Duality, Magnetic space group and their applications to quantum phases and phase transitions on bipartite lattices in several experimental systems
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y Duality, Magnetic space group and their applications to quantum phases and phasetransitions on bipartite lattices in several experimental systems
Jinwu Ye
Department of Physics, The Pennsylvania State University, University Park, PA, 16802 (Dated: October 29, 2018)By using a dual vortex method, we study phases such as superfluid, solids, supersolids andquantum phase transitions in a unified scheme in extended boson Hubbard models at and slightlyaway from half filling on bipartite optical lattices such as honeycomb and square lattice. We alsomap out its global phase diagram at T = 0 of chemical potential versus the ratio of kinetic energyover the interaction. We stress the importance of the self-consistence condition on the saddle pointstructure of the dual gauge fields in the translational symmetry breaking insulating sides, especiallyin the charge density wave side. We find that in the translational symmetry breaking side, differentkinds of supersolids are generic possible states slightly away from half filling. We propose a new kindof supersolid: valence bond supersolid ( VB-SS). In this VB-SS, the density fluctuation at any site isvery large indicating its superfluid nature, but the boson kinetic energies on bonds between two sitesare given and break the lattice translational symmetries indicating its valence bound nature. Weshow that the quantum phase transitions from solids to supersolids driven by a chemical potentialare in the same universality class as that from a Mott insulator to a superfluid, therefore haveexact exponents z = 2 , ν = 1 / , η = 0 with a logarithmic correction. Comparisons with previousquantum Monte-Carlo (QMC) simulations on a square lattice are made. Implications on possiblefuture QMC simulations in both bipartite lattices are given. All these phases and phase transitionscan be potentially realized in ultra-cold atoms loaded on optical bipartite lattices. Then we applyour results to investigate the reentrant ”superfluid” in a narrow region of coverages in the secondlayer of He adsorbed on graphite and the low temperature phase diagram of Hydrogen physisorbedon Krypton-preplated graphite ( H /Kr/graphite ) near half filling. We suggest that He and H lattice supersolids maybe responsible for the experimental signals in the two systems. Finally, wesuggest Cooper supersolid is repressible for the phase diagram of La − x Ba x CuO near x = 1 / I. INTRODUCTION.
The Boson Hubbard model with various kinds of in-teractions, on all kinds of lattices and at different fillingfactors is described by the following Hamiltonian : H = − t X
2, the boson modelEqn.1 can be mapped to a ”generalized” anisotropic S = 1 / : H = − t X
2. So novel phase transi-tions such as SF to CDW or SF to VBS can be realizedonly in such cases. Recently, the dual vortex method( DVM ) developed in was greatly expanded to studyEqn.1 on a square lattice just at such generic filling fac-tors in . It turns out that the DVM at q ≥ q = 1 case due to the cru-cial involvement of the non-commutative projective spacegroup at q ≥
2. The general procedures are the follow-ing. After performing the similar charge-vortex dualitytransformation as in , the authors in obtained a dualtheory of Eqn.1 in term of the interacting vortices ψ a hopping on the dual lattice subject to a fluctuating dual ” magnetic field”. The average strength of the dual ”magnetic field ” through a dual plaquette is equal to theboson density f = p/q . This is similar to the Hofstadterproblem of electrons moving in a crystal lattice in thepresence of a magnetic field . The projective represen-tation of the space group (PSG) dictates that there areat least q -fold degenerate minima in the mean field en-ergy spectrum ( when q = 1, the PSG just reduces to theusual commutative space group ). Near the supercon-ductor to the insulator transition, the most important ψ a fluctuations will be near these q minima which canbe labeled as ψ l , l = 0 , , · · · , q − q di-mensional representation of the PSG. In the continuumlimit, the final effective theory in terms of these q orderparameters should be invariant under this PSG. In thesuperfluid state < ψ l > = 0 for every l , while in the insu-lating state < ψ l > = 0 for at least one l . In the insulatingstate, there must exist some kinds of charge density wave(CDW) or valence bond solid ( VBS) states which maybe stabilized by longer range interactions or possible ringexchange interactions included in Eqn.1. The CDW orVBS order parameter was constructed to be the mostgeneral bilinear and gauge invariant combinations of the ψ l .In a recent unpublished preprint , the author stud-ied all the possible phases and phase transitions in theEBHM slightly away from half filling q = 2 on bipar-tite lattices such as honeycomb and square lattice. Itwas found in that the dual vortex method at d = 2can achieve many important results which are very diffi- cult to achieve from the direct boson picture. Althoughsquare lattice has been studied by various analytic andnumerical methods before, the boson Hubbard model ona honeycomb lattice was not studied in any details byboth analytic and Quantum Monte-Carlo (QMC) meth-ods before the preprint . It is not a Bravais lattice, somay show some different properties than those in squarelattice. Experimentally, the honeycomb lattice could alsobe easily realized in ultra-cold atomic experiments to bediscussed in section II. The honeycomb lattice is also therelevant lattice for adatom adsorption on substrates tobe discussed in section VIII. I found that a supersolid (SS ) state exists only away from half filling . A super-solid in Eqn.1 is a state with both superfluid and solidorder. A supersolid in Eqn.2 is defined as the simulta-neous orderings of ferromagnet in the XY component (namely, < b i > = 0 ) and CDW in the Z component. Re-cently, by using the torsional oscillator measurement, aPSU’s group lead by Chan observed a marked 1 ∼
