is indeed the case.

C. Collapse of monopole string tension

If the state occurring forµc(L) < µ < 1 is indeed aquantumU(1)-liquid, it should possess gapped, deconfinedmonopole excitations8. We can study the motion of thesemonopoles indirectly through the thought experiment usedto construct configurations in different flux sectors in Sec-tion II D. As illustrated in Fig. 4, separating the twomonomers (monopoles) created by removing a dimer, de-stroys flippable hexagons along the “string” of dimers con-necting the monomers. In the crystalline state, where the po-tential energy dominates, one can expect that there is a finiteenergy cost of such an operation, proportional to the lengthofthe string. Where the string defect threads a periodic bound-ary of the cluster, its energy cost can be calculated as the gapto the ground state with one additional unit of flux.

We therefore define the “string tension” as the energy dif-ference between the ground state in the zero-flux sector (E0)and a sector with a single string defect (E1), divided by the

length of the string (L)

κ1 =E1 − E0

L=

∆1

L(21)

For the string defects considered,L is simply the linear di-mension of the cluster. This string tension is an indirect mea-sure of the attractive, confining force between monopole ex-citations. It must therefore vanish in the deconfined quantumU(1)-liquid, but will be finite in theanyordered state.

In Fig. 8 we plot simulation result for the monopole stringtensionκ1 as a function ofµ for the same set of clusters andrange of parameters as Fig.6. Deep inside the ordered state,for negative values ofµ, classical effects dominate. A singlestring defect in a perfectly orderedR-state destroys2L flip-pable hexagons at a potential energy cost of2L|µ|. This lin-ear relationship betweenκ1 and|µ| for µ . 0 is clearly visiblein GFMC simulation results for the larger clusters. Howeverthe string tension drops rapidly as we approach the transitionfrom theR-state into the competing phase atµ = µc(L), andfor large clusters it is essentially zero forµc(L) < µ ≤ 1.

The coincidence between the jump in the order parametermR [Fig. 6] and the collapse in the string tensionκ1 [Fig. 8]lends yet more support to the proposed form of the phase dia-gram for the QDM on a diamond lattice [Fig.1]. In particular,the absence of a string tension forµc(L) < µ < 1 rules outthe possibility that the competing phase is a different orderedstate, for example a three-dimensional analogue of the res-onating plaquette phase found in quantum dimer and quantumice models in two dimensions3,32,33. However we still need torule out the possibility that the competing phase is a differentform of quantum liquid, by confirming explicitly that it is in-deed the quantumU(1)-liquid we have been seeking. This isaccomplished below.

D. Birth of a U(1)-liquid

Within the field-theoretical scenario for a three dimensionalQDM on a bipartite lattice, the quantumU(1)-liquid phase“grows” out of the RK point8,12. Exactly at the RK point, theground states in all flux sectors are degenerate. Away from theRK point, in a finite size system, the ground state in the zero-flux sector has the lowest energy, and the (finite-size) energygaps to the other low-lying flux sectors scale as∆φ ∼ c2φ2/L[cf. Eq. (14)]. By construction,c vanishes at the RK point.

Since the exact ground state is known at the RK point, wecan calculate the finite size gaps∆φ (and implicitly the speedof light c) within perturbation theory about the RK point.Writing

HQDM = HRK − δµNf (22)

whereHRK is defined by Eq. (5), δµ = 1−µ, and the operatorNf is defined by Eq. (3), we find

∆φ,MC = ∆Nφ,MCδµ+O(δµ2) (23)

where

∆Nφ,MC = 〈Nf〉RKφ,MC − 〈Nf〉

RK0,MC (24)

10

0 0.05 0.1 0.151/L

0

0.05

0.1

0.15

coef

ficie

nt φ2

0 10 20 30 40 50

φ2

0

1

2

3

4

5

6

-∆N

φ

43210242000345654888192

0 10 20 30 40 50

φ2

0

0.05

0.1

0.15

-∆N

φ / φ2

(a)

(b) (c)

(a)(a)(a)

FIG. 9: (Color online) Reduction in the average number of flippableplaquettes∆Nφ, as calculated in simulations at the RK point. (a)−∆Nφ plotted as a function ofφ2 for cubic clusters of linear dimen-sionL = 4 (432 bonds) toL = 16 (8192 bonds). (b) The samedata set plotted as−∆Nφ/φ

2 vs φ2 to extract the leading depen-dence onφ. (c) Finite-size scaling of the the coefficient ofφ2 foundfrom fits to−∆Nφ for clusters of linear dimensionL [cf. interceptto ordinate axis in (b), above]. The linear collapse of the coefficientof φ2 with 1/L is consistent with the prediction of the effective fieldtheory for a quantumU(1) liquid Eq. (14).

is the difference in the average number of flippable plaque-ttes between the zero-flux sector and the sector with fluxφ,calculated at the RK point, within the maximally-connectedsubsector of the Hilbert space,λc = MC. The maximally flip-pableR-state belongs to the zero-flux sector, and we thereforeanticipate that∆Nφ < 0. If, as supposed, a quantumU(1)-liquid grows adiabatically from the RK point, then we mustfind

∆Nφ ∼ −φ2

L(25)

In Fig. 9 we show results for∆Nφ obtained from simula-tion of a series of [100] cubic clusters with linear dimensionranging fromL = 6 (432 bonds) toL = 16 (8192 bonds), forflux ~φ = (φ, 0, 0), plotted as a function ofφ2. Simulationswere performed within the maximally-connected subsector ofstates connected to theR-state forφ = 0, and within the sub-sector associated with a given number of string defects forhigher values of flux. Due to large finite size effects, the ex-pected linear relationship between−∆Nφ andφ2 is only vis-ible for clusters with more than 2000 bonds. However the co-efficient ofφ2 can be extracted from a polynomial fit to∆Nφ

0 10 20 30 40 500

2

4

6

∆ φ / (

µ−1)

0 10 20 30 40 50

φ2

0

0.05

0.1

∆ φ / [

(µ−1

) φ2 ]

perturbation theory

µ = 0.99µ = 0.95µ = 0.9

(b)

(a)

FIG. 10: (Color online) Comparison of the energy gaps∆φ foundin GFMC simulations with the predictions of a perturbation theoryabout the RK point [Eq. (22)], for a range ofµ within the pro-posed quantumU(1)-liquid phase. (a) Energy gap∆φ normalizedto −δµ = µ− 1, to extract the leading dependence onµ, and plot-ted as a function ofφ2 [cf. Eq. (14)]. (b) Energy gap∆φ normalizedto −δµ× φ2, to extract the leading dependence on bothµ andφ,plotted as a function ofφ2 [cf. Eq. (14)]. All results are for a 1024-bond cluster with cubic symmetry.

as a function ofφ, at givenL. This is found to be proportionalto 1/L, as shown in the inset to Fig.9. From these results weobtain

c2 ≈ 0.6× δµ (26)

(in the present units) confirming that an incipient quantumU(1)-liquid is present at the RK point. This result is specificto the QDM on a diamond lattice. We note that simulations ofthe QDM on a cubic lattice yield quite different results34.

