PHYSICAL REVIEW B VOLUME 55, NUMBER 5 1 FEBRUARY 1997-I
Superfluids and supersolids on frustrated two-dimensional lattices
Ganpathy Murthy* Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218
Daniel Arovas Physics Department 0319, University of California at San Diego, La Jolla, California 92093-0319
Assa Auerbach Department of Physics, Technion IIT, Haifa, Israel ͑Received 24 July 1996͒ We study the ground state of hard-core bosons with nearest-neighbor hopping and nearest-neighbor inter- actions on the triangular and kagome´ lattices by mapping to a system of spins (Sϭ1/2), which we analyze using spin-wave theory. We find that the both lattices display superfluid and supersolid ͑a coexistence of superfluid and solid͒ order as the parameters and filling are varied. Quantum fluctuations seem large enough in the kagome´ system to raise the interesting possibility of a disordered ground state. ͓S0163-1829͑97͒09605-7͔
INTRODUCTION † ϩ the transverse magnetization density via ͗ai ͘ϭ͗Si ͘, while - is the magnetic suscepץ/nץthe boson compressibility Kϭ H. Liu and Fisher identified four phases ofץ/M zץA supersolid is a state of matter which simultaneously tibility ϭ exhibits both solid and superfluid properties. That is to say, it interest: ͑i͒ a normal fluid, in which the magnetization is displays both long-ranged positional order as well as finite uniform and in the zˆ direction, ͑ii͒ a normal solid, in which superfluid density and, naı¨vely, off-diagonal long-ranged or- the magnetization lies along zˆ yet is spatially modulated at der ͑ODLRO͒. The intriguing suggestion by Andreev and some wave vector k, ͑iii͒ a superfluid, in which the magne- Lifshitz1 that vacancies in solid 4He might Bose condense in tization is uniform and has a component which lies in the the vicinity of the melting line has, to our knowledge, never x-y plane, and ͑iv͒ a supersolid, in which there simulta- been experimentally verified.2,3 Nonetheless, a sizeable lit- neously exists a nonzero transverse component to the mag- erature has developed on the theoretical properties of netization MЌ , as well as a spatial modulation of the longi- 4–16 supersolids. In two dimensions, the physics of tudinal magnetization M z . ͑An incompressible normal fluid Josephson-junction arrays17 has also stimulated the theoreti- is also called a Mott insulator.͒ cal study of supersolids. Most of the work is based on the If one relaxes the hard-core constraint in favor of a finite contributions of Matsuda and Tsuneto18 and of Liu and on-site repulsion U, one obtains the Bose-Hubbard model Fisher,19 who established some key concepts in the theory of lattice-based supersolid models. Of central importance is the † † mapping between a hard-core lattice Bose gas and a spin-1/2 ϭϪt͚ ͑ai ajϩajai͒Ϫ͚ ni H ͗ij͘ i quantum magnet: 1 ϩ Ϫ ϩ † ϩ Ϫ † z 1 U͚ ni͑ni 1͒ V͚ ninj . ͑3͒ a S , a S , n ϭa a S Ϫ . ͑1͒ 2 i ͗ij͘ i ↔ i i↔ i i i i↔ i 2 This model has been extensively studied since the seminal Thus, an occupied site is represented by an up spin, while an 20 empty site is represented by a down spin. An interacting work of Fisher et al., who considered the model with Vϭ0 hard-core lattice Bose gas with nearest-neighbor hopping t, in the context of a superconductor-insulator transition. A study of this model in the presence of disorder has led to an nearest-neighbor repulsion V, and chemical potential is 21 thereby equivalent to the anisotropic Sϭ1/2 Heisenberg understanding of the Bose glass. On a two-dimensional model square lattice, and for VÞ0, the model was studied at Tϭ0, for both finite and infinite U, by Scalettar et al.,14 Bruder, Fazio, and Scho¨n,6 and van Otterlo and co-workers7,8 using z z x x y y z ϭ ͓JʈSiSjϩJЌ͑SiSjϩSi Sj͔͒ϪH Si , ͑2͒ mean-field theory and quantum Monte Carlo techniques. To H ͚ ͚i ͗ij͘ summarize their results, no supersolid phase is observed at where JʈϭV is an antiferromagnetic longitudinal exchange, half filling (͗n͘ϭ1/2), where a first-order transition occurs JЌϭϪ2t is a ferromagnetic transverse exchange, and Hϭ as a function of V between superfluid ͑small V͒ andaNe´el Ϫ(1/2)zV is an external magnetic field ͑z is the lattice co- solid ͓large V, kϭ͑,͔͒. ͑Large next-nearest-neighbor re- ordination number͒. The spin model exhibits a global U(1) pulsion VЈ stabilizes a striped phase, the colinear solid.͒ symmetry with respect to rotations about the zˆ axis, which of Away from half filling, there is no normal solid phase, and course means total particle number conservation in the boson the transition is instead from superfluid to supersolid. The language. The boson condensate order parameter is related to supersolid phase exhibits both a peak in the static structure
0163-1829/97/55͑5͒/3104͑18͒/$10.0055 3104 © 1997 The American Physical Society 55 SUPERFLUIDS AND SUPERSOLIDS ON FRUSTRATED... 3105 factor S͑k͒ at the Ne´el vector ͑and properly proportional to the lattice volume͒, and a nonzero value of the superfluid density s . Again, next-nearest neighbor VЈ can stabilize a striped supersolid phase with anisotropic s . One can also obtain Mott insulating phases with fractional filling in the presence of next-nearest neighbor interactions. In this paper, we will investigate the properties of the model in Eq. ͑2͒ on frustrated two-dimensional lattices. We are motivated by the fascinating interplay between frustra- tion, quantum fluctuations, order, and disorder which has been seen in quantum magnetism. Frustration enhances the effects of quantum fluctuations. Indeed, as early as 1973, Fazekas and Anderson23,24 raised the possibility that for such systems, quantum fluctuations might destroy long-ranged antiferromagnetic order even at zero temperature. In many cases, frustration leads to an infi- nite degeneracy at the classical ͑or mean field͒ level not as- sociated with any continuous symmetry of the Hamiltonian itself. In these cases, it is left to quantum ͑or thermal͒ fluc- tuations to lift this degeneracy and select a unique ground state,25,26 sometimes with long-ranged order. Our models ex- hibit both a depletion ͑but not unambiguous destruction͒ of FIG. 1. Mean-field phase diagram for the triangular lattice. The order due to quantum fluctuations, as well as the phenom- kagome´ lattice phase diagram differs only by a rescaling of h. enon of ‘‘order by disorder.’’ Heavy lines denote first-order transitions, light lines second-order In our work, we will choose the units of energy to be Jʈ , transitions, and dashed lines denote linear instabilities. writing ⌬ϵt/VϭJЌ/2Jʈ , and hϵH/Jʈ . We will be follow- ing closely the analysis of the anisotropic triangular lattice ¨ ͑where S has been set equal to 1/2͒ as a function of ⌬ for the antiferromagnet by Kleine, Muller-Hartmann, Frahm, and z Fazekas ͑KMFF͒,27 who performed a mean-field ͑Sϭϱ triangular lattice. It is clear that the quantum corrected S is limit͒ and spin-wave theory ͑order 1/S corrections to mean- very close to zero for all ⌬, reflecting the fact that at hϭ0 field͒ analysis. Contemporaneously with KMFF, Chubukov the lattice is half-filled. Two sublattices acquire large correc- and Golosov28 derived the spin-wave expansion for an iso- tions due to quantum fluctuations ͑even in the Ising limit ⌬ 0͒, while the third has only small quantum corrections. tropic Heisenberg antiferromagnet in a magnetic field, while → Sheng and Henley29 obtained the spin-wave theory for the This is very similar to the fully antiferromagnetic case stud- ied by KMFF. Therefore, even at Sϭ1/2 the solid order sur- anisotropic antiferromagnet in the absence of a field. x The mean-field phase diagram is shown in Fig. 1 ͑both the vives. The off-diagonal order