2% su-perfluid component even in bulk solid He at ∼ . K .If this experimental observation indicates the existenceof He supersolid remains controversial . However, itwas established by spin wave expansion and quantumMonte-carlo (QMC) simulations that a supersolidstate could exist in an extended boson Hubbard model(EBHM) with suitable lattice structures, filling factors,interaction ranges and strengths.It was also explicitly pointed out in that the DVMdeveloped in holds only in the superfluid ( SF ) and thevalence bond solid ( VBS ) side where the saddle point ofthe dual gauge field can be taken as uniform, however, itfails in the charge density wave ( CDW ) side where thesaddle point of the dual gauge field can not be taken asuniform anymore. So not only the fluctuations, but alsothe average values of the dual gauge fields are differenton both sides. This is in sharp contrast to the super-fluid to the Mott transition at and near f = n describedby Eqns.A2 and A4 in the appendix. As explained be-low Eqn.A2, the average density of bosons is the samein both the SF and the Mott insulating side, namely, ittakes the integer n on both sides, so the average strengthof the dual magnetic field can be taken as a uniform zeroon both sides, of course, the fluctuations of the gaugefield are completely different on both sides. However, inthe CDW side which breaks the translational symmetry,special care is needed to choose a correct saddle point ofthe dual gauge field in the CDW side to make the the-ory self-consistent, so a different action is needed in theCDW side . This paper is an expanded version of theunpublished preprint . In this expanded version, (1) bypushing the DVM to slightly away from commensuratefilling factors and (2) also extending the DVM explicitlyto the lattice symmetry breaking CDW side by choos-ing the corresponding self-consistent saddle points of thedual gauge field, I will map out the global phase diagramof the EBHM on bipartite optical lattices such as hon-eycome and square lattice at and near half filling andalso investigate superfluid, solid, especially supersolidsand quantum phase transitions in the phase diagram ina unified scheme.The DVM is a magnetic space group ( MSG ) symmetry-based approach which, in principle, can beused to classify all the possible phases and phase tran-sitions after choosing correct saddle points for the dualgauge fields. But the question if a particular phase willappear or not as a ground state depends on the spe-cific values of all the possible parameters in the EBHMin Eqn.1, so it can only be addressed by a microscopicapproach such as Quantum Monte-Carlo (QMC) simula-tions. The DVM can guide the QMC to search for partic-ular phases and phase transitions in a specific model. Fi-nite size scalings in QMC in a specific microscopic modelcan be used to confirm the phases and the universalityclasses of phase transitions discovered by the DVM. Thetwo methods are complementary to each other and bothare needed to completely understand phases and phasetransitions in Eqn.1.The rest of the paper is organized as following, In sec-tion II, I will explicitly derive the generators of the mag-netic space group ( also called projective space group )for q = 2 in a honeycomb lattice, construct the effectiveactions which are invariant under this MSG, also writedown both the charge density wave and the valence bondorder parameters to characterize the symmetry breakingsides in the insulating regimes. In section III and IV, Iwill describe the phases and phase transitions driven bythe competition of kinetic energy and repulsive potentialenergy in Fig. 4 ( represented by the parameter r inFig. 2 and Fig.3 ) along the horizontal axis at the com-mensurate fillings and the phases and phase transitionsdriven by the chemical potential µ along the vertical axisslightly away from the commensurate fillings in Ising andeasy-plane limit respectively. We also stress that satis-fying the self-consist condition for the dual gauge fieldin the Ising limit is absolutely necessary to achieve cor-rect answers in the CDW side. In section V, we studythe EBHM in a square lattice and also discuss the stilldisputed so called deconfined quantum critical point inthe easy-plane limit. In section VI, we compare our re-sults achieved by the DVM with some available QMCresults and make implications on possible future QMCsimulations in both lattices. Then we study the applica-tion of our results on 3 different experimental systems:ultra-cold atoms with long range interactions on opti-cal lattices in both square and honeycomb lattice in sec-tion VII, adatom adsorption on different substrates suchas possible reentrant supersolids in the second layer of He adsorbed on graphite and Hydrogen adsorbed onKrypton-preplated graphite in honeycomb lattice in sec-tion VIII, possible cooper pair supersolid in high temper-ature superconductor La − x Ba x CuO in square latticein section IX. finally, we reach conclusions in section X.In the appendix, we review the boson-vortex duality atinteger filling f = n in which I will stress the role of thedual gauge field and its connection to the more generalcase of f = p/q developed in and the main text of this paper. II. MAGNETIC SPACE GROUP, EFFECTIVEACTION AND ORDER PARAMETERS IN THEDUAL VORTEX PICTURE.