In Fig.10we compare the results of the perturbation theoryEq. (23) with GFMC calculations made for a set of parametersbordering on the RK point, for a cluster of 1024 diamond lat-tice bonds. The energy gaps obtained from GFMC are plottedas∆φ/(1 − µ) so to compare with perturbation theory, andas a function ofφ2 so as to extract the leading behaviour oftheU(1)-liquid phase. Both sets of data are found to show the

11

1 2 3 4 5 6 7φ

0.1

1

∆ φ /

[φ2 (1

−µ) ]

µ = 0µ = 0.2µ = 0.4µ = 0.5µ = 0.6µ = 0.7µ = 0.8µ = 0.9µ = 0.95µ = 0.99pert. theory

FIG. 11: (Color online) First order quantum phase transition fromthe orderedR-state into the quantumU(1)-liquid, as imaged in thefinite size spectrum of a 1024-bond cluster. GFMC simulationresultsfor the energy gap∆φ are plotted as∆φ/[φ

2(1−µ)] so as to collapsethe spectra for theU(1)-liquid phase. The phase transition is visibleas an abrupt qualitative change in the form of the spectra forµ = 0.7.The results of perturbation theory about the RK point (blackline andpoints) are shown for comparison.

sameφ2 dependence on flux sector, confirming the existenceof a quantumU(1)-liquid bordering the RK point.

E. A phase transition in a spectrum

We are now in a position to draw a set of strong conclusionsabout the zero-temperature phase diagram of the QDM on adiamond lattice (cf. Fig.1). For

µ < µc = 0.75± 0.02

the ground state of the model is an orderedR-state. Forµc < µ < 1 it is a quantumU(1)-liquid with linearly dispers-ing photon excitations and deconfined magnetic monopoles.The quantum phase transition between these two phases atµ = µc is first order. Forµ → 1−, the quantumU(1)-liquidis adiabatically connected with the RK point.

In Fig. 11we present the results of GFMC simulations of acubic 1024-bond cluster, for a range of parameters0 ≤ µ ≤ 1spanning both theR-state and the quantumU(1)-liquid. Weplot the energy gaps∆φ, rescaled as∆φ/[φ

2(1− µ)] so as tocollapse all data within theU(1)-liquid phase. Deep withinthe ordered phase the∆φ should be proportional to both thelinear dimension of the cluster and the flux (number of stringdefects), i.e.

∆Rφ ∼ L× φ

A plot of ∆φ/φ2 at fixedL should therefore show a clear

distinction between aU(1)-liquid (∆φ/φ2 ∼ const.), and an

ordered phase (∆φ/φ2 ∼ 1/φ). For this cluster size a clear

division is observed betweenµ ≥ 0.7 (liquid) andµ ≤ 0.6(ordered). This is unambiguous evidence of a phase transition

( (( ((b)(a)

even odd

even

odd

A

C

E

B

F

F

FIG. 12: (Color online) (a) Block-structure of the Hamiltonianfor the quantum dimer model on a diamond lattice, for an 128-bond cluster with a fixed value of the flux~φ = (0, 0, 0), followingTable III. Connected dimer configurations break up into differentsub-ensembles. These can be divided into two groups, accordingto whether their transition graphs contain an odd or even numberof loops. Both even and odd subsectors contain isolated configu-rations which do not participate in the dynamics of the model, anddo not contribute to the ground state forµ < 1. (b) Contrastingblock-structure of the Hamiltonian for the quantum dimer model ona square lattice for a finite size cluster with a fixed value of the flux~φ = (0, 0). All dimer configurations which contribute to the groundstate forµ < 1 belong to a single connected block. The remaining,isolated, dimer configurations do not participate in the dynamics ofthe model.

from a linearly confining phase to a quantumU(1)-liquid, im-aged in a spectrum. The abrupt, qualitative change in the spec-tra forµ = 0.6 confirms that this phase transition is first-orderin character.

IV. ERGODICITY AND CONSERVATION LAWS WITHINA GIVEN FLUX SECTOR

A. Lessons from exact diagonalization

Since GFMC uses on matrix elements of the QDM Hamil-tonian [Eq. (1)] to connect different dimer configurations, itis guaranteed to respect the flux quantum numbers definedin SectionII D. However, not all dimer configurations withthe same magnetic flux are connected by matrix elements ofthe Hamiltonian. Clearly, therefore, GFMC simulations foragiven flux arenotergodic, in the sense that they do not exploreall dimer configurations which might, in principle, contributeto the ground state for that flux sector. It is therefore veryimportant to consider how the Hilbert space for a given flux~φ breaks up into different subsectors, and to ensure that sim-ulations are carried out in the appropriate sub-sector(s).Tothis end, much can be learnt from the exact diagonalization ofsmall clusters.

In total there are 115,150,848dimer coverings of theL = 2,128-bond, cubic cluster. Of these, 11,835,352 have zero flux(φ = 0). These zero-flux configurations include 1440 isolatedconfigurations with no flippable plaquettes, and 3072 config-urations with a single flippable plaquette which connect onlyto a single other configuration. The remaining 11,830,840

12

configurations can be grouped into a small number of subsec-tors which are connected by the matrix elements of Eq. (1).These are listed in TableIII , and illustrated schematically inFig. 12(a). This situation should be contrasted with the muchsimpler structure of the Hilbert space in the QDM on a squarelattice, where all connected dimer configurations for a givenvalue of flux~φ = (φx, φy) are associated with a single blockof the Hamiltonian — cf. Fig.12(b).

TABLE III: Subsectors of the Hilbert space of the QDM on a dia-mond lattice with zero flux (φ = 0) for a cubic, 128-bond cluster.Subsectors are listed in order of increasing dimension, with isolatedconfigurations and subsectors consisting of only two configurationsomitted. The largest, maximally-connected, subsector (A)containstheR-states. Further subsectors can be obtained through cyclicper-mutations of dimers on closed loops — the 8-link loop operationshown in Fig. 13 provides access to subsector B. Subsectors can beclassified according to the number and type of loops in their transi-tion graph.

subsector

label

λc

multiplicity dimension

number of

loops in

transition

graph (mod 2)

longest

irreducible

loop in

transition graph

A 1 7794744 0 2

B 1 3468032 1 8

C 3 131584 0 10

D 3 45184 0 14

E 8 3376 0 14

F 192 40 1 12

For this 128-bond cluster, about two-thirds of the configura-tions withφ = 0 belong to a maximally-connected subsector,labelled A in TableIII , which includes theR-states. The vastmajority of the remaining configurations — about one-thirdof the total — are found in one other subsector, labelled B inTableIII . Subsector B can be accessed from the maximallyconnected subsector by acting on a randomly chosen configu-ration with the update shown in Fig.13— the cyclic permuta-tion of dimers around a closed, 8-link loop. Interestingly,this8-link permutation is the sub-leading term for dimer dynamicsin the derivation of the QDM from Eq. (7) or Eq. (8).

Subsectors C, D, E and Fcannotbe accessed from sub-sector A using the 8-link loop alone. However theycanbe reached by cyclically permuting dimers within suitablychosen configurations belonging to subsector A, on loops oflength 10, 14, 14 and 12, respectively. Since the number andlength of possible such loops grows with the size of the sys-tem, we anticipate that the number of subsectors in the Hilbertspace grows with system size. We put these observations on amore formal footing below.

(b)(a)

FIG. 13: (Color online) 8-link “loop” used to connect dimer config-urations within subsectors A and B of the Hilbert space of theQDMon a diamond lattice [cf. Table III]. (a) 8-link loop of bondswithinthe cubic unit cell of the diamond lattice. Three distinct 8-link loopsof this type are possible, each of which appears octagonal when pro-jected into an appropriate [100] plane. Where exactly 4 bonds areoccupied by dimers, this loop is “flippable”, i.e. a new stateobey-ing the dimer constraint can be obtained by cyclicly permuting thosedimers around the loop. (b) equivalent 8-link loop of sites within thecubic unit cell of the pyrochlore lattice.

1

2

(b)

3

1

(c)

2

(a)

1

FIG. 14: (Color online) Schematic representation of all possiblechanges in the topology of a transition graph which follow from thecyclic permutation of dimers on a hexagonal 6-link loop. Redlinesdenote loop segment with a single dimer. Blue lines denote a loopsegment with an odd number dimers. All loop segments, whether redor blue, start and finish on sites belonging to different sublatticiesof the (bipartite) diamond lattice, denoted here by filled oremptycircles. The number of loops associated with each section ofthetransition graph is shown in the middle of the hexagonal plaquette.This number changes by either 0 or 2.

B. A hidden quantum number

The way in which the Hilbert space of the QDM on a dia-mond lattice breaks up into distinct blocks suggests the exis-tence of further, conserved quantities besides magnetic flux φ.We can gain more insight into this problem, and understandthe action of the 8-link loop shown in Fig. (13) by studyingtransition graphs.

13

3

3

1

2

(c)

(d)

(e)

4

(a)

2

(b)

21

1

2

FIG. 15: (Color online) Schematic representation of all possiblechanges in the topology of a transition graph which follow from thecyclic permutation of dimers on an 8-link loop. The number ofloopsof the transition graphs changes by either 1 or 3.

A transition graph is the set of directed loops formed byoverlaying two dimer configurations on a bipartite lattice,anddrawing arrows from A to B sublattice on those bonds whichare occupied by a dimer in the first configuration, and from Bto A sublattice on those bonds which are occupied by dimersin the second configuration3. Cyclicly permuting the dimerson the loops of the transition graph converts the first configu-ration into the second (and vice versa). By construction, theseloops cannot intersect one another — although one loop maybe threaded through another in a three-dimensional lattice.

A pair of dimer configurations are connected if (and onlyif) we can undo all the long loops in their transition graph us-ing only matrix elements of Eq. (1) — cyclic permutationsof dimers on a 6-link hexagonal plaquette — so that onlytrivial “loops” occupying a single bond remain. These triv-ial loops have length 2, and the maximum number of trivialloops achievable is equal to the number of dimers in the sys-tem — an even number for all of the clusters considered in thispaper. In Fig.14 we enumerate the three possible, topologi-cally distinct, changes in the transition graph associatedwiththe cyclic permutation of dimers on an hexagonal plaquette.In each case the total number of loops in the transition graphchanges by an even number — either 0 or 2.

Cyclic permutations of dimers on an 8-link loop have asomewhat richer structure, illustrated in Fig.15. In this case,the total number of loops in the transition graph changes by

an odd number — either 1 or 3. Consequently, the action of8-link loop cannot be undone using matrix elements of theHamiltonian [Eq. (1)] alone.

More formally, we can associate a unique transition graphwith each distinct dimer configuration by choosing one partic-ular dimer configuration as reference (for present purposes, anR-state is a convenient choice). The number of loops in thistransition graph, modulo 2, is then a conserved quantity withinthe dynamics of the QDM Eq. (1). Furthermore, it must beeven in the subsector connected with theR-state. In TableIIIwe classify the all of the different subsectors of the zero-fluxsector of the 128-bond cubic cluster containing more than twoconfigurations according to this new, Ising, quantum number.

Examination of transition graphs also provides some insightinto the other, smaller, subsectors of the Hilbert space, la-belled C, D, E and F in TableIII . Empirically, we find thatthese can be accessed from the maximally-connected subsec-tor containing theR-states (subsector A) through the cyclicpermutation of dimers on suitably chosen loops of length 10,12, and 14. Closed loops in the transition graph can generallybe contracted through the repeated permutation of dimers on6-link loops of the type found in the Hamiltonian Eq. (1). Byconstruction,all non-trivial loops in the transition graph fortwo configurations within subsector A can be removed in thisway, leaving only trivial loops of length 2. The different sub-sectors B—F listed in TableIII can therefore be classified ac-cording to the longest irreducible loop in their transitiongraphwith a (suitably chosen) state within subsector A. The lengthsof the irreducible loops are listed in TableIII .

It is interesting to note that the Ising quantum number asso-ciated with the number of loops in the transition graph is evenwhere the longest irreducible loop is of length 2, 10 or 14,and odd where the longest irreducible loop is of length 8 or12. We infer (but have not proved) that topological selectionrules of the form illustrated in Fig.14apply for updates basedon loops of length 6, 10, 14. . . , and of the form illustrated inFig.15for updates based on loops of length 8, 12, 16. . . Cyclicpermutations on loops of length 6, 10, 14. . . are precisely theform of dynamics permitted for (spinless) fermionic modelsof the form Eq. (7), while bosonic models also permit cyclicpermutations on loops of length 8, 12, 16. . . .35 We thereforespeculate that the number of loops in the transition graph willbe a conserved quantity (mod 2) for the most general QDMHamiltonian derived from a fermionic model on a pyrochlorelattice, butnot for the equivalent bosonic or spin model.

Clearly, more questions could be asked about the non-ergodicty of the QDM on a diamond lattice. What, for ex-ample, are the the conserved quantities which distinguish thesmaller subsectors of the Hilbert space ? And how do thenumber, multiplicity, and dimension of these subsectors growwith the size of the system ? For the purposes of this paper,however, the most pressing need is to understand the impact ofthis non-ergodicity on the simulations described in Section III .It is this to which we now turn.