parameter S is reduced in triangular and kagome´ lattices have the same mean-field phase diagram up to a rescaling of h͒. Notice that the super- solid phase appears in a broad region of ⌬ and filling. The reason the supersolid is so robust is that the lattice frustrates a full condensation into a solid. Generically, frustrated lat- tices might be good places to look for this phase. Let us briefly concentrate on hϭ0 before describing the entire phase diagram. We will be assuming a three sublattice structure throughout. The mean-field state is then described by three polar and three azimuthal angles: (A ,B ,C , A ,B ,C), and is invariant under uniform rotation of the azimuths. Due to the ferromagnetic coupling in the x-y spin direc- tions the mean-field solution is always coplanar. Just as in KMFF, there is a one-parameter family of degenerate mean- field solutions in the zero-field case ͑originally found by Mi- 30 yashita and Kawamura ͒. The A sublattice polar angle A may be chosen as the free parameter; spin-wave theory ͑SWT͒ is necessary to lift the degeneracy and uncover the true ground state. Figure 2 shows the ground-state energy in SWT as a function of A for the triangular lattice at ⌬ϭ0.25. Using SWT we also compute the fluctuations of the spins, and the consequent quantum-corrected magnetization and the solid and ODLRO order parameters. Figure 3 illustrates FIG. 2. Ground-state energy of the triangular lattice at ⌬ϭ0.25 these quantities in mean field and to leading order in SWT as a function of A . The minimum is quadratic. 3106 GANPATHY MURTHY, DANIEL AROVAS, AND ASSA AUERBACH 55
berg limit ⌬ϭϪ1 ͑for a partial set of references, see Refs. 31–38͒. For the fully antiferromagnetic case there are local motions of the spins that move the system on the degeneracy submanifold, leading to a much larger ground-state degen- eracy for the kagome´ lattice than for the triangular lattice. However, for ⌬у0, the ferromagnetic transverse interaction eliminates the possibility of these local motions, resulting in a ground-state degeneracy parametrized only by A , just as in the triangular lattice. We carried out SWT for the two long-range ordered con- figurations shown in Fig. 4—a three sublattice ‘‘qϭ0’’ state, and a nine sublattice )ϫ) structure,35,36 respectively. These states have the same mean-field ground-state energy. It will turn out that two of the three sublattices have the same spin orientation in the ground state for both lattices. With this proviso, note that the qϭ0 structure has one twofold axis and a mirror plane ͑point group C2v͒, while the )ϫ) struc- ture has a sixfold axis and a mirror plane ͑point group C6v͒. Once again SWT selects the true ground state. When quan- tum fluctuations are accounted for, we find that the )ϫ) structure always has lower energy than the qϭ0 structure, to FIG. 3. Classical and quantum magnetizations for the triangular the numerical accuracy of our calculations. More impor- lattice, Sz and Sx per site, as a function of ⌬ at the true ground state. tantly, the fluctuations of the spins on six of the nine sublat- The full quantum-corrected S is always close to zero, while the tices diverge in the limit h 0, as shown in Fig. 5. This z → quantum-corrected Sx is always nonzero. divergence is the consequence of a flat ͑dispersionless͒ mode at zero energy as h 0. Higher-order terms in the spin wave magnitude by quantum corrections, but goes to zero only as expansion will lift this→ mode and remove the divergence.25,39 ⌬ 0. Supersolid order survives quantum fluctuations for the However, the fluctuations remain large for Sϭ1/2, as we triangular→ lattice, at least in this order of SWT. estimate in Sec. III. This indicates that quantum fluctuations Let us now consider the kagome´ lattice. There is a quali- may be strong enough to wash out any order, including tative difference between the antiferro-ferromagnetic case ODLRO, on two of the sublattices, which raises the intrigu- considered here, ⌬у0, and the fully antiferromagnetic case ing possibility of a partially disordered ground state at Tϭ0. ⌬р0, which has been exhaustively explored for the Heisen- It must be emphasized that we have not demonstrated that
FIG. 4. The two long-range- ordered structures on the kagome´ lattice. ͑a͒ the qϭ0 structure and ͑b͒ the R3 structure. 55 SUPERFLUIDS AND SUPERSOLIDS ON FRUSTRATED... 3107
with a spin-S at each site. We then represent each spin as a classical vector of magnitude S, S␣ϭS⍀␣, where ⍀ˆ is a unit vector in three dimensions. We rescale the magnetic field by S and write
2 z z x x y y z MF /S ϭ͚ ⍀i⍀jϪ⌬͚ ͑⍀i⍀jϩ⍀i ⍀j͒Ϫh͚ ⍀i . H ͗ij͘ ͗ij͘ i ͑4͒
In the mean-field solution, all spins lie in the x-z plane. Furthermore, it turns out that the triangular and both kagome´ structures have the same mean-field energy, to within a con- stant factor ͑up to a separate rescaling of h in the case of the kagome´ structures͒, so we will consider all three cases simul- taneously. Note that coplanar states have been selected at the mean-field level for the kagome´ lattice, in contrast to the fully antiferromagnetic case.
A. Zero field Consider first mean-field ground states on the triangular and kagome´ lattices that have a three-sublattice structure. FIG. 5. Classical and quantum-corrected values of the magneti- The three specific cases are the usual sublattice structure on zations Sz and Sx per site as a function of h at ⌬ϭ0.25 for the R3 the triangular lattice, the qϭ0 structure on the kagome´ lat- structure on the kagome´ lattice. Notice the divergence as h 0. ´ → tice, and the )ϫ) structure on the kagome lattice. The nine This is an artifact of SWT. sublattices of the )ϫ) structure are organized into three groups (A,B,C) of three, so that an A site has two B and two this is so: the fluctuations could be correlated between dif- C neighbors. Thus, the energy per site, in units of S2,is ferent sites, there could be long-range order with a larger unit cell, a condensed array of vortices, etc. This problem merits e ϵE /NS2 further study, with, e.g., quantum Monte Carlo methods. MF MF
Let us now turn to a fuller description of Fig. 1, where ϭ͑cosAcosBϩcosBcosCϩcosCcosA͒ four types of mean-field states are present ͑solid lines indi- cate first-order transitions͒. We adopt the nomenclature of Ϫ⌬͑sinAsinBϩsinBsinCϩsinCsinA͒ ͑5͒ Ref. 14 ͑see Fig. 12 of this reference for comparison͒.At high fields h the system is in a Mott phase—incompressible on the triangular lattice and zKag/zTriϭ2/3 this value on the kagome´ lattice. and fully polarized. As the field is lowered, for any ⌬Ͼ0, the 30 system enters a compressible superfluid phase ͑SF͒, with Miyashita and Kawamura have shown that there is a one-parameter family of degenerate ground states for this AϭBϭCϾ0. A first-order transition from the superfluid to an incompressible Ne´el solid ͑NS͒ at filling fraction 2/3 classical problem for arbitrary ⌬, which does not seem to be related to any obvious symmetry of the model. Following ͓magnetization per site M zϭϮ(1/3)S͔ occurs for ⌬Ͻ1/2. Fi- nally, the supersolid phase ͑SS͒ exists for ⌬Ͻ1/2 between KMFF, and writing iϭiϪ(AϩBϩC), and defining the the two symmetry-related Ne´el solid lobes. There is a tri- two-dimensional vectors iϭ͑sini ,cosi͒, we can write the critical point at (⌬*,h*)ϭ[(1/2),3]. Quantum fluctuations mean-field energy in terms of the two-component vector will modify these mean-field phase boundaries. Since the ϵAϩBϩC : superfluid state, we have found, benefits the most from spin- wave energy corrections, it will encroach on its neighbors as 1 1 e ϭ ͑1Ϫ⌬͒͑2Ϫ3͒ϩ ͑1ϩ⌬͒ . ͑6͒ S decreases from ϱ. Increasing h tends to suppress quantum MF 4 2 y fluctuations.