In this section, we will extend the DVM in to studythe EBHM Eqn.1 in honeycomb lattice at and slightlyaway from q = 2. Honeycomb lattice ( solid line inFig.1 ) is not a Bravais lattice, so may show some dif-ferent properties than a square lattice. The dual lat-tice of the honeycomb lattice is a triangular lattice (dashed line in Fig.1 ). Two basis vectors of a primi-tive unit cell of the triangular lattice can be chosen as ~a = ˆ x, ~a = − ˆ x + √ ˆ y, ~a d = ~a + ~a as shown in Fig.1.The reciprocal lattice of a triangular is also a triangularlattice and spanned by two basis vectors ~k = k ~b + k ~b with ~b = ˆ x + ˆ y √ ,~b = √ ˆ y satisfying ~b i · ~a j = δ ij . Thepoint group of a triangular lattice is C v ∼ D whichcontains 12 elements. The two generators can be cho-sen as C = R π/ , I . The space group also includesthe two translation operators T and T along ~a and ~a directions respectively. The 3 translation operators T , T , T d , the rotation operator R π/ , the 3 reflection op-erators I , I , I d , the two rotation operators around thedirect lattice points A and B : R A π/ , R B π/ of the MSGare worked out in the following.In the Landau gauge ~A = (0 , Hx ), the mean fieldHamiltonian for the vortices hopping in a triangular lat-tice in the presence of f flux quanta per triangle in thetight-binding limit is : H v = − t X ~x [ | ~x + ~a >< ~x | + h.c. + | ~x + ~a > e i π fa < ~x | + h.c. + | ~x + ~a d > e i π f ( a +1 / < ~x | + h.c. ] (3)where ~x = a ~a + a ~a denotes lattice points of the tri-angular lattice. Note that the total vortex Hamiltonian H v = H v + V where V is the interaction between vor-tices. Because V does not contain any Aharonov-Bohm(AB) phase factor from the non-trivial A µ background,so the magnetic space group is completely determined bythe vortex kinetic term H v . V always commutes withthe generators of the MSG given by Eqns. 4,5,10. Sowhen constructing the representation of the MSG, wecan ignore the V term without losing any generality.The T , T , T d and the rotation operator R π/ of thePSG are worked out in . Here we listed them in slightlydifferent notations: T = X ~x | ~x + ~a > e i π fa < ~x | T = X ~x | ~x + ~a >< ~x | T d = X ~x | ~x + ~a d > e i π f ( a +1 / < ~x | R π/ = X ~x e i πf ( a − a a ) | a − a , a >< a , a | (4)It can be shown that they all commute with H . How-ever, they do not commute with each other T T = ωT d , T T = ω T T .After performing some algebras, we found three of the6 reflection operators in the point group C v : I = K X ~x e − i πfa | a − a , − a >< a , a | I = K X ~x e − i πfa | − a , a − a >< a , a | I d = K X ~x e − i πf a a | a , a >< a , a | (5)where K is the complex conjugate operator which make I , I , I non-unitary operators. Note that in contrastto the reflection operators in square lattice, the phasefactors in Eqn.5 are crucial to ensure they commute withthe Hamiltonian Eqn.3.The eigenvalue equation Hψ ( ~k ) = E ( ~k ) ψ ( ~k ) leads tothe Harper’s equation in the triangle lattice :( e − ik + e − i ( k + k +2 πf (2 l − ) ψ l − ( k , k )+( e ik + e i ( k + k +2 πf (2 l − ) ψ l +1 ( k , k )+2 cos( − k + 2 πf l ) ψ l ( k , k ) = E ( ~k ) ψ l ( k , k ) (6)where l = 0 , , · · · , q − q = 2 case where there isonly one band E ( ~k ) = − t (cos k +cos k − cos( k + k )).Obviously, E ( k , k ) = E ( − k , − k ) = E ( k , k ). Thereare two minima at ~k ± = ± ( π/ , π/ ψ ± . By using theexpressions of the rotation and reflection operators inEqns. 4, 5, we find the two fields transform as: T , T : ψ ± → e ∓ iπ/ ψ ± ; T d : ψ ± → − e ∓ i π/ ψ ± R π/ : ψ ± → ψ ∓ ; I α : ψ ± → ψ ∗∓ , α = 1 , , d (7)where the transformations under T α , R π/ were alreadyderived in . Note that R π/ plays the same role as the Z exchange symmetry between ψ + and ψ − .The quadratic terms of the effective action is simplythe scalar electrodynamics as in the square lattice case: L = X α = ± | ( ∂ µ − iA µ ) ψ α | + r | ψ α | + 14 F µν (8) where F µν = ǫ µνλ ∂ ν A λ is the strength of the non-compact gauge field A µ .It is easy to show that there are only 2 independentquartic invariants under the above transformations:Λ = | ψ + | + | ψ − | , Λ = | ψ + | | ψ − | (9)The direct honeycomb lattice is a non-Bravais latticewhich contains two lattice points A and B per direct unitcell, it is useful to work out the rotation operators aroundthe direct lattice points A and B which contain new sym-metries not included in the rotation operator around adual triangular lattice point R π/ : R A π/ = X ~x e i πf ( a − a a − a − a ) | − a + 1 , a − a >< a , a | R B π/ = X ~x e i πf (( a − − a a +2 a ) | − a + 1 , a − a + 1 >< a , a | (10)They act on the two vortex fields ψ ± as: R A π/ : ψ ± → e ∓ iπ/ ψ ± ; R B π/ : ψ ± → e ± iπ/ ψ ± (11)It is ease to see the two operators Λ , Λ in Eqn.9 arealso invariant under Eqn.11. Finally we reach the mostgeneral quartic term invariant under all the above trans-formations in Eqns.7, 11: L = γ ( | ψ + | + | ψ − | ) − γ ( | ψ + | − | ψ − | ) (12)In the square lattice, In Landau gauge, a unitary trans-formation to the permutative representation is needed toreach Eqn.12. Here in the gauge chosen in Eqn.3, ψ ± are automatically in the permutative representation. Animportant and subtle point is how to construct bosondensity order parameters to characterize the symmetrybreaking patterns in the direct lattice in terms of the dualvortex fields in the dual lattice. In , the density orderparameter was constructed to be the most general gaugeinvariant and bilinear combinations of the vortex fields ψ l . Then it was evaluated at the direct lattice points,links and dual lattice points to represent boson density,kinetic energy and amplitude of the ring exchanges re-spectively. The dual lattice of a square lattice is still asquare lattice with the same lattice constant. The linkpoints of a square lattice also form a square lattice withlattice constant √
2. Putting the direct, dual and linklattices together forms a square lattice with lattice con-stant 1 /
2. Although like a square lattice, the honeycomblattice is also a bi-partisan lattice consisting of two in-terpenetrating triangular sublattices A and B , its duallattice is a triangular lattice which is a frustrated one (Fig.1), its link points form a Kagome lattice. Putting thethree lattices together forms a very complicated lattice.So the density wave order parameters in the honeycomblattice may not be evaluated in the three lattices. Indeed,when trying to identify the density operators in the insu-lating state, we find the density operators proposed in does not apply anymore in the honeycomb lattice. Inthe following, by studying how gauge invariant bilinearvortex fields transform under the complete PSG, we canidentify both the boson density and boson kinetic energyoperators in the direct lattice. We need evaluate thesequantities only at dual lattice points to characterize theCDW and VBS orders in the direct lattice. Let’s lookat the generalized density order operators which charac-terize the symmetry breaking patterns in the insulatingstates. In the low energy limit, the vortex field is: ψ ( ~x ) = e − i~k + · ~x ψ + ( ~x ) + e − i~k − · ~x ψ − ( ~x ) (13)Intuitively, the generalized density operator ρ ( ~x ) = ψ † ( ~x ) ψ ( ~x ) can be written as: ρ ( ~x ) = ψ † + ψ + + ψ †− ψ − + e i ~Q · ~x ψ † + ψ − + e − i ~Q · ~x ψ †− ψ + (14)where ~Q = 2 π/ , A and B ( Fig. 1). So Eqn.14 should containboth the information on the boson densities on sites Aand B and the boson kinetic energy on the link betweenA and B ( or XY exchange energy < S + A S − B + h.c. > in the spin language ). The single boson Green func-tion hopping on the direct lattice is related to the gauge-invariant single vortex Green function on the dual latticein a highly non-local way . Fortunately, the two bosonquantities such as the density and the kinetic energy mayhave simple local expressions in terms of the dual vortexfields. By studying how the operators in Eqn. 14 trans-form under 7 and 11, in the scaling limit, we can identifythese quantities as ( up to a unknown prefactor) : ρ A = ψ † + ψ + , ρ B = ψ †− ψ − K AB = e i ~Q · ~x ψ † + ψ − + e − i ~Q · ~x ψ †− ψ + (15)where ~x stands for dual lattice points only .Moving slightly away from half filling f = 1 / mean dual magnetic field H ∼ δf = f − / inside the SF phase , the most general action invariantunder all the MSG transformations upto quartic terms is: L SF = X α = a/b | ( ∂ µ − iA µ ) ψ α | + r | ψ α | + 14 e ( ǫ µνλ ∂ ν A λ − πδf δ µτ ) + L (16)where A µ is a non-compact U (1) gauge field. Uptothe quartic level, with correspondingly defined ψ a/b in asquare lattice, Eqn.16 is the same as that in the squarelattice derived in . Because Eqn.16 is a long wavelengtheffective action, the relations between the phenomenolog-ical parameters in Eqn.16 and the microscopic parame-ters in Eqn.1 are not known. Fortunately, we are still able aa a d A B ABAB YX ZYZ Y Z
FIG. 1: (a) Bosons at filling factor f are hopping on a hon-eycomb lattice ( solid line ) which has two sublattices A and B . Vortices are hopping on its dual lattice which is a triangu-lar lattice ( dashed line ) which has three sublattices X, Y, Z .In the easy-plane limit shown in Fig.2c, one of the three VBSstates on the honeycomb lattice is shown by the thick bondsin the figure. The other two VBS can be obtained by R A π/ or R B π/ . to classify some phases and phase transitions and makesome concrete predictions from Eqn.16 without knowingthese relations. In the following, we assume r < III. ISING LIMIT If γ >
0, the system is in the Ising limit, the meanfield solution is ψ a = 1 , ψ b = 0 or vice versa. The systemis in the CDW order. We can see how ρ A , ρ B transformunder Eqns. 7 and 11: T α , R A/B π/ : ρ A/B → ρ A/B ; R π/ , I α : ρ A ↔ ρ B (17)These transformations confirm that ρ A and ρ B indeedcan be identified as the boson density operators at directsublattices A and B . In this section, we discuss at andaway from half filling respectively. A. SF to CDW transition at half filling δf = 0 . If r <