14

C. Can we trust simulations ?

It is clear from the analysis above that QDM on a diamondlattice isnot ergodic within a given flux sector. The analy-sis which leads to the phase diagram Fig1 rests on simula-tions performed within the (maximally connected) subsectorof states connected to theR-state by the matrix elements ofEq. (1), and on the systematic construction of configurationswith finite flux using “string” defects’ [cf. Fig. (4)]. Beforewe can have full confidence in these results we must thereforeaddress the following question :

What bias do we introduce into our results by simulatingonly within these subsectors of the Hilbert space ?

The enumeration of states for an 128-bond cluster confirmsthe common-sense expectation that we can access the largestsubsector of the Hilbert space for zero flux by seeding simula-tions from anR-state. Moreover, we now understand how toextend these simulations to configurations with an odd num-ber of loops in their transition graph, through use of an 8-linkloop. However there remain a non-vanishing fraction of dimerconfigurations — about 5% for the 128-bond cubic cluster[TableIII ] — which can only be reached by other, more com-plex updates.

It seems reasonable to expect that these unsociable con-figurations will (a) contribute little to the ground state wavefunction and (b) constitute a vanishing fraction of the totalnumber of dimer configurations in the thermodynamic limit.However this reasonable expectation must be approached withsome caution, not least because, in the absence of a full un-derstanding of conserved quantities, it is hard to assess howthese neglected corners of the Hilbert space grow with systemsize.

We can most easily get a handle on the consequences ofnon-ergodicity at the RK point, where simulations for groundstate properties amount to performing classical averages overall dimer configurations within a given flux sector. These canbe performed in several different ways

1. As in SectionIII D , using “local” updates based on thematrix elements of the Hamiltonian.

2. Using a “loop” algorithm developed for the simulationof ice models13,36.

3. Using a “worm” algorithm based on movement ofmonomers37.

In the “loop” algorithm, a directed random walk alongbonds is used to construct a self-intersecting path in whichall flux arrows associated with dimers have the same sense.The dimers which lie along this path are each cycled one linkfurther along, reversing the sense of the flux around the closed“loop”, while preserving the dimer constraint.

In the “worm” algorithm, a single dimer is removed, and thetwo monopoles created are allowed to diffuse freely aroundthe lattice until they meet, at which point the missing dimeris re-introduced. We have checked explicitly that “loop” and“worm” simulations lead to identical results for the QDM onthe diamond lattice, and concentrate here on comparing re-sults obtained using “local” and “loop” updates.

0 5 10 15 20 250

0.05

0.1

0.15

0.2

norm

aliz

ed n

umbe

r of

con

figur

atio

ns all 0-flux sectors

connected to the R-state

50 60 70 80 90 100N

f

0

0.02

0.04

0.06

(a)

(b)

FIG. 16: (Color online) Histogram of the number of flippable pla-quettes in different sets of dimer configurations, calculated at RKpoint. (a) Results for an 128-bond cubic cluster, calculated withinin the entire zero-flux sector (blue, shaded columns), and inthe sub-sector connected to theR-state (unshaded columns). (b) Equivalentresults a for an 1024-bond cluster.

A necessary condition for the existence of a quantumU(1)-liquid in a QDM is that the average number of flippableplaquettes in the dimer configurations belonging to a givenflux sector scale as∆Nφ = Nf(~φ) − Nf(~0) ∼ −φ2/L[Eq. (25)]. In Fig. 16(a) we compare the probability distri-bution for the number of flippable plaquettesNf calculatedfor all dimer configurations within the zero-flux sector (green,shaded columns) with that calculated within the maximallyconnected subsector containing theR-state (columns withoutshading). Identical results were obtained both within simula-tion, using “loop” updates to explore all dimer configurations,and “local” updates to access to the maximally-connectedsub-sector, and within exact diagonalization, by explicit enumer-ation of all dimer configurations. The probability distributionfor the number of flippable plaquettes is markedly differentinthe two cases, and the mean value ofNf clearly higher for thesubset of configurations connected with theR-state.

At first sight this disagreement might seem rather alarming,but it is simply a finite-size effect. In Fig.16(b) we presentthe results of an identical study for a larger, 1024-bond clus-ter. The same, Gaussian, probability distribution with mean〈Nf〉 = 73.27 and varianceσNf

= 6.21 is found for boththe full set of dimer configurations sampled by the “loop” al-gorithm, and the maximally-connected subsector explored bythe “local” updates. Similar gaussian probability distributionsare found in sectors of the Hilbert space with finite flux, and

15

-1 -0.5 0 0.5 1µ

0

0.2

0.4

0.6

0.8

1

orde

r pa

ram

eter

432, R-connected432, after 8-loop1024, R-connected1024, after 8-loop

FIG. 17: (Color online)µ-dependence of the order parametermR

for theR-state, within two different subsectors of the Hilbert space:starting from theR-state in fluxφ = 0 subsector (open symbols)and in the subsector that is connected to it by an 8–site loop (solidsymbols). Results are calculated using Variational Monte Carlo sim-ulation for cubic clusters with 432 (triangles) and 1024 (circles)diamond-lattice bonds.

are again independent of the method of simulation.There are two obvious possible explanations as to why the

same probability distribution should be found for averagescalculated at the RK point in the full set of dimer configu-rations, and those calculated within its maximally-connectedsubsector. One is that the relative size of the maximally-connected subsector grows with system size, so that it comesto dominate all averages in the thermodynamic limit. Theother is that all large blocks of the Hilbert space (within agiven flux sector) are statistically very similar. This might oc-cur if, for example, configurations in different subsectorsdif-fered only by some local defect which could not be removedusing matrix elements of the Hamiltonian, but which had noimpact on the bulk properties of the state.

To distinguish between these alternatives we can exam-ine how quantum Monte Carlo simulation results in the twolargest subsectors of the Hilbert space depend on the subsec-tor λc in which they are carried out. In Fig.17 we presentthe results of a Variational Monte Carlo study of the orderparameter〈mR〉λc

, for a cubic 432 and 1024-bond clusters.Simulations were carried out within

i) the subsector with fluxφ = 0 containing theR-state;

ii) the subsector with fluxφ = 0 obtained by acting with an8-link loop update on a randomly chosen state from thesubsector containing theR-state.