We will present each of the above results in more detail in Therefore eMF , while nominally depending on the three the rest of this paper. Section I concentrates on the mean- angles i , actually depends only on two combinations of field theory and the mean-field phase diagram. Section II them, leaving one parameter free. We can then parametrize describes the selection of the true ground state by quantum the degenerate ground states by A , by defining Bϭ⑀Ϫ␦, fluctuations at hϭ0, and the form of the spin-wave excita- and Cϭ⑀ϩ␦, where tions for arbitrary h. Section III presents the suppression of order by quantum fluctuations. We end with our conclusions, tanA connections to experimental work, and open questions. tan⑀ϭϪ , ͑7͒ ⌬ I. MEAN-FIELD THEORY Ϫ⌬ cosA The mean-field limit is obtained by setting Sϭϱ. For- cos⑀ϭ , 2 2 mally, we first generalize the model from Sϭ1/2 to a model ͱ1Ϫ͑1Ϫ⌬ ͒cos A 3108 GANPATHY MURTHY, DANIEL AROVAS, AND ASSA AUERBACH 55
The phase diagram has already been shown in Fig. 1. Let us keep at a particular value of ⌬ and turn up the field h.At zero field there are two regimes, the superfluid with no solid order for ⌬Ͼ1/2, and the supersolid with both solid and ODLRO (Tϭ0) for ⌬Ͻ1/2. For ⌬Ͼ1/2, the ground state remains a uniform superfluid, though M z becomes nonzero, as h is increased. The spins cant at an angle ϭcosϪ1[h/6(1ϩ⌬)]. The energy in this phase is
h2 e͓SF͔ϭϪ3⌬Ϫ . ͑10͒ MF 12͑1ϩ⌬͒
Eventually, for hϾ6(1ϩ⌬), every site has the maximum possible Sz, and the system is in the Mott insulator ͑MI͒ phase, with each site fully occupied with one boson. Borrow- ing from spin-wave results derived in Sec. II, the linear in- stability of the SF phase occurs at
hi1ϭ6ͱ͑1ϩ⌬͒͑1Ϫ2⌬͒. ͑11͒ Next focus on a specific ⌬Ͻ1/2, and increase the field h from zero. At zero field, the one-parameter degeneracy of the mean-field ground states has to be lifted by quantum fluctua- FIG. 6. Optimal AϭB and C as a function of h for ⌬ϭ0.25. Note the continuous approach to collinearity at the supersolid-solid tions, which select a particular A . However, it turns out that transition, and the discontinuous change of the angles at the solid- we can recover this A by considering nonzero h and taking superfluid transition. the limit h 0, which gives → 1Ϫ2⌬ ⌬ cos2 ϭ . ͑12͒ cos␦ϭ . A 1Ϫ⌬2 2 2 ͑1Ϫ⌬͒ͱ1Ϫ͑1Ϫ⌬ ͒cos A It seems surprising that the ground state selected by quantum It is easy to verify that A can lie in the range fluctuations can be predicted by an entirely classical calcula- tion. A plausible ͑though nonrigorous͒ argument will be pro- 1 1Ϫ2⌬ 1 Ϫ1 vided for this in terms of spin-wave theory in the next sec- cos рAр . ͩ 1Ϫ⌬ ͱ1Ϫ⌬2 ͪ 2 tion. As h is increased from zero, A and C change ͑recall As ⌬ increases from zero, the range of possible A is AϭB throughout͒. At a certain critical field hc1(⌬), SC compressed, and the differences ͉BϪA͉, ͉CϪA͉ shrink, points exactly along the Ϫzˆ direction ( ϭ), while S until at ⌬ϭ1/2 the ground state is colinear with C A,B point along the zˆ direction (AϭBϭ0). This critical field AϭBϭCϭ(1/2)—a featureless superfluid. For ⌬Ͻ1/2, h can be analytically determined by the following consid- in the case of zero field, the true ground state will be selected c1 eration: the point Aϭ0, Cϭ is always a stationary point by quantum fluctuations. Minimizing eMF gives x of the mean-field energy. However, for hϽhc1 it is a saddle ϭ(⌬ϩ1)/(⌬Ϫ1), yϭ0, and point, while for hϾhc1 it is a minimum. Therefore the sec- ond derivative matrix of e ( , ) should have a zero 1Ϫ⌬ϩ⌬2 MF A C e ϭϪ . ͑8͒ eigenvalue at hϭhc1. Setting the determinant to zero gives MF ͩ 1Ϫ⌬ ͪ 3 h ͑⌬͒ϭ ͑2ϩ⌬Ϫͱ4Ϫ4⌬Ϫ7⌬2͒ . ͑13͒ B. Nonzero field c1 2 Turning on a field h adds an energy For hϽhc1(⌬), we are in the supersolid phase. Just above 1 hc1, however, the system is exactly 2/3 filled, and its mag- ⌬eMFϭϪ h͑cosAϩcosBϩcosC͒ ͑9͒ netic susceptibility is zero ͑the compressibility of the corre- 3 sponding boson system is zero͒. There is no superfluid order per site and lifts the degeneracy described in the previous and the system is again a Mott insulator, the Ne´el solid. It subsection, producing a unique mean-field ground state ͑un- will be seen in Sec. III that the exact filling of 2/3 survives like in the Heisenberg antiferromagnet28͒. We find that mini- quantum fluctuations. Since the superfluid order goes con- mization generally leads to a state where ͑without loss of tinuously to zero below the transition, it is clear that this generality͒ AϭBÞC in the supersolid phase. The results transition is second-order within in mean-field theory. The are plotted in Fig. 6. A rescaling of the field for the kagome´ energy of the Ne´el solid phase is lattice ͓h ϭ͑2/3͒h ͔ makes the entire mean-field phase Kag Tri h diagram identical, and we have therefore shown only the e͓NS͔ϭϪ1Ϫ . ͑14͒ triangular lattice results. MF 3 55 SUPERFLUIDS AND SUPERSOLIDS ON FRUSTRATED... 3109
One can furthermore determine that the Ne´el solid phase is assumed to lie on the x-z plane, we can do this by a rotation linearly stable for 3/2(2ϩ⌬Ϫͱ4Ϫ4⌬Ϫ7⌬2)рhр3/2(2 about the y axis. Labeling the local frame spins with a tilde, ϩ⌬ϩͱ4Ϫ4⌬Ϫ7⌬2). we have Further increasing h, we find a critical field x ˜x ˜z SRϭcosSRϩsinSR , hc2͑⌬͒ϭ2͑1ϩ⌬͒ϩ4ͱ͑1ϩ⌬͒͑1Ϫ2⌬͒, ͑15͒ Sy ϭ˜Sy , ͑17͒ beyond which the canted superfluid becomes energetically R R favored. The transition is first order since it is far from any z ˜x ˜z linear instabilities. Finally, a second-order line at SRϭϪsinSRϩcosSR , hϾhc3(⌬)ϭ6(1ϩ⌬), signals the boundary between super- where the subscript R labels a Bravais lattice site, a basis fluid and fully polarized Mott phases. element, and is A,B,C depending on the mean-field orien- When ⌬ϭ1/2, we can solve analytically to find tation of the sublattice. The triangular lattice and the qϭ0 structure on the kagome´ lattice have three sublattices, while 1 ´ cos ϭ h, the )ϫ) structure on the kagome lattice has nine. A 3 We now describe the spin operators in terms of Holstein- Primakoff bosons 1 cosCϭϪ h, ͑16͒ ˜ϩ † † † Ϫ1/2 3 SRϭRͱ2SϪRRϭͱ2SRϩ ͑S ͒, O ˜Ϫ † † Ϫ1/2 3 1 SR ϭͱ2SϪR RR ϭͱ2SRϩ ͑S ͒, ͑18͒ e͓SS͔ϭϪ Ϫ h2. O MF 2 18 ˜z † S ϭ RϪS. This is exactly the same as the energy of the SF phase at R R ⌬ϭ1/2 ͓which is characterized by all the angles satisfying The Hamiltonian ͑restoring h for generality͒ is now written cos ϭ(1/9)h͔, which marks this vertical line as a first-order in Fourier space as line. Note the tricritical point at ⌬ϭ1/2, hϭ3, where two MN first-order lines ͑with infinite slope͒ meet a second-order line † 0 swϭE0ϩ͑1/2͒S͚ :⌿ ͑k͒ ⌿͑k͒:ϩ͑S͒ ͑with finite slope͒. H k ͩ NMͪ O The situation is completely symmetric with respect to the ͑19͒ sign of h, with the Ne´el solid phase now existing at 1/3 filling for hϽ0. At the mean-field level, the ⌬ axis is a ͑note the normal ordering͒ where the energy E0 is given by first-order transition line up to ⌬ϭ1/2, since Sz is discontinu- 1 ous across it. However, for Sϭ1/2 it apparently becomes E ϭ͑N/K͒S2 z X Ϫ h . ͑20͒ 0 2 ͚ Ј Ј ͚ continuous, at least for the triangular lattice. ͫ Ј ͬ The phase diagram has some similarities to the classic picture of the Mott lobes surrounded by superfluid described Here, N