0, the system is in the CDW order which couldtake checkboard ( π, π ) order or a stripe order . Eqn.16is an expansion around the uniform saddle point < ∇ × ~A > = f = 1 / r <
0, a different saddle pointwhere < ∇ × ~A a > = 1 − α for sublattice A and < ∇ × ~A b > = α for sublattice B with α < / ψ b in the staggered dual magnetic field at δf = 0, theeffective action inside the CDW state is: L C − CDW = | ( ∂ µ − iA bµ ) ψ b | + ˜ r | ψ b | + ˜ u | ψ b | + · · · + 14˜ e ( ǫ µνλ ∂ ν A bλ ) (18)where ˜ r <
0, so the system is in the CDW state where < ψ b > = 0. Eqn.18 is essentially the same as Eqn.A2 in q = 1 case. Note that because < ψ a > = 0, the gauge field ~A a is always massive, so it does not appear in Eqn.18.As explained in the appendix, this indicates the densityfluctuations in sublattice A is suppressed, so can be takenas fixed.Due to the change of the saddle point structures, thetransition from the SF to the CDW driven by the horizon-tal axis ( quantum fluctuation r ) in Fig.2b is a strong firstorder transition. If we assumed that Eqn.16 at δf = 0in the Ising limit could be used to describe this first or-der transition, then the CDW side could be described by < ψ a > = 0 , < ψ b > = 0 or vice versa , but no phaseswith < ψ a > = 0 , < ψ b > = 0. However, as shown inEqn.18, in the CDW side, < ψ a > = 0 , < ψ b > = 0, soboth gauge fields ~A a and ~A b are massive, the density fluc-tuations in both sublattice A and B are suppressed, soboth can be taken as fixed. This fact is due to the changeof saddle point structure across the SF to the CDW tran-sition. So we conclude that due to this change of saddlepoint structure of the dual gauge fields across the SF tothe CDW, Eqn.16 at δf = 0 in the Ising limit can notbe used to describe this first order transition. However,it does give a qualitative indication of this strong firstorder transition. This is expected, because in a strongfirst order transition, two separate dual actions Eqn.16in the SF side and Eqn.18 are needed to describe the twosides separately. Note that in the direct picture, the SFbreaks the U (1) symmetry, the CDW breaks the latticesymmetry, so the two sides break two completely differentsymmetries, it can only be first order. B. Charge Density Wave (CDW) supersolid awayfrom half filling δf = 0 . Now we look at the effects of the in-commensurability δf = f − / r >
0, it isknown the SF is stable against the change of the chemicalpotential ( or adding bosons ). In the CDW side where˜ r <
0, moving slightly away from half filling f = 1 / mean dual magnetic field H ∼ δf = f − / L IC − CDW = | ( ∂ µ − iA bµ ) ψ b | + ˜ r | ψ b | + ˜ u | ψ b | + · · · + 14˜ e ( ǫ µνλ ∂ ν A bλ − πδf δ µτ ) (19)where the vortices in the phase winding of ψ b should beinterpreted as the the boson number . Note that be-cause < ψ a > = 0 and remains non-zero in the presenceof δf , the gauge field ~A a is always massive and remainsmassive in the presence of δf , so it still does not appearin Eqn.18. This indicates the density fluctuations in sub-lattice A remains suppressed, so can be taken as fixed inthe presence of δf .Eqn.19 is essentially the same as Eqn.A4 which has thestructure identical to the conventional q = 1 component Ginzburg-Landau model for type-II ” superconductors ”in a ”magnetic” field. It was shown in that for typeII superconductors, the gauge field fluctuations will ren-der the vortex fluid phase intruding at H c between theMessiner and the mixed phase ( see Fig. 2a). For param-eters appropriate to the cuprate superconductors, thisintrusion occurs over too narrow an interval of H to beobserved in experiments. In the present boson problemwith the nearest neighbor interaction V > π, π ) CDW state at f = 1 /
2, alongthe dashed line driven by the vertical axis ( chemicalpotential µ ) in Fig.2b, this corresponds to a CDW su-persolid (CDW-SS) state intruding between the commen-surate CDW state at f = 1 / / δf which could be stabilized by fur-ther neighbor interactions in Eqn.1. The first transitionis in the z = 2 , ν = 1 / , η = 0 universality class firstdiscussed in , while the second is a 1st order transition.We expect the intruding window at our q = 2 systemis much wider than that of the q = 1 system. In theCDW states, < ψ b > = 0, so the dual gauge field ~A bµ inEqn.19 is massive. In the CDW-SS state, < ψ b > = 0,there is the gapless superfluid mode represented by thedual gauge field A bµ . The CDW-SS has the same ( π, π )diagonal order as the C-CDW. The identified universalityclass of the CDW to the CDW-SS transition has manyphysical implications. For example, near the C-CDWto the CDW-SS transition, the superfluid density shouldscale as ρ s ∼ | ρ − / | ( d + z − ν = | ρ − / | = δf witha logarithmic correction. The logarithmic correction willbe calculated in a future publication . There must be atransition from the CDW-SS to the SF inside the win-dow driven by the quantum fluctuation r in the Fig.2b.The universality class of this transition is likely to be firstorder and will be investigated further in a future publi-cation . Combining the results in III-A and III-B leadsto the global phase diagram Fig.2b in the Ising limit . IV. EASY-PLANE LIMIT. If γ <