Within the crystalline, ordered phase, an 8–loop update re-duces themR relative to the subsector containing theR-state.This reduction in order parameter scales as1/L3 — as wouldbe expected for a local defect into an ordered state — and van-ishes in the thermodynamic limit. Meanwhile the energy costof the 8–loop update is a constant in the ordered state. Thisbehaviour should be compared with the topological string de-fect used to change flux sector acting in theR-state ordered

0 10 20 30 40 50

φ20

1

2

3

4

5

6

-∆N

φ

R-connectedafter 8-loop

FIG. 18: (Color online) The difference in the average numberofflippable plaquettes∆Nφ [Eq. (24)], calculated at the RK point fordimer configurations in different subsectors of the Hilbertspace.Green circles denote results for configurations connected to theR-state by matrix elements of Eq. (1) and string defects. Maroon dia-monds denote results for configurations containing an additional 8-link “loop” defect. All results are for a cubic 1024-bond cluster.

phase, whose effect on the order parameter scales as1/L2,and which creates a finite size energy gap∝ L.

In marked contrast, the simulation results for〈mR(µ)〉, inthe quantumU(1)-liquid phase are independent of the subsec-tor within which they are calculated. Intuitively, we imaginethat as theR-state order melts into theU(1)-liquid phase, sodefects in the order parameter melt, too, leaving only an invis-ible partition in the Hilbert space.

We have repeated this analysis for more general local de-fects in theR-state, seeding simulations from configurationsgenerated using randomly chosen loops with 10, 12, 14, 16,18 and 20 links, applied states within the subsector contain-ing theR-state. After simulation starting from a configurationobtained using one 10-, 14- or 18-link loop we recover thesame values of〈mR(µ)〉 as in the maximally connected sectorcontaining theR-state. Meanwhile simulates started from aconfiguration obtained using an 12-, 16- or 20-link loop ex-hibit the same values of〈mR(µ)〉 as those in the second largesubsector accessed using the 8-link loop.

This is consistent with our analysis of the transition graphs:the 8-loop cannot be undone by Hamiltonian matrix updatesand so we remain in a subsector not connected toR-state,for which the average value of the order parameter has to belower. Loops of length 10, 14, 18... can typically be reducedto 6-link loops so that the resulting configuration remains inthe maximally connected sector, while loops of length 12, 16,20... can be reduced to an 8-link and 6-links loops, leading tothe second large subsector. Although we do not have a pre-cise understanding of the origin of the additional subsectorsfound in the 128-bond cluster [TableIII ], from our simulationfor larger systems we tentatively conclude that the statisticallyaccessible configurations with an even(odd) number of loopsin their transition graph are of the type A and B (i.e. connected

16

to theR state directly or by a single 8–link loop).It is also interesting to compare the changes in the liquid

phase ground state energy in detail. The ground state energyin all subsectors must be zero at the RK point. In the liquidphase bordering on the RK point, the gaps between states withdifferent magnetic flux should scale as∆φ ∼ φ2/L [Eq. (14)].This in turn implies that, at the RK point,∆Nφ ∼ −φ2/L[Eq. (25)]. In Fig.18we present the results of simulations car-ried out in maximally-connected subsector for eachφ, and inthe subsector connected to it by a single, local, 8-link “loop”defect. The differences in the values of∆Nφ obtained in thetwo subsectors are smaller than the error bars on the simula-tion.

This analysis represents only a partial solution to the prob-lem of how to understand different subsectors within theHilbert space of a three-dimensional QDM. However for thepurposes of this paper, we gain confidence that GFMC simu-lations performed in the maximally-connected subsector, fol-lowing the prescriptions of SectionII and SectionIII , do givereliable estimates of the ground state properties of the QDMon a diamond lattice.

We can trust these estimates in the ordered,R-state, be-cause the alternative subsectors correspond to excited states,and not to the ground state. And we can trust the estimates inthe quantumU(1)-liquid phase because in that case all statis-tically significant subsectors contribute equally to the groundstate. Thus, while the QDM on a diamond lattice is not er-godic within a given flux sector, this lack of ergodicity has nopathological consequences for the ground state phase diagramas we have calculated it. Our initial analysis suggests thatthesame may not be true of the QDM on a cubic lattice34.

V. CONCLUSIONS

In this paper we set out to show that a three-dimensionalquantum dimer model (QDM) — the QDM on a bipartite dia-mond lattice — could support a quantum liquid ground state,whose excitations include linearly dispersing photons andde-confined magnetic monopoles of an underlyingU(1) gaugetheory. We have identified the order parameter associatedwith the competing, ordered “R-state” and, through use ofquantum Monte Carlo simulation, have shown that it is finiteonly where the ratioµ of kinetic to potential energy terms inthe QDM Hamiltonian is less thanµc = 0.75 ± 0.02. Wehave further shown that (a) the string-tension associated withseparating a pair of monomers (magnetic monopoles) van-ishes forµ > µc, and (b) the finite-size energy spectrum forµc < µ ≤ 1 is consistent with the predictions of the Maxwellaction of conventional quantum electromagnetism.

We conclude that the QDM on a diamond lattice undergoesa first-order phase transition from an orderedR-state into aquantumU(1) liquid phase atµ = µc. This liquid phaseoccupies an extended region of parameter space, terminatingat the RK pointµ = 1. This completes the analysis outlinedin our earlier Letter [Ref. 14], and confirms the validity of thephase diagram Fig.1, previously proposed in Refs 8,12 on thebasis of field-theoretical arguments.

We have also explored how the Hilbert space of the QDMon a diamond lattice breaks up into different, disconnected,subsectors, and identified a new, Ising, quantum number as-sociated with the number of loops in the transition graph of agiven dimer configuration. We argue that the lack of ergod-icity within any given flux sector does not have pathologicalconsequences for the ground state phase diagram, as foundby quantum Monte Carlo simulations. We further show thatthe same ground state phase diagram is obtained, regardlessof whether the dimers are treated as bosonic or fermionic ob-jects.

Taken together, these results would seem to set the exis-tence of a quantumU(1) liquid phase in the QDM on a di-amond lattice beyond reasonable doubt. However the veryfact that such a state exists in this model begs all manner ofother questions. Do other quantum dimer models on bipar-tite lattices in three dimension — for example the cubic lat-tice — also support quantumU(1) liquids, as conjectured inRef. 8 ? What are the nature and consequences of the ad-ditional quantum numbers associated with the non-ergodicityof three-dimensional quantum dimer models ? Which othermodels can support a quantumU(1) ground state, and forwhat range of parameters ? An obvious candidate in this caseis the quantum loop model studied in Ref. 13. Here we findevidence for a quantumU(1) liquid for −0.5 < µ < 1, whichwill be presented elsewhere38.

The greatest excitement, however, would attend to findinga quantumU(1) liquid phase in experiment, and identifyingits deconfined, fractional excitations. Here more quantita-tive analysis is needed, both to pin down the characteristicsof these excitations, and to bridge the gap between idealizedmodels such as Eq. (1), and real materials or artificial assem-blies of cold atoms. Since the coherence temperature of thequantumU(1) liquid will be low in either case, both finitetemperature effects16, and the consequences of competing in-teractions, need to be explored in more detail.