is the total number of lattice sites, K is the number by Fisher, Weichman, Grinstein, and Fisher ͑FWGF͒.20 of sublattices, and we define the quantities However, there are important differences. FWGF considered X ϭcos cos Ϫ⌬ sin sin , local Hubbard repulsion, whereas our extended Bose- Ј Ј Ј Hubbard model we consider affords the possibility of incom- Y ϭsin sin Ϫ⌬ cos cos , ͑21͒ pressible Mott phases at fillings 0, 1/3, 2/3, and 1. The Ne´el Ј Ј Ј solid phase found by Scalettar et al., for example, exists at h ϭh cos , filling 1/2. Fractional fillings have also been observed in the square lattice with frustrating longer range interactions in and zЈ is the number of Ј sublattice neighbors each Ref. 6. Also, the transitions from the fractional filling MI sublattice site has. The vector k lives in the first Brillouin phases to the ͑canted͒ superfluid are first order. Finally, and zone of the reciprocal lattice, and most notably, the entire region between the two fractional † † † † Mott lobes is taken over by the supersolid, and the super- ⌿ ͑k͒ϭ͓1͑k͒,2͑k͒,...,K͑k͒, solid gives way to the superfluid only beyond a hopping tϾ(1/2)V. 1͑Ϫk͒,2͑Ϫk͒,...,K͑Ϫk͔͒. ͑22͒ The matrices M and N have diagonal and off-diagonal ele- II. SPIN-WAVE THEORY ments given by We now develop the spin-wave theory ͑SWT͒ for this problem. If hÞ0, there is a unique ground state ͑up to per- M ͑k͒ϭhϪ͚ zЈXЈ , ͑23͒ mutations of the sublattices͒. When hϭ0, the ground-state Ј manifold is parametrized by A , and the other angles, B and ͑in general not equal͒, can be determined from and ⌬ 1 C A M ͑k͒ϭ ͑Y Ϫ⌬͒f ͑k͒, using Eq. ͑7͒. We implement SWT in the usual way: by first Ј 2 Ј Ј performing local rotations of the spins so that the mean-field directions point along the local z axis. Since all the spins are N͑k͒ϭ0, 3110 GANPATHY MURTHY, DANIEL AROVAS, AND ASSA AUERBACH 55
1 Ϫ N ͑k͒ϭ ͑Y ϩ⌬͒f ͑k͒, ϭ D͓ , , , , , ͔e A, Ј 2 Ј Ј Z ͵ A B C A B C ͑27͒ ץ :where the function f Ј(k) is given by the following sum ϭ d ͚ iS cos͑r,͒ ͑r,͒ϩ ͓,͔ , ͪ Hץ A ͵ ͩ r Ј f Ј͑k͒ϭ͚ exp͑ik•␦͒, ͑24͒ ␦ where stands for the spin Hamiltonian of Eq. ͑4͒ written in terms ofH and . The first term is the Berry phase contribu- where the prime on the sum indicates that the sum is over tion to the path integral, which also makes S cos the mo- nearest-neighbor vectors connecting a sublattice site to a Ј mentum canonically conjugate to . sublattice site. Let us concentrate on just the kϭ0 modes. There are three Now we perform a Bogoliubov transformation, which variables and three conjugate variables. We choose to amounts to finding a rank 2K matrix T satisfying T†⌳Tϭ⌳, call one of the variables corresponding to an overall with 0 rotation of all the spins around the z axis, and call the re- maining s, 1 , and 2 . 1KϫK 0KϫK ⌳ϭ , ͑25͒ Choose any particular ground state labeled by A . The ͩ 0 Ϫ1 ͪ KϫK KϫK Hamiltonian for small deviations from the ground-state con-
Ϫ1 figuration is now given by as well as ⌳T ⌳ swTϭ, a non-negative diagonal matrix with identical upperH left and lower right blocks. The ͑k͒ 1 1 are the spin-wave frequencies. The spin-wave correction to T T ϭ M ϩ M , ͑28͒ the ground-state energy is given by H 2 2
T T K where ϭ(␦ ,␦ ,␦ ) and ϭ( , , ). The first 1 A B C 0 1 2 row and column of M are zero since does not appear in ⌬Eϭ S͚ ͚ ͓͑k͒ϪM ͑k͔͒. ͑26͒ 0 2 k ϭ1 the Hamiltonian. The general procedure for finding the normal modes is the following: A. Ground-state selection ͑i͒ Diagonalize the 2ϫ2 block of M and rescale the re- Figure 2 shows the ground-state energy, including spin- sulting eigenvectors so that the rescaled M Ј becomes a unit wave corrections, for the triangular lattice at hϭ0 as a func- matrix in the 2ϫ2 block. Call the rescaled variables Ϯ . tion of A . Since the classical energy is independent of A ͑ii͒ Use the Berry’s phase terms to identify the canoni- all the variation comes from the SW correction. It is clear cally conjugate momenta to 0 , Ϯ as P0 , PϮ , respectively. that there are two possible values of which minimize the A ͑iii͒ Re-express the matrix M as a matrix M P . ground-state energy, one of them lying at the edge of the ͑iv͒ Since 0 is cyclic, its canonically conjugate momen- allowed range of . However, the two minima turn out to A tum P0 is conserved. Treat P0 as a constant and form linear be physically identical, and correspond to a relabeling of the combinations PϮЈ ϭPϮϩ␣ϮP0 such that the off-diagonal sublattices. We call the minimizing value of which lies A terms containing P0 are eliminated. A diagonal term multi- away from the edge of its allowed range * . 2 A plying P 0 remains. * The value of A can be determined by purely classical ͑v͒ Now diagonalize the lower 2ϫ2 block of M P . The z arguments, by extremizing the value of S ϰAϩBϩC two eigenvalues of M P are the squares of the energies of the along the degeneracy submanifold. We have checked that normal modes of the Hamiltonian. The third normal mode this is so by a comparison of the analytic expression A* corresponds to a uniform rotation of all the spins around the Ϫ1 2 ϭcos ͱ(1Ϫ2⌬)/(1Ϫ⌬ ) with the ground-state energy z axis coupled to a change in Sz , and its energy is always 2 curves obtained from SWT. We now provide an argument zero regardless of the coefficient of P 0. indicating why this might be the case. The above is true even if there is no ground-state degen- Our argument will be the following: For a generic A eracy. If there is ground-state degeneracy in the subspace there is only one mode of zero energy at kϭ0, whereas if the matrix M P has a null eigenvector v0 . For generic A , P0 -Aϭ0 there are two zero modes ͑to spin-wave order͒. has nonzero overlap with v0 . In this case the above proceץ/Szץ 2 The ground-state energy is the sum of the energies of all the dure always produces a zero coefficient for P 0. Thus there is modes, and does not depend on just the zero modes. How- still only one mode of zero energy, the Sz mode. Another ever, we think it plausible that having more modes of zero way of seeing this is to recognize that as long as P0 and v0 energy at kϭ0 drags down the energies of all the k modes, have nonzero overlap, one can always rescale v0 so that it thus reducing the full ground-state energy. Of course this becomes canonically conjugate to 0 . One then has to 40 argument is not rigorous, and there may be counterex- modify Ϯ to keep them independent of v0 in the Poisson amples that we are not aware of. bracket sense. Let us go on to show the first statement about the number However, if P0 and v0 have zero overlap, the null vector of modes of zero energy. We only sketch the argument here: must lie in the subspace of PϮ . This means one of the ei- We treat the issue with more generality and greater detail in genvalues of the lower 2ϫ2 block of M P must be zero, im- the Appendix. It is helpful to think in terms of the coherent plying that there is another zero mode of the Hamiltonian, 22 states path integral apart from the Sz mode. The condition for P0 and v0 to have 55 SUPERFLUIDS AND SUPERSOLIDS ON FRUSTRATED... 3111
FIG. 7. The Brillouin zone of the triangular lattice and the reduced zone for the sublattice structure.