0, the system is in the easy-plane limit,the mean field solution is ψ a = e iθ a , ψ b = e iθ b , then ρ A = ρ B = 1, so the two sublattices remain equivalent.This limit could be reached by possible ring exchange in-teractions in Eqn.1. In the following, we discuss at andaway from half filling respectively. A. SF to VBS transition at half filling δf = 0 . Because the two sublattices remain equivalent, the uni-form saddle point < ∇× ~A > = f = 1 / K AB = cos( ~Q · ~x + θ − )where θ − = θ a − θ b . Let’s look at how the kinetic energy (a) µ C−CDW r<0, γ>0 (b) SF C−VBSVB−SSIC−VBSCDW−SSIC−CDW (c) r r µ SF VortexLiquid T r<0, γ<0 H LatticeVortex SC
FIG. 2: (a) The phase diagram slightly away from f = 1described by Eqn.A4 in the appendix is the same as type-IIsuperconductors in an external magnetic field H where thereshould be a vortex liquid state intruding between the Messinerstate and the vortex lattice state. But the intruding regimeis too narrow to be seen in type-II superconductors . (b)and (c) are the zero temperature global phase diagrams ofthe chemical potential µ versus r in Eqn.16 in the honeycomblattice. (b) The Ising limit γ > f = 1 / / δf .The CDW-SS has the same lattice symmetry breaking as theC-CDW. (c) The Easy-Plane limit γ < f = 1 / / δf . The VB-SS has the same lattice symmetry breakingas the C-VBS. The thin ( thick ) line is the 2nd ( 1st ) ordertransition. The 1st order transition in the Ising ( Easy-plane) limit is strong ( weak ) one. (b) and (c) are drawn onlyabove half filling. µ → − µ corresponds to below half filling. K transform: T α : K → cos( ~Q · ( ~x − ~a α ) + θ ) R π/ : K → cos( ~Q · ~x − θ ); I α : K → KR A/B π/ : K → cos( ~Q · ~x + θ ∓ π/
3) (20)Note that K transforms differently under R π/ and I α ,because the latter are anti-unitary operators. Thesetransformations confirm that K indeed can be identifiedas the boson kinetic energy or the XY exchange energyoperators at the direct lattice.Upto the quartic order, the relative phase between ψ + and ψ − is undetermined. Higher order terms are neededto determine the relative phase. It is clear to see thereare only 3 sixth order invariants: C = | ψ + | + | ψ − | C = ( | ψ + | + | ψ − | ) | ψ + | | ψ − | C = λ [( ψ ∗ + ψ − ) + ( ψ ∗− ψ + ) ] = λ cos 3 θ (21)where θ = θ + − θ − . Especially C is invariant under R A/B π/ Obviously, only the last term C can fix the rel-ative phase. This term corresponds to the 3-monopoleoperator in the spin language.If λ > , θ = (2 n + 1) π/ π/ , π, − π/
3. The kineticenergy K = 1 , − , X, Y, Z where x + x = 0 , , ~Q · ~x =0 , π/ , − π/ K at X, Y, Z , we find K X = cos θ, K Y = cos( θ + 2 π/ , K Z =cos( θ − π/ X , then by the translation, we can getall the other bonds in sublattices Y and Z with the sameorientation. The two bonds on the other two orientationscan be reached by rotations listed in Eqn. 20. By thisway, we can get all the bonds in the whole direct lattice.So the system is in the Valence Bond Solid (VBS ) state,one VBS is shown in Fig.1.If λ < , θ = 2 nπ/ , π/ , − π/
3. The kinetic en-ergy K = 2 , − , − λ >
0. Because the sign of thekinetic term can be changed in a bi-partisian lattice bychanging the sign of b i in one of the two sublattices inEqn.1 (or can be changed by changing the sign of S xi , S yi in one of the two sublattices, but keeping S zi untouched inEqn.2 ), but the product of the sign around a hexagon isfixed. The two cases have the same sign product, so canbe transformed to each other by the transformation. Thiscan also be understood by observing that C in Eqn.21behaves like a hopping term in 3 power, so its sign can bechanged by the transformation. This is in sharp contrastto the square lattice to be discussed in section V whereone sign leads to a columnar dimer, the other leads to aplaquette pattern. B. Valence Bond Supersolid away from half filling δf = 0 . Now we look at the effects of the in-commensurability δf = f − / r > r <
0, slightly away from the half-filling, Eqn.16 becomes: L V BS = ( 12 ∂ µ θ + − A µ ) + 14 e ( ǫ µνλ ∂ ν A λ − πδf δ µτ ) + · · · + ( 12 ∂ µ θ − ) + 2 λ cos 3 θ − (22)where θ ± = θ a ± θ b .Obviously, the θ − sector is massive ( namely, θ a and θ b are locked together ) and can be integrated out. Assum-ing λ >
0, then θ − = π . Setting ψ + = e iθ + ∼ ψ a ∼ − ψ b in Eqn.16 leads to Eqn.19 with ˜ u = 2 γ , so the discus-sions on Ising limit case following Eqn.19 also apply. Inthe present boson problem with possible ring exchangeinteractions in Eqn.1 which stabilizes the VBS state at f = 1 /