VI. ACKNOWLEDGEMENTS

It is our pleasure to acknowledge helpful conversationswith Kedar Damle, Yong-Baek Kim, Andreas Lauchli,Gregoire Misguich, Roderich Moessner, and Arno Ralko.We are particularly indebted to Federico Becca for ad-vice and encouragement with simulation techniques. Thiswork was supported under EPSRC grants EP/C539974/1 andEP/G031460/1, Hungarian OTKA grants K73455, U.S. Na-tional Science Foundation I2CAM International Materials In-stitute Award, Grant DMR-0645461, and the guest programsof MPI-PKS Dresden and YITP, Kyoto.

Appendix A: Absence of a fermionic sign problem

All results forHQDM are obtained under the assumption thatthe dimers are bosonic objects, i.e., exchanging two dimersdoes not yield a phase. However, as we show in this Appendix,

17

the same effective model can be used to describe the fermioniccase.

FIG. 19: (Color online) Labeling of the sites of the pyrochlore latticeby enumerating one of the two types of tetrahedra and four sublat-tices on each tetrahedron. This labeling allows us to show that theHamiltonianHQDM has no fermionic sign problem even if the under-lying particles are fermions.

Let us now assume that the dimers are objects which arecreated (annihilated) by fermionic operatorsc†i,σ (ci,σ). Theindexi ∈ {1, . . . , n} refers to a tetrahedron and the indexσ ∈{1, 2, 3, 4} refers to the site on that tetrahedron (see Fig.19).Since the diagonal partNf of the Hamiltonian inHQDM isnot affected by changing to fermions, we only focus on thering-exchange processes. These can be written in terms of thefermionic operators as

Kf =∑

(

c†ia,σacia,σ′

a

c†ib,σbcib,σ′

b

c†ic,σccic,σ′

c

+H.c.)

,

(A1)where the sum is taken over all hexagons, andia < ib < ic.We thus choose only one of the twelve different ways of howthe three fermions can hop around the hexagon.

We show now that all twelve ways, which are resulting fromclockwise/counter-clockwise processes as well as different se-quences, have a positive matrix element. (i) Fermion configu-rations which result from clockwise or counter-clockwise pro-cesses are related by an even number of fermion exchanges,e.g. (1, 2, 3) ↔ (2, 3, 1), and thus have the same sign. Inpassing we mention that this is generally true for ring-hoppingprocesses which involve anodd number of fermions, whileprocesses with anevennumber of fermions have the oppositesign and thus cancel each other. (ii) the matrix-element doesnot depend on the sequence in which the fermions hop. Thiscan be seen if we choose “ordered” fermion configurationswith respect to the tetrahedra, i.e., we fill the vacuum|0〉 by

|c〉 = c†iN ,σN. . . c†i2,σ2

c†i1,σ1|0〉. (A2)

Because of the hardcore-dimer constraint, there is at most onefermion on each tetrahedroni which then has an arbitraryσ.In Hamiltonian (A1), the fermionic operators come in pairs foreachi and thus there is always an even number of fermionicoperators which need to be exchanged to evaluate the matrixelements. Thus, the sign does not depend on the sequence in

which the operator pairs are applied and all non-zero matrixelements are equal to+1. The argument works only for thedimer model. If we consider a related loop model on the samelattice, there are two fermions per tetrahedron (i.e., two dimersattached to each site) and the sign problem can no longer beavoided using this method.

Appendix B: Landau theory for R-state

In this paper we studied the zero-temperaturequan-tum phase transition from the orderedR-state into aquantumU(1)-liquid. However we note that the finite-temperature, phase transitions between ordered and classicalU(1) “Coulomb” phases in three-dimensionalclassicaldimermodels have also proved be a very interesting7,39–41. Thesephase transitions do not fall into the conventional Landauparadigm, since the high temperature Coulomb phase exhibitsthe power-law decay of correlation functions characteristic ofa U(1) gauge theory, rather than the exponential decay pre-dicted by a Ginzburg-Landau theory of the low-temperatureordered state.

In this context it is interesting to note that the present orderparameter symmetry permits a third-order invariant in a Lan-dau action, as well as the usual quadratic and quartic terms :

∆R = a

6∑

i=1

m2

R,i

+b (mR,1mR,2mR,3 +mR,4mR,5mR,6)

+c1

6∑

i=1

m4

R,i + c2

(

6∑

i=1

m2

R,i

)2

+c3

[

(

m2

R,1 +m2

R,2 +m2

R,3

)2

+(

m2

R,4 +m2

R,5 +m2

R,6

)2]

+c4(

m2

R,1m2

R,4 +m2

R,2m2

R,5 +m2

R,3m2

R,6

)2

(B1)

In the ordered phase with〈mR〉 = m 6= 0, the cubic termselects eight possible configurations. Forb < 0 these havethe form given in TableIV, and are of course the eightR-stateconfigurations, which become exact ground states of the quan-tum problem in the limitµ → −∞. Forb > 0 the cubic termdoes not select a valid (static) dimer configuration.

Appendix C: Comparison of variational wave functions

In this Appendix we explore the transition from solid to liq-uid phases of the quantum dimer model on a diamond latticeusing a variety of variational wave functions. This is an es-sential preliminary to Green’s function Monte Carlo (GFMC)simulations, which rely on importance sampling of configura-tions to achieve convergence in a finite time. Except wherestated otherwise, the calculations presented here were per-formed using variational Monte Carlo, within the stochastic

18

TABLE IV: Eight R-states selected by the cubic invariantb < 0 inEq. (B1), expressed in terms of the coefficients of the order parameterEq. (18), within an ordered state withm = 〈mR/

√3〉. The four

R-states labelledζ = 1, . . . , 4 have the opposite chirality from thoselabelledζ = 5, . . . , 8.

ζ mR,1 mR,2 mR,3 mR,4 mR,5 mR,6

1 m m m 0 0 0

2 m m −m 0 0 0

3 −m m −m 0 0 0

4 −m −m m 0 0 0

5 0 0 0 m m m

6 0 0 0 m −m −m7 0 0 0 −m m −m8 0 0 0 −m −m m

reconfiguration algorithm31,42, for a 1024-bond cluster withcubic symmetry.

0 0.1 0.2 0.3 0.4 0.5α

-3

-2

-1

0

1

µ

12843210242000

FIG. 20: (Color online) Value ofµ for which the one-parametervariational wave function|ψVar

α 〉 [cf. Eq. (C1)] has the lowest energy,as a function of the variational parameterα. Results are plotted forcubic clusters ranging in size from 128 to 2000 bonds.