Aϭ0 along the degeneracy B. Spin-wave dispersionsץ/Szץ zero overlap is identical to direction, which is the same as extremizing S . z 1. Triangular lattice ,AÞ0, there is only one zero modeץ/Szץ In brief, if Aϭ0, there are two zero modes at kϭ0. A The Brillouin zone ͑BZ͒ of the triangular lattice is shownץ/Szץ whereas if rigorous derivation is supplied in the Appendix. in Fig. 7, with the lattice and reciprocal-lattice vectors being We have done explicit calculations to verify all these statements for the triangular lattice. If one computes the fre- e1ϭa͑1,0͒, quencies of the kϭ0 modes at a nonoptimal A , one finds one mode of zero energy ͑the Sz mode͒, and two modes of 1 ͱ3 e ϭa , , nonzero energy, neither of which is exactly along the degen- 2 ͩ 2 2 ͪ eracy direction. However, at precisely the optimal points, ͑30͒ there appear two modes with zero energy, one of which is the 4 ͱ3 1 Sz mode and the other exactly the degeneracy mode. The G1ϭ ,Ϫ , third mode still has nonzero energy. aͱ3 ͩ 2 2ͪ In particular, the argument makes no assumptions about z the interactions other than that they should conserve total S . 4 So this result should hold even for site dilution or longer G2ϭ ͑0,1͒, range interactions. Of course, for site dilution, one should aͱ3 focus on a particular realization of randomness and look at where a is the lattice spacing. Sz over the degeneracy subspace. In general, our argument is As usual, we first concentrate on hϭ0. Figure 8 shows that the search for the true ground state can be restricted to the spin-wave dispersion for ⌬Ͼ1/2. Since there is only one the points on the degeneracy submanifold where all con- sublattice, there is only one mode. However, plotting it on served quantities commute in the Poisson bracket sense with the reduced BZ of the sublattice problem forces us to fold it the generators of motion along the degeneracy submanifold. back and represent it as three modes. In this scheme, there is This criterion can fail if the kÞ0 modes do not follow the one gapless mode, which is the Goldstone mode correspond- behavior of the kϭ0 modes. Also, if there are more con- ing to the density fluctuations of the bosons. However, as ⌬ served quantities than degeneracy directions, not all of them approaches 1/2, a ‘‘roton’’ minimum develops, precisely at may be extremized at the true ground state. the wave vectors corresponding to the sublattice structure, as With and taking on the value quoted above, we find A B demonstrated in Fig. 9. At exactly ⌬ϭ1/2, this becomes a that is at the extreme edge of its allowed range: c gapless mode, heralding the transition to the supersolid phase. 1 1Ϫ2⌬ Now consider the dispersion at hϭ0 in the SS phase, cos*ϭϪ . ͑29͒ c 1Ϫ⌬ ͱ1Ϫ⌬2 which is shown in Fig. 10. There are two gapless modes 3112 GANPATHY MURTHY, DANIEL AROVAS, AND ASSA AUERBACH 55
FIG. 8. Triangular lattice SW dispersion for ⌬ϭ0.75, hϭ0. FIG. 10. Triangular lattice SW dispersion in the SS phase at ⌬ϭ0.25, hϭ0. within SWT. One is the standard density fluctuation, whereas at the transitions. The SWT for the canted spin phase can be the other corresponds to the degeneracy mode. The degen- analyzed analytically, since there is only one sublattice. We eracy mode will be shown to acquire a gap to higher order in find that the SW dispersion is 1/S in the next subsection.25,39 There is a third ‘‘optical’’ mode to complete the count of the sublattice degrees of free- h2 ͑k͒ϭzS ⌬͑1Ϫ␥ ͒ ⌬ϩ 1Ϫ ␥ , dom. The energy scale of the two low-lying modes is ⌬, ͱ k ͫ ͩ 36͑1ϩ⌬͒ͪ kͬ while the optical mode has an energy scale of 1. As we move ͑31͒ towards ⌬ϭ0 the two low-lying modes get softer, until they 1 become completely flat at ϭ0. Ј Ϫik•␦ ⌬ ␥kϭ ͚ e , However, when we turn on a field the degeneracy mode z ␦ becomes gapped, as shown in Fig. 11. Let us now investigate where zϭ6 for the triangular lattice, and the prime on the the modes at nonzero field in the other phases, in particular sum restricts ␦ to nearest-neighbor vectors. We show an ex- ample for ⌬ϭ0.25, hϭ6.0 in Fig. 12. We have Ϫ1/2р␥kр1,
FIG. 9. Triangular lattice SW dispersion for ⌬ϭ0.51, hϭ0. Note the quadratic minimum which is nearly gapless. This is the FIG. 11. Triangular lattice SW dispersion in the SS phase at linear instability that leads to the SS phase. ⌬ϭ0.25, hϭ0.5. The degeneracy mode has become gapped. 55 SUPERFLUIDS AND SUPERSOLIDS ON FRUSTRATED... 3113
FIG. 12. Triangular lattice SW dispersion in the canted spin FIG. 14. Triangular lattice SW dispersion in the MI phase at ͑CS͒ phase at ⌬ϭ0.25, hϭ6.0. Note the single gapless density ⌬ϭ0.25, hϭ1.5. mode. ͑including spin-wave corrections͒ in the Ne´el solid phase is which leads to the linear instability of the SF phase, which is independent of h, which demonstrates its incompressibility, denoted hi1 in Eq. ͑11͒. Note that the mean-field energies of and hence the exact filling of 2/3 throughout this phase. the various phases lead to first-order instabilities which make 0؍the linear instability irrelevant except at ⌬ϭ0 and ⌬ϭ1/2. 2. kagome´ lattice q We next turn to the transition line between the supersolid The kagome´ lattice is a triangular Bravais lattice with a ´ and Neel solid phases, denoted by hc1(⌬), a second-order three element basis. If a is the nearest-neighbor separation, line. Figure 13 shows the SW dispersion at the transition for then the Bravais lattice constant is ˜aϭ2a. The spin-wave ⌬ϭ0.25, hϭ0.9738. The soft density mode is now at kϭ0 theory dispersions are quite similar to those of the triangular and has a quadratic dispersion instead of the usual linear one. lattice, with the modes being softer ͑because of the lower As we increase h and enter the Ne´el solid phase, the density coordination͒. Note that while the Bravais lattice is triangu- mode becomes gapped, as Fig. 14 shows. The total energy lar, the symmetry is reduced ͑to C2v͒ and the dispersion curves do not have zero slope at the zone edge ͑the X point͒. Figures 15, 16, and 17 show the dispersions in the superfluid, the supersolid, the Ne´el solid state, respectively. A notewor- thy feature is the presence of a zero-energy mode at hϭ0 along the ⌫M direction in the supersolid phase. This mode disperses and has nonzero energy except along the ⌫M di- rection.
3. kagome´ 9 sublattice structure The )ϫ) structure on the kagome´ lattice is described by a triangular Bravais lattice of lattice constant ˜aϭ2)a with a nine element basis. The elementary lattice vectors are shown in Fig. 4. The spin-wave dispersions are unremark- able except for a flat mode whose energy vanishes as ͱh in the small h limit. This is connected to the degeneracy mode, which is local up to harmonic order, and will be discussed in the next subsection. All the other features are quite similar to those of the triangular lattice as shown in Figs. 18, 19, and 20.