2, along the dashed line driven by the verticalaxis ( chemical potential µ ) in Fig.2c, this correspondsto a VBS supersolid (VB-SS) state intruding betweenthe commensurate VBS ( C-VBS) state at f = 1 / / δf as shown in Fig.2c. In the C-VBS state described in thesubsection A, < ψ a > = − < ψ b > = 0 , < ψ † a ψ a > = <ψ † b ψ b > = − < ψ † a ψ b > = 0, the gauge field A µ is mas-sive due to the Higgs mechanism. In the VB-SS state, < ψ a > = < ψ b > = 0, but < ψ † a ψ a > = < ψ † b ψ b > = − <ψ † a ψ b > = 0, so there is a VBS order characterized by theorder parameter K AB = cos( ~Q · ~x + θ − ) which is the sameas the C-VBS, while the gauge field A µ is massless whichstands for the gapless superfluid mode inside the VB-SS.So this VB-SS has both the VBS order and the super-fluid order which justifies its name. So in this VB-SS, thedensity fluctuation on each site is very large signalizingits superfluid nature, while the VBS order is fixed signal-izing its VBS nature . In this IC-VBS state, δf valencebonds shown in Fig.1 is slightly stronger than the others,these δf slightly stronger bonds also form a dilute latticeon top of the underlying C-VBS lattice. Again, the firsttransition from the C-VBS to the VB-SS is in the z =2 , ν = 1 / , η = 0 universality class, while the second fromthe VB-SS to the IC-VBS is 1st order. Near the C-VBSto the VB-SS transition, the superfluid density shouldscale as ρ s ∼ | ρ − / | ( d + z − ν = | ρ − / | = δf with log-arithmic corrections. The nature of the transition fromthe VB-SS to the SF where < ψ a > = < ψ b > = 0 and < ψ † a ψ b > = 0 inside the window driven by the quantumfluctuation r in the Fig.2c is likely to be weakly first orderand will be studied further in a future publication. Com-bining (4a) and (4b) leads to the global phase diagramFig.2c. V. SQUARE LATTICE.
As said below Eqn.16, with correspondingly defined ψ a/b in a square lattice, upto the quartic level, Eqn.16 isthe same as that in the square lattice derived in . A. Ising limit
So the phase diagram in the Ising limit Fig.3b remainsthe same as Fig.2b. The SF to the C-CDW transition at δf = 0 along the horizontal axis is a strong first orderone. Away from the half filling, along the dashed line inFig.3b, the IC-CDW can be stabilized only by very longrange interactions in Eqn.1, if it is not stable, then Fig.3breduces to Fig. 4b ( which is the Fig.14 in ). B. Easy-plane limit
In the Easy-plane limit, as shown in , the lowest orderterm coupling the two phases θ a/b is C sq = λ cos 4 θ , sothe C term in Eqn.22 need to be replaced by C sq . If λ is positive ( negative), the VBS is Columnar dimer (plaquette ) pattern .By using duality analysis and QMC simulation, the au-thor in suggested that in the easy plane limit, the q = 2 (a) µ C−CDW r<0, γ>0 (b) SF C−VBSVB−SSIC−VBSCDW−SSIC−CDW (c) r r µ SF VortexLiquid T r<0, γ<0 H LatticeVortex SC
FIG. 3: Phase diagram in square lattice. (a) is identical to2a and presented here just for completeness. (b) is similarto Fig.2b on the honeycomb lattice in the Ising limit. In theeasy plane limit (c), there is a possible 2nd order transitionbetween the SF and the C-VBS through a so called deconfinedquantum critical point. However, this has been disputed in .. component scalar electrodynamics Eqn.16 at δf = 0along the horizontal axis is a second order transitionthrough a so called ”de-confined QCP”. If this is in-deed the case, then C sq = λ cos 4 θ is irrelevant, thereis a possible 2nd order transition between the SF andthe C-VBS through a so called deconfined quantum crit-ical point as shown in Fig.3c. Away from the half fill-ing, along the dashed line in Fig.3c, the phase transi-tions are the same as those in the honeycomb latticeFig. 2c. The transition from the VB-SS to the SF where < ψ a > = < ψ b > = 0 and < ψ † a ψ b > = 0 inside the win-dow driven by the quantum fluctuation r in the Fig.3c isalso through line of deconfined quantum critical points.Recently, some connections are made between the cor-relation functions ( which lead to transport propertiessuch as conductivity ) near the deconfined quantum crit-ical point in Fig.3c and those in N = 8 supersymmet-ric SU ( N ) Yang-Mills gauge theory in the large N limitat d = 2 . These correlation functions were calculatedthrough AdS × S /CF T connection . The particle-vortex self-duality in Fig.3c and corresponding electro-magnetic self-duality in AdS × S /CF T put strongconstraints on correlation functions in the two modelsrespectively.However, recently, by using QMC simulations at muchlarger systems, the authors in pointed out that theQMC study in violates the hyperscaling and concludedthat there is no such deconfined QCP, so the transitionfrom SF to the VBS driven by the horizontal axis ( quan-tum fluctuation r ) is a weak first order one. If this isindeed the case, the phase diagram in the easy-plane limitremains the same as in a honeycomb lattice Fig.2c. VI. IMPLICATION ON QUANTUMMONTE-CARLO (QMC) SIMULATIONS.