Perhaps the simplest wave function we can consider is

|ψVarα 〉φ,c = exp [αNf ] |ψRK〉φ,c (C1)

whereα is the single variational parameter associated withthe total number of flippable hexagonsNf , and |ψRK〉φ,c isthe RK point ground state function, i.e. the equally weightedsum of all dimer configurations within a given subsector ofthe Hilbert space. In the discussion of ground state proper-ties which follows, we will consider the maximally connectedsubsector for~φ = 0, which comprises all dimer configurationsconnected to anR-state by matrix elements of Eq. (1).

We can view Eq. (C1), as the Jastrow form which followsfrom Eq. (22). For this reason we anticipate that this wavefunction will work well in the vicinity of the RK point, but willtend to over-emphasize any liquid phase occurring forµ→ 1.The wave function|ψVar

α 〉φ,c has the advantage that the value

of µ which minimizes the ground state energy for a givenαcan be found by evaluating averages at the RK point

µ =1

2〈NfKf +KfNf〉RK − 〈Kf〉RK〈Nf〉RK

〈N 2

f〉RK − 〈Nf〉2RK

(C2)

whereKf is defined by Eq. (2). In Fig. 20 we present resultsfor µ as a function ofα for cubic clusters ranging in size from128 to 1024 diamond lattice bonds. It is tempting to interpretthe abrupt change in∂µ/∂α at α ≈ 0.4, µ ≈ −0.5 as evi-dence of a first order phase transition. However comparisonwith exact diagonalization, and with other variational wavefunctions below, suggests that|ψVar

α 〉 gives very poor energiesaway from the RK point, and is only really useful forµ→ 1.

0

0.2

0.4

0.6

0.8

1

orde

r pa

ram

eter

Ψα

Ψβ

Ψαβ

Ψαβγ

-1 -0.5 0 0.5 µ

-0.25

-0.2

-0.15

-0.1

-0.05

0

Ene

rgy

per

site

(a)

(b)

FIG. 21: (Color online) Comparison of different variational wavefunctions for 1024-bond cluster: (a) order parameter and (b) energyper site: (green)|ψVar

α 〉 [cf. Eq. (C1)] (orange)|ψVarβ 〉 [cf. Eq. (C3)];

(red) |ψVarαβ〉[cf. Eq. (C4]); (black)|ψVar

αβγ〉 [cf. Eq. (20)].

We can construct another simple one-parameter variationalwave function based on the order parametermR for theR-state

|ψVarβ 〉φ,c = exp[βmR]|ψRK〉φ,c (C3)

This wave function gives a good account of the orderedR-state, and exhibits a clear first order phase transition transitionfrom as a jump in〈mR〉 for µ ≈ 0.25 in a 1024-bond cubiccluster — cf. Fig.21(a). However from Fig.21(b) we can seethat|ψVar

α 〉 has a lower energy approaching the RK point.Seeking the advantages of both terms, we try a two-

parameter wave function

|ψVarαβ 〉φ,c = exp[αNf + βmR]|ψRK〉φ,c (C4)

19

0 0.1 0.2 0.3 0.4 0.5order parameter

-0.07

-0.065

-0.06

-0.055

-0.05

-0.045

ener

gy

from an ordered configurationfrom a disordered configuration

0.2 0.25 0.3 0.35 0.4 0.45 0.5 µ

0

0.1

0.2

0.3

0.4

0.5

orde

r pa

ram

eter

FIG. 22: (Color online) Quantum hysteresis — variational MonteCarlo results for|ψVar

αβγ〉 [cf. Eq. (20)] in a 1024-bond cluster. (left)energy per site as a function of the order parametermR, (right) orderparameter as a function ofµ. Simulations converge to different localminima for a range of values of0.28 < µ < 0.43, according towhether they are started from an ordered or a disordered state.

This gives lower energies overall, and predicts a transitionfrom solid to liquid forµ ≈ 0.2 in a (1024 bonds). Howeverthis wave function does a poor job of describing the quantummelting of theR-state.

In order to better model this, we introduce correlations be-tween individual flippable plaquettes, on which all dynamicsin the QDM depend

|ψVarαβγ〉φ,c = exp[αNf + βmR +

∑

ij

γijτiτj ]|ψRK〉φ,c

(C5)

Hereτi = 1 if the plaquettei is flippable andτi = 0, oth-erwise. For complete generality we should also allow for thechirality of the flippable plaquette, and consider a separateγijfor each indistinguishable pair of plaquettes in the cluster. Forthe smallest 128 bond cluster there are 16 of these, and thenumber grows to 84 in the 1024-bond cluster. However inpractice we do not find that including chirality brings signifi-cant gains in energy, and we restrict our calculations to a setof about 40γij .

This wave function has a significantly lower energy thanany of the alternatives considered above for−0.5 < µ < 0.25,and predicts a transition forµ ≈ 0.4 (1024-bond cluster —black lines in Fig.21). The wave function also treats fluctu-ations accurately enough to capture some a part of the hys-teresis associated the quantum melting of a solid. In Fig.22we show the order parameter and energy for the 1024-bondcluster, obtained using|ψVar

αβγ〉, seeding VMC from differentstates. Starting from an ordered state, we obtain an orderedstate solution forµ . 0.42, while starting from a disorderedconfiguration we could see a minimum forµ & 0.3. These re-sults confirm the first order character of the phase transition,and extend the usefulness of|ψVar

αβγ〉 as a guide function forGFMC some way into the region where it predicts a liquidphase.

-1 -0.5 0 0.5 1µ

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

ener

gy p

er s

ite

1024, gfmcPade approximantsperturbation theory

FIG. 23: (Color online) Energy per site as a function ofµ. GFMCresults for 1024-bond cluster (points) are compared to Padeapprox-imants for series expansion around an ordered state (solid lines) andperturbation expansion in the vicinity of the RK point (dashed lines).

-1 -0.8 -0.6µ

1

1.5

2

2.5

κ 1

Pade432, gfmc1024, gfmc

FIG. 24: (Color online) Monopole string tensionκ1 = ∆1/L, plot-ted for a range ofµ spanning the orderedR-state phase. Solid linesdenote different Pade approximants to the series expansion about anR-state configuration [cf. Eq. (D2)]. Points denote the results ofGFMC simulations of 432-bond (red) and 1024-bond (green) cubicclusters.

Appendix D: Series expansion about the orderedR-state

There are two limits in which the exact ground state ofEq. (1) is known. One is the RK pointµ = 1, where theground state is the equally weighted sum over all dimer con-figurations. The other isµ → −∞, where the ground state isone of eight degenerateR-state configurations. In the secondcase we can construct a perturbation theory in1/µ by expand-ing fluctuations about an orderedR-state. We find that theground state energy (per bond) in the zero-flux sector behaves

20

as

E0

N=

µ

4+

1

24

1

µ−

229

47520

1

µ3

+37811909

48926592000

1

µ5+O

(

1

µ7

)

. (D1)

The first term in this expansion counts the number of flippableplaquettes in theR-state (a quarter of the number of bonds).Subsequent terms are derived combinatorially, by countingthenumber of possible ways of returning to the the initialR-state,using a given number of applications ofKf and following therecipe of the Rayleigh-Schodinger perturbation theory. Thiscan be calculated for a finite-size cluster, or an infinite lattice;the results quoted here are for an infinite lattice. The resultingseries Eq. (D1) contains only terms of odd order in1/µ.