C. Gap of the degeneracy mode FIG. 13. Triangular lattice SW dispersion at the SS-MI transi- The degeneracy mode appears gapless in SWT ͑and in tion at ⌬ϭ0.25, hϭ0.9738. Note the quadratic dispersion of the fact appears flat in the )ϫ) structure on the kagome´ lat- gapless density mode. tice͒, but acquires a gap to higher order in the SW 3114 GANPATHY MURTHY, DANIEL AROVAS, AND ASSA AUERBACH 55
´ FIG. 15. kagome qϭ0 SW dispersion in the CS phase at FIG. 17. kagome´ qϭ0 SW dispersion in the MI phase at ⌬ϭ0.25, hϭ4. ⌬ϭ0.25, hϭ1. expansion.25,39 We will first treat the triangular lattice, and 0 go on to the case of the kagome´ lattice. ϭ D D0 Z ͵ A 0 1 2 0 2ץ Triangular lattice 0 .1 ϫexpϪ d iS cosA ϩ S K͑ ͒ ͪ 2ץ ͩ ͵ Let us concentrate on the kϭ0 degeneracy mode of the triangular lattice. The easiest way to derive the gap is in the 22 Ϫ Ј͓,͔ spin coherent-state path-integral language introduced in the ϫ ͵ ͟ D͓k ,k͔e A , ͑32͒ previous section. kÞ0 Now imagine separating the path integral by first integrat- where Ј does not include the explicitly written terms in- A ing all the modes except the kϭ0 degeneracy mode ͑call the volving 0 in the preceding factor. Since the classical 0 0 0 angles A and ͒. ground-state energy does not depend on A, only the Berry phase term and a ‘‘potential’’ energy for 0 appear there
FIG. 16. kagome´ qϭ0 SW dispersion in the SS phase at ⌬ϭ0.25, hϭ0. Note the flat character of the mode along the ⌫M FIG. 18. kagome´ R3 SW dispersion in the CS phase at ⌬ϭ0.25, direction. hϭ4. 55 SUPERFLUIDS AND SUPERSOLIDS ON FRUSTRATED... 3115
as opposed to the order S2 potentials generated classically for the other modes. Thus the gap is order ͱS, compared to the energies of order S seen in SWT. Now for the specific details. We concentrate on the neigh- 0 0 borhood of A* , the optimal value of A, and write AϭA* 0 ϩ␦A . It is easy to see that the that couples to this is 0 ϭA͑kϭ0͒ϪB͑kϭ0͒. The eigenvector corresponding to 0 0 this mode is (A ,B ,C)ϭ[(1/2) ,Ϫ(1/2) ,0]. For the triangular lattice the classical ‘‘potential’’ energy of a 0 deformation yields
sinB* Kϭ3⌬ sin2* 1ϩ . ͑33͒ A ͩ 2 sin*ͪ A Choosing the particular value ⌬ϭ0.25 for illustration, we fit the curve in Fig. 2 near its minimum to obtain the term in the 2 effective action (1/2)SC(␦A) , for which we obtain CӍ6. Of course, a ‘‘potential’’ term will also be generated by the integration of the rest of the modes, but since it is order S,it can be neglected compared to the order S2 term already present classically. We will choose the ‘‘coordinate’’ as Q ϭS sin*0. The Berry phase term now looks like FIG. 19. kagome´ R3 SW dispersion in the SS phase at ⌬ϭ0.25, A * 0 ), which enables us to identify Pϵ␦A asץ/ ץ)hϭ0. Note the completely flat zero-energy mode. S sinA ␦A the conjugate momentum. We now write the action as
Q 1 1ץ -expanded to lowest leading order͒. If one neglects the inte͑ Sϭ d iP ϩ SCP2ϩ K˜Q2 , ͑34͒ ͪ 2 2ץ ͩ ͵ gration over the remainder of the modes, the degeneracy mode appears to have a potential energy but no kinetic en- ˜ ergy. This is analogous to a particle of infinite mass, and where Kϭ3⌬͓1ϩ(1/2)(sinC*/sinA*)͔. We can now find the from the simple harmonic oscillator formula ϭͱK/M, the gap as the harmonic frequency of this oscillator: oscillator energy is zero. This is why the mode appears gap- ˜ less. 0ϭͱSCK. ͑35͒ However, as seen from Fig. 2, the integration over the rest ´ of the modes ͑to leading order in the spin-wave expansion͒ The numerical value is 0Ϸ2.3ͱS. The kagome lattice 0 structure introduces new considerations, which we now ad- creates an effective potential which has a dependence on A, rendering the mass M finite and the mode gapped. Since the dress. effective potential is produced by SWT, it will be of order S, 2. kagome´ lattice The important difference that occurs for the kagome´ lat- tice is that the effective action for the A has a linear cusp at the minimum instead of a quadratic minimum, as shown in Figs. 21 and 22 for the qϭ0 and )ϫ) structures, respec- tively. This kind of minimum has previously been seen for the kagome´ lattice in the fully antiferromagnetic Heisenberg case.41,42 We assume that the degeneracy mode is nondis- persing, which, at spin-wave order, is approximately true for the qϭ0 structure, and exactly true for the )ϫ) structure. This implies that the modes are local, and we assume this to be true even after quantum fluctuations have lifted them from zero energy. We can now interchange the roles of and as P and Q and consider the quantum mechanics of the Hamil- tonian
P2 ϩ͉Q͉, ͑36͒ 2M where is of order S. This leads to a gap in the degeneracy mode of order S2/3. A lifting of the degeneracy mode of order S2/3 has previously been seen in Refs. 43 and 44 in the Heisenberg case. FIG. 20. kagome´ R3 SW dispersion in the MI phase at ⌬ϭ0.25, Since the kagome´ )ϫ) structure is always lower in en- hϭ1. ergy, we will concentrate on it in the following, leaving a 3116 GANPATHY MURTHY, DANIEL AROVAS, AND ASSA AUERBACH 55
gons must be mediated by the bordering C , and must there- fore be third order or higher in ␦A͑r͒. This means that to quadratic order the A fluctuations of the hexagon do not interact with the rest of the lattice. It is these local excitations that produce the flat mode. Note that even classically, the energy of this distortion is not zero if higher orders in ␦A are included. This is analogous to the flat mode in the qϭ0 structure of the isotropic kagome´ antiferromagnet, where once again, the mode is flat only to harmonic order.35 Consider this degeneracy mode in the presence of a small field. It is easy to show that the field energy ͑per nine site unit cell͒ near the optimal angles is
3 ͑2Ϫ⌬͒2 ␦E ϭ hS2 cos ͑␦͒2. ͑37͒ field 2 ͑1Ϫ⌬͒ A We do not need to consider the bond energies of the spins to this order for the following reasons: ͑i͒ One can decompose the deviation from the optimal A , B , C into ␦Bϭ␦BAϩ␦BЈ , and ␦Cϭ␦CA ϩ␦CЈ , where the first part represents the change in B and C along the degeneracy direction, due to the change in A , FIG. 21. Quantum energy correction versus A for the kagome´ and the primed part corresponds to a change orthogonal to qϭ0 structure. Note the cusp at the minimum. the degeneracy direction.
͑ii͒ The optimal ␦BЈ , ␦CЈ in the presence of the field are detailed analysis of the qϭ0 structure to a future publication. of order h. This is because the bond energy is quadratic in
The degeneracy mode which was merely gapless at kϭ0on ␦BЈ , ␦CЈ while the field energy is linear in these quantities. the triangular lattice, but of nonzero energy at every other k, Therefore, the energy difference due to the bonds will be of becomes completely flat at zero energy on the kagome´ lattice order h2, which can be neglected in comparison to the order with the )ϫ) structure. The reason is fairly straightfor- h field energy at small fields. ward. Consider an ABABAB hexagon in Fig. 4͑b͒, isolated ͑iii͒ There is no bond energy associated with a change from the rest of the lattice by a ring of C sites. For the along ␦A . optimal angles it is easy to deduce that if AϭA*ϩ␦, then The local mode conjugate to this A mode is ͓͑1/2͒␦, 2 BϭA*Ϫ␦ and that C changes only to order ␦ . Any Ϫ͑1/2͒␦,0͔. The ‘‘potential’’ energy ͑per nine-site unit cell͒ coupling between the ␦A͑r͒ of neighboring ABABAB hexa- of this mode can be calculated from the classical Hamil- tonian exactly as in the triangular lattice case, and is
3 sinC* Uϭ S2⌬2sin2* 2ϩ . ͑38͒ 2 A ͩ sin*ͪ A The Berry phase term for this unit cell, 0 0 ), allows us to identify Qϭ3S sinA* asץ/ ץ)3S sinA*␦ the coordinate, and Pϭ␦ as its conjugate momentum. Then
͑S,h,⌬͒ϭ˜͑⌬͒Sͱh, ͑39͒ 2 sinC* ͑2Ϫ⌬͒ ˜͑⌬͒ϭ 2ϩ ⌬ cos*. ͱͩ sin*ͪ ͑1Ϫ⌬͒ A A This gap goes to zero as ͱh in the limit h 0, and the mode becomes gapless in addition to being flat.→ So far we have used the field to stiffen the degeneracy mode. Now consider the effect of quantum fluctuations at hϭ0. We can find the effective potential to lowest leading order by computing the ground-state energy for various A in SWT. This is plotted for ⌬ϭ0.25 in Fig. 22. We assume that the mode will remain dispersionless, and therefore local, even when lifted from zero energy by quantum fluctuations.
FIG. 22. Quantum energy correction versus A for the kagome´ This assumption allows us to extract an effective Hamil- R3 structure. Once again the leading-order potential has a linear tonian for each local degeneracy mode, which has been writ- cusp. ten down in Eq. ͑35͒, where we identify the momentum as 55 SUPERFLUIDS AND SUPERSOLIDS ON FRUSTRATED... 3117
0 Pϭ3S sinA* and the conjugate coordinate as Qϭ␦, and where the numerical value of is obtained from the figure ͑for ⌬ϭ0.25͒:
M Ϫ1ϭ0.1944, ͑40͒ ϭ0.025S. We use a Gaussian trial wave function to obtain the approxi- mate ground-state energy of this Hamiltonian, yielding
272M Ϫ1 1/3 Ϸ Ϸ0.02S2/3. ͑41͒ 0 ͩ 128 ͪ Of course, we expect this mode to disperse, but the calcula- tion of the dispersion43,44 is much harder than the one pre- sented here, and will be pursued in future work.
III. EFFECT OF QUANTUM FLUCTUATIONS ON ORDER PARAMETERS One of the motivations for this work has been to see if quantum fluctuations can disorder the system, creating a spin FIG. 23. Quantum fluctuation corrections to the magnitudes of liquid, which would correspond to a ordinary liquid ͑with the spins on the A and C sublattices as a function of ⌬ at hϭ0 for nonzero viscosity͒ for the bosons. To leading order in SWT, the triangular lattice. one can compute the average values of the spins as small, the two models have no simple relationship with each ͗S ͘ϭ͑͗a†a ͘ϪS͒⍀ˆ . ͑42͒ i i i MF other since the triangular lattice is not bipartite. x The calculation sketched out in Sec. II also produces the Although total S decreases as ⌬ decreases, it seems that explicit Bogoliubov transformation, which can then be used superfluid order persists all the way, vanishing only when to find the expectation values of bilinears. More explicitly, in ⌬ϭ0. In Fig. 23 we plot the quantum fluctuation corrections terms of the matrix T which implements the transformation to the magnitudes of the spins on the three sublattices ͑since BϭC we plot only two values͒.As⌬ 0 the A sublattice 2K remains stiff, while the B and C sublattice→ spins get reduced † † to about half their classical value for Sϭ1/2 . Thus, super- ͗ei ͑k͒e j͑k͒͘ϭ ͚ T␣i͑k͒T j␣͑k͒, ͑43͒ ͑ ͒ ␣ϭKϩ1 solid order on the triangular lattice survives quantum fluc- tuations at the spin-wave level. where it is understood that i and j run from 1 to K ͑the Let us now turn to Fig. 24, which shows the classical and number of sublattices͒, while ␣ runs from Kϩ1to2K, the quantum order parameters as a function of ⌬ for hϭ3. For rank of T. Let us now turn to the different cases. ⌬Ͻ1/2 it is clear that though there are quantum fluctuation corrections to each of the spins, the total Sz is not corrected, A. Triangular lattice thus leaving the filling exactly 2/3. For ⌬Ͼ1/2, there are Figure 3 shows a plot of the classical and quantum- fluctuation corrections which do not affect the nature of the corrected total Sz and total Sx, for hϭ0. It is clear that the phase. Once again, we will concentrate on the lower energy quantum-corrected value of the magnetization is very close )ϫ) structure on the kagome´ lattice, leaving an analysis of to zero, independent of ⌬. Exact diagonalizations of finite the qϭ0 structure to future work. clusters for the anisotropic antiferromagnetic case ⌬Ͻ0 ͑Ref. 45͒ suggest strongly that this is an exact statement. That is, ´ the exact ground state at hϭ0 has zero longitudinal magne- B. kagome 9 sublattice structure tization M z . An interesting fact about the triangular lattice Figure 5 presents the naive results for the occupation allows us to map the ⌬Ͼ0 problem ͑our model͒ to the ⌬Ͻ0 numbers of the different sublattices in SWT for the )ϫ) model, which is the anisotropic antiferromagnet solved in structure at ⌬ϭ0.25 as a function of h. Notice that as h 0 SWT by KMFF. For very small ⌬ one can work to linear the fluctuation corrections diverge. This is a consequence→ of order in ⌬, which means that one is working within the set of the flat mode, whose energy goes to zero as h 0. In reality, antiferromagnetic Ising model ground states. For nearest- as we have seen in the previous section, the energy→ flat mode neighbor spin flips, the Marshall sign property is obeyed by will be lifted in higher orders of SWT. While a quantitatively the wave functions, arising from a partition of the set of accurate analysis requires the computation of the dispersion ground states into disjoint even and odd states.46 This means of this degeneracy mode, we can make a rough estimate of that for very small ⌬ there is evidence that the boson system the fluctuation corrections to the sublattice spins by assum- would be exactly half-filled in the true ground state. How- ing that the mode remains flat at the value 1 calculated in ever, to any higher order in ⌬, or numerically for ⌬ not so Sec. II C 2. 3118 GANPATHY MURTHY, DANIEL AROVAS, AND ASSA AUERBACH 55
will, weakly͒ in order to estimate the quantum fluctuations in the sublattice spins. We will also assume that none of the matrix elements change substantially. Thus the energy of the mode is the only significant factor. This means that we can estimate the contribution to the fluctuation correction to the spins by applying a field hQ such that the field-induced en- ergy is the same as the energy induced by quantum fluctua- tions:
2 h Ϸ SϪ2/3. ͑47͒ Q ˜ 2
Numerically, we find for ⌬ϭ0.25 that hQϭ0.000 57. This implies boson occupations of nAϭnBϭ1.66, and nCϭ0.04. Since the boson occupations are larger than the value Sϭ1/2, we find the fluctuations to be larger than the mean-field value of the spin. Thus, despite the lifting of the flat mode due to quantum fluctuations, we find that fluctuations may be large enough to disorder the mean-field ordering on the A and B sublattices. However, it is not certain that large fluctuations imply disorder. Furthermore, it is difficult to understand how the C sublattice could remain stiff and ordered if the other FIG. 24. Classical and quantum values of Sz and Sx for the triangular lattice at hϭ3 as a function of ⌬. Note that for ⌬Ͻ1/2 we two lattices are disordered. Strictly speaking, what this result are in the MI phase, and the total magnetization is uncorrected from tells us is that we have reached the limitations of spin-wave its mean-field value of 2/3. theory.
We need to determine the reduction in the magnitude of CONCLUSIONS AND OPEN QUESTIONS each sublattice spin due to quantum fluctuations. In terms of the equilibrium position * and the deviations from equilib- We have found a broad region of parameter space where rium ␦ and ␦ we can write the magnitude of the spin as supersolid order is stable to leading order in mean-field theory. The inclusion of spin-wave corrections modifies this 1 1 P2 Q2 picture differently for the triangular and kagome´ lattices. The ͉͗S͉͘ϭS 1Ϫ ͑␦͒2Ϫ sin2*2 ϭ 1Ϫ Ϫ , ͩ 2 2 ͪ ͩ 2 18S2 ͪ supersolid is quite robust on the triangular lattice. Our inter- ͑44͒ pretation of the supersolid corresponds with the ideas of An- 1 0 dreev and Lifshitz, with hopping vacancies undergoing where we have once again used Pϭ3S sinA* and QϭA , Bose condensation. The role of lattice frustration is to pre- and assumed a nondispersing degeneracy mode. vent all the particles from condensing into a solid. We know from the harmonic oscillator that We have also come across a general ͑though nonrigorous͒ argument which limits the search for the true ground state in 2 a system with ground-state degeneracy: Look for the points ͗P ͘0ϭ Ϫ1 , 2M at which global conserved quantities commute in the Poisson ͑45͒ bracket sense with the generators of motion on the degen- eracy subspace. ͗Q2͘ ϭ , 0 2K On the kagome´ lattice we find the )ϫ) structure to be more stable than the qϭ0 structure at all parameter values. and in the case hÞ0, we find Fluctuations seem much stronger here, and may even be able to destroy the long-range order assumed in mean-field 1 ⌬͑2ϩsinC*/sinA*͒ ͉͗S͉͘ϭSϪ theory. This is a fertile region for numerical and experimen- ͩͱcos*͓͑2Ϫ⌬͒2/1Ϫ⌬͔ tal work. 12Sͱh A To what experimental systems might these considerations 2 h cosA*͓͑2Ϫ⌬͒ /1Ϫ⌬͔ be applicable? One might be able to construct an array of ϩ . ͑46͒ 47 2 ͱ ⌬͑2ϩsin*/sin*͒ ͪ Josephson junctions which satisfy the conditions necessary A A for the existence of a supersolid. This means that the charg- Clearly there is a divergence as h 0, which is seen in ing energy of each grain should be very high, and the Fig. 5. Of course, there will also be a→ contribution indepen- nearest-neighbor charging energy should be higher than the dent of h due to the other modes not associated with the Josephson coupling between the neighboring grains. Further- degeneracy. more, only pair hopping should be relevant, which implies Now consider the case hϭ0. As shown in the previous temperatures low compared to the bulk superconducting Tc . section, the flat mode will be lifted from zero energy by Of course, since this is a two-dimensional system, the Bose quantum fluctuations and acquire an energy of order S2/3.We condensate disappears for TÞ0, but power-law ODLRO is will assume that the mode does not disperse ͑even though it expected to remain. 55 SUPERFLUIDS AND SUPERSOLIDS ON FRUSTRATED... 3119
ץ Another experimental system to which these results might 4 H ϭ0 ͑A2͒ ͪ xץ ͩ be relevant is He on graphite. A variety of orderings and transitions are known to occur as a function of temperature X͑͒ 48 and coverage. Also, steps in the superfluid density have for all ϭ1,...,2N, where parametrizes the degeneracy been seen as a function of coverage ͑for multiple layers͒ for submanifold, a curve in phase space defined by X͑͒. The 49 4He on graphite, and have been interpreted as resulting degeneracy submanifold is one dimensional in this example. 50 from correlation effects. Differentiating with respect to gives As it stands, this work is not applicable to the question of 4 2N 2 X͑͒ץ ץ -supersolidity in He on a smooth substrate. In order to ap proach the continuum one would have to consider very low ͚ H ϭ0, ͑A3͒ ץ ͪ xץ xץ ͩ ϭ1 densities on the lattice, as well as long-range interactions X͑͒
is a null eigenvector of the Hessianץ/Xץ see Ref. 15 for an example of a continuum approach͒.We i.e., the vector͑ close with a number of important open questions. matrix evaluated at the point X͑͒. SWT seems sufficient for the triangular lattice, but not for As an example, consider the function the kagome´ lattice. A formalism that can consider ordered 1 and disordered states in a unified manner is necessary, per- 2 2 51,36 ͑r,͒ϭ ͑r Ϫ1͒ ͑aϩb cos͒, ͑A4͒ haps a variant of the large-N approaches. Second, even H 4 for the triangular lattice, the question of vortices in the ground state is open. The spin-wave ground state has vorti- whose minimum is at rϭ1 independent of . Note that is not cyclic in , and its conjugate momentum ͑say r2,by ces, but they occur in nearest-neighbor pairs. A fruitful way H to incorporate vortices might be to consider the effective definition͒ is not conserved as Noether’s theorem does not theory of the gapless modes only, and include their interac- apply. Nonetheless, there is a one-parameter family of tion with vortices in a semiclassical manner. ground states, parametrized by . The Hessian matrix, evalu- The original picture of Ref. 20 has been confirmed by a ated at rϭ1, is strong-coupling expansion.10 Strong-coupling expansions 52,53 2͑aϩb cos͒ 0 and variational approaches are complimentary to the Hϭ ͑A5͒ spin-wave expansion, and it would be interesting to investi- ͩ 00ͪ gate our model by these approaches. r2 0 .)( )ϭ(1) is a null eigenvectorץ/ץ) The effects of nonzero temperature on systems with and the vector ground-state degeneracy can be very nontrivial. In particular, Now consider the Gaussian fluctuations about the ground quantum and thermal ground-state selection effects may state. We are interested in the partition function Z Ϫ ϭ͐ ͓p,q͔e A, which is obtained from the action func- compete to produce a sequence of phases and transitions as D the temperature is raised.29 Clearly, thermal effects have to tional be elucidated before our results can be directly applied to an ប piץ experimental situation. Finally, it is intriguing to harken back ϭ ϩ 24 d iប͚ qi ͑p,q͒ , ͑A6͒ ͪ Hץ to the question raised by Anderson and Fazekas and ask A ͵0 ͩ i whether it is possible for a liquid state to be stable at zero where iϭ1,...,N. We expand about one of the ground states, temperature. writing piϭPiϩ␦pi , qiϭQiϩ␦qi , and, using summation convention, ACKNOWLEDGMENTS ប ប ប It is a pleasure to thank John Clarke, Subir Sachdev, Steve ϭ ␦p T ␦p ϩ ␦p R ␦q ϩ ␦q Rt ␦p 2 i ij j 2 i ij j 2 i ij j Kivelson, Dung-hai Lee, Shoucheng Zhang, Zlatko Te- H sanovic, and Dan Rokhsar for valuable discussions. We are ប ϩ ␦q V ␦q ͑A7͒ especially grateful to Chris Henley for detailed comments on 2 i ij j draft versions and enlightening discussions. This research was supported in part by NSF Grant No. DMR-9311949 with ͑G.M.͒. G.M. and D.P.A. are also grateful for the hospitality 2ץ 1 2ץ of the Technion at Haifa, where this work was initiated. 1 T ϭ H , R ϭ H , ͪ qץ pץͩ p ͪ ij បץ pץͩ ij ប i j P,Q i j P,Q APPENDIX: CONTINUOUS DEGENERACY NOT ARISING 2 ץ FROM A SYMMETRY 1 V ϭ H , ͑A8͒ ͪ qץ qץͩ ij ប Consider a Hamiltonian function ͑x͒, where i j P,Q H and Rt is the transpose of R. Thus, the integrand of is A )␦x , whereץxϭ͑p1 ,p2 ,...,pN ,q1 ,q2 ,...,qN͒ ͑A1͒ given by (ប/2)␦x(HϩiJ is the vector of ͑dimensionless͒ coordinates and momenta. TR Hϭ ͑A9͒ What does it mean that the ground state is continuously de- ͩ Rt Vͪ generate? Clearly, there must exist a one-parameter family of ground states, i.e., and 3120 GANPATHY MURTHY, DANIEL AROVAS, AND ASSA AUERBACH 55
Piץ 0NϫN 1NϫN i⌰ץ Jϭ . ͑A10͒ S sin⌰iץ ͪ Ϫ1 0 ͩ ͑A12͒ץ NϫN NϫN N ϭ ϭ Qץ 1͘ ͉ The matrix H is brought to diagonal form by the basis i ͩ 0 ͪ ͪ Nץ ͩ change S:
⍀ϩ 0 and S†HSϭ , ͑A11͒ ͩ 0 ⍀Ϫͪ Piץ where Ϯϭdiag( Ϯ ,..., Ϯ) is a diagonal matrix, and 0Nץ ⍀ 1 N † ͉N ͘ϭ ϭ . ͑A13͒ ͪ Q ͩ1ץ S JSϭJ preserves the symplectic structure ͑commutation re- 2 lations͒. Note that the diagonalized Hamiltonian is i N ͪ ץ ͩ t ϩ t Ϫ t t 2 (1/2)(p ⍀ pϩq ⍀ q) rather than, say (1/2)(p pϩq ⍀ q). The two are, of course, related by a canonical transformation, Ϯ Note that the ͉N2͘ eigenvector containsa1ineach of its provided the k are nonzero.͒ The eigenvectors come in Ϯ lower N entries. When we compute the overlap ͗N2͉J͉N1͘, pairs ͉ k ͘ which satisfy we obtain JH͉ϩ͘ϭϪϩ͉Ϫ͘, k k k N z Mץ i⌰ץ Ϫ Ϫ ϩ ͗N2͉J͉N1͘ϭϪ S sin⌰i ϭ , ͑A14͒ ץ ץ JH͉k ͘ϭϩk ͉k ͘, i͚ϭ1 as well as the orthonormalization condition where M z is the total zˆ component of the magnetization: z N ͗ϩ͉J͉ϩ͘ϭ͗Ϫ͉J͉Ϫ͘ϭ0, M ϵ͚ iϭ1S cos⌰i . Unless the magnetization M z is extrem- k l k l ized with respect to , there is a finite overlap, and by ap- Ϫ ϩ ϩ Ϫ propriate normalization one can choose ͉N1͘ and ͉N2͘ to be Ϫ͗k ͉J͉l ͘ϭ͗k ͉J͉l ͘ϭ␦kl , a ͉Ϯ͘ pair. When the magnetization is extremized, however, where k,lϭ1,...,N. The N energy eigenvalues are then given the overlap is zero which means that these two null right ϩ Ϫ by ekϭបͱk k . eigenvectors belong to separate pairs, and there are at least Now consider the case where the degeneracy subspace is two zero modes. two dimensional, parametrized by the coordinates ͑,͒.In Therefore, the two modes are independent only when z z -ϭ0, that is to say, when the magnetization M is exץ/ Mץ our spin-wave theory, we have that qiϭi and piϭS͑1Ϫcosi͒, and the invariance is realized as some tremized. In the general case both the conserved quantity and relation among the colatitutes ⌰i() along the ground-state the generator of motion along the degeneracy subspace may submanifold, while the invariance is simply expressed as have components along both p and q subspaces. It is then ⌽i()ϭ⌽i(0)ϩ. The matrix H ͑and hence JH͒ has two easy to see that the two vectors will have zero overlap only if known null right eigenvectors—call them ͉N1͘ and the conserved quantity commutes ͑in the Poisson bracket ͉N2͘—corresponding, respectively, to the and invari- sense͒ with the generator of motion along the degeneracy ances. We have that subspace.
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