The EBHM Eqn.1 of the hard core bosons on squarelattice with V and V interactions was studied by theQMC simulations in , the authors found a stable striped( π,
0) and (0 , π ) SS ( Fig.4 ). A stable ( π, π ) SS canonly be realized in the soft core boson case . But the (a) µ /V FullEmptySFSF t/V
Striped−SS EmptySFSFFull (π,π)
Solid /V µ FIG. 4: Phase diagrams of the EBHM Eqn.1 in the hardcore limit U = ∞ achieved from the QMC in : (a) withthe nearest neighbor interaction V >
0, the CDW orderingwavevector is ( π, π ). The corresponding ( π, π ) supersolid isunstable against the phase separation. However, it becomesstable in the soft core limit (b) V = 0, but next nearestneighbor interaction V >
0, the CDW ordering wavevector is( π,
0) or (0 , π ) which is called stripe phase. The correspondingstripe supersolid is stable even in the hard core limit. So thereis a narrow window of stripe SS intervening between the stripesolid and the SF. The universality class of the stripe solid tothe stripe SS in (b) was not studied in . The thin (thick)line is a 2nd (1st ) order transition. nature of the CDW to supersolid transition has neverbeen addressed. Our results in the Fig.3 show that theCDW to the SS transition must be in the same univer-sality class of Mott to superfluid transition with exactexponents z = 2 , ν = 1 / , η = 0 with a logarithmiccorrection. It is important to (1) confirm this predic-tion by finite size scaling through the QMC simulationsin square lattice for (0 , π ) and ( π,
0) supersolid in bothhard core and soft core case in the Fig.4 and ( π, π ) su-persolid in the soft core case only (2) do similar thingsin honeycomb lattice to confirm Fig.1. Some of our re-sults in Fig.3 are indeed confirmed in very recent QMCsimulations on soft core bosons in honeycomb lattice .(3) To Eqn.1 with U = ∞ , V >
0, adding a ring ex-change term − K s P ijkl ( b † i b j b † k b l + h.c. ) where i, j, k, l label 4 corners of a square in the square lattice and − K h P ijklmn ( b † i b j b † k b l b † m b n + h.c. ) where i, j, k, l, m, n la-bel 6 corners of a hexagon in the honeycomb lattice tostabilize the C-VBS state at half filling ( Note that if K s , K h >
0, the QMC are free of sign problems), thenconfirm the prediction on C-VBS to VB-SS transitionin Fig.2c and Fig.3c. We expect that the VB-SS phaseshould be stable against a phase separation in both hardand soft core cases. The second transition ( CDW-SS toIC-CDW in Fig.2b and Fig.3b and the VB-SS to IC-VBSin Fig. 2c and Fig.3c ) is hard to be tested in QMC,because some very long range interactions are needed tostabilize the IC-CDW or the IC-VBS state. They arefirst order transition anyway.In fact, one of the predictions in this paper on the scal-ing of the superfluid density ρ s ∼ | ρ − / | was alreadyfound in the striped ( π,
0) solid to striped supersolid tran-sition by QMC in Sec.V-B of Ref. . As shown in sectionIII-B, there should be a logarithmic correction to thescaling of ρ s , it remains a challenge to detect the loga- rithmic correction by a high precision QMC. Of course,the superfluid density is anisotropic ρ xs > ρ ys in the ( π, . Although the authors in suggested it isa 2nd order transition, they did not address the univer-sality class of the transition.In the following three sections, I will discuss the ap-plications of the results achieved in the previous sectionson 3 different experimental systems. VII. ULTRA-COLD ATOM ON SQUARE ANDHONEYCOMB OPTICAL LATTICES
Recently, Bose-Einstein condensation (BEC) was re-alized in ultra-cold atomic gases ( for a review , see ).Superfluid to Mott insulator transition was also observedin optical lattices of ultra-cold alkali atoms . Atomicphysicists are constructing effective various kinds of 2dand 3d optical lattices using laser beams and then loadeither ultra-cold fermion or boson atoms at different fill-ing factors on the lattices. They may tune the param-eters in Eqn.1 to realize different phases and quantumphase transitions . The optical honeycomb lattice ge-ometry Fig.1 and square lattice could be realized in fu-ture ultra-cold atomic experiments. The challenge is toachieve longer range interactions than the onsite interac-tion. Very exciting perspectives to achieve longer rangeinteractions have been opened by recent experiments oncooling and trapping of polar