The series forE0/N [Eq. (D1)] can be continued to valuesof µ relevant to our simulations by means of Pade approxi-mants. In Fig.23 we compare several different Pade approx-imants to Eq. (D1), with the ground state energy per bondfound in GFMC simulations of a 1024-bond cluster. We findessentially perfect agreement between our simulations results

and the results of perturbation theory forµ . −0.8, at whichpoint the different Pade approximants start to diverge. Someof the Pade approximants remain in agreement with the sim-ulations up toµ ≈ −0.5. For comparison, we also includeresults forE0/N derived in an expansion about the RK point,as described in SectionIII D . Again, the agreement betweenperturbation theory and simulation is found to be very good inthe limit µ→ 1.

Forµ → −∞, it is also possible to perform a series abouta state with unit-flux, i.e. anR-state with a single string exci-tation. From Eq. (21) we find the monopole string tension tobe

κ1 = −2µ+1

15

1

µ−

71243

1995840

1

µ3

+1044561213049

41535007113600000

1

µ5+O

(

µ−7)

. (D2)

In Fig. 24 we compare the perturbational expansion andGFMC simulation results for the string tensionκ1. The pre-diction of the perturbation theory forµ . −1 matches well tothe results of simulations forµ & −1.

1 M. E. Fisher, Phys. Rev.124, 1664 (1961)2 P. W. Kastelyn, Physica27, 1209 (1961)3 D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett.61,2376

(1988).4 G. Misguish and C. Lhuillier, inFrustrated Spin Systems, Ed

H. T. Diep, World Scientific, Singapore (2004).5 R. Moessner and S. L. Sondhi, Phys. Rev. Lett.86, 1881 (2001).6 G. Misguich, D. Serban and V. Pasquier, Phys. Rev. Lett.89,

137202 (2002).7 D. A. Huse, W. Krauth, R. Moessner, and S. L. Sondhi, Phys. Rev.

Lett. 91, 167004 (2003).8 R. Moessner and S. L. Sondhi, Phys. Rev. B68,184512 (2003).9 P. W. Leung, K. C. Chiu and K. J. Runge, Phys. Rev. B54, 12938

(1996).10 F. Trousselet, D. Poilblanc and R. Moessner, Phys. Rev. B78,

195101 (2008)11 R. Moessner, S. L. Sondhi and P. Chandra, Phys. Rev. B 64,

144416 (2001).12 D. L. Bergman, G. A. Fiete, and L. Balents, Phys. Rev. B73

134402 (2006).13 M. Hermele, M. P. A. Fisher, and L Balents, Phys. Rev. B69,

064404 (2004).14 O. Sikora, F. Pollmann, N. Shannon, K. Penc, and P. Fulde Phys.

Rev. Lett.103, 247001 (2009).15 P. Fulde, K. Penc and N. Shannon, Ann. Phys. (Leipzig)11, 892

(2002)16 A. Banerjee, S. V. Isakov, K. Damle, and Y. B. Kim, Phys. Rev.

Lett. 100, 047208 (2008).17 J. F. Nagle, Phys. Rev.152, 190 (1966); J. Math. Phys.7, 1485

(1966)18 D. L. Bergman, R. Shindou, G. A. Fiete, and L. Balents, Phys.

Rev. Lett96 097207 (2006);erratumibid 97, 139906 (2006)19 J. Ruostekoski, Phys. Rev. Lett.103, 080406 (2009)20 F. Vernay, A. Ralko, F. Becca, and F. Mila, Phys. Rev. B 74,

054402 (2006).21 R. Youngblood, J. D. Axe, and B. M. McCoy, Phys. Rev. B21,

5212 (1980).22 C. L. Henley, Phys. Rev. B71, 014424 (2005).23 S. V. Isakov, R. Moessner and S. L. Sondhi, Phys. Rev. Lett.95,

217201 (2005).24 C. L. Henley, Annual Review of Condensed Matter Physics1, 179

(2010).25 T. Fennel, P. P. Deen, A. R. Wildes, K. Schmalzl, D. Prab-

hakaran, A. T. Boothroyd, R. J. Aldus, D. F. McMorrow and S.T. Bramwell, Science326, 415 (2009)

26 N. Trivedi and D. M. Ceperley, Phys. Rev. B41, 4552 (1990).27 M. Calandra Buonaura and S. Sorella, Phys. Rev. B57,11446

(1998).28 A. Ralko, M. Ferrero, F. Becca, D. Ivanov, and F. Mila, Phys. Rev.

B. 71, 224109 (2005).29 A. Ralko, M. Ferrero, F. Becca, D. Ivanov, and F. Mila, Phys. Rev.

B. 74, 134301 (2006).30 A. Ralko, M. Ferrero, F. Becca, D. Ivanov, and F. Mila, Phys. Rev.

B 76, 140404(R) (2007).31 S. Sorella, Phys. Rev. B64, 024512 (2001).32 A. Ralko, D. Poilblanc, and R. Moessner, Phys. Rev. Lett100,

037201 (2008)33 N. Shannon, G. Misguich and K. Penc, Phys. Rev. B69,

220403(R) (2004)34 O. Sikora, N. Shannon, F. Pollmann, K. Penc, and P. Fulde, un-

published.35 D. Schwandt, M. Mambrini, and D. Poilblanc Phys. Rev. B81,

214413 (2010).36 M. E. J. Newman and G.T. Barkema,Monte Carlo Methods in

Statistical Physics, Clarendon Press, Oxford (2007).37 G.T. Barkema and M. E. J. Newman, Phys. Rev. E57, 1155

(1998).38 N. Shannon, O. Sikora, F. Pollmann, K. Penc, and P. Fulde,

arXiv:1105.419639 F. Alet, G. Misguich, V. Pasquier, R. Moessner, and J. L. Jacobsen

Phys. Rev. Lett.97, 030403 (2006).40 S. Powell and J. T. Chalker Phys. Rev. Lett.101, 155702 (2008)

21

41 G. Chen, J. Gukelberger, S. Trebst, F. Alet, and L. Balents, Phys.Rev. B80, 045112 (2009).

42 M. Capello, F. Becca, M. Fabrizio, S. Sorella and E. Tosatti,Phys.Rev. Lett.94, 026406 (2005).

43 We note that these, arbitrary, units differ from those defined inEq. (17) by a factor4/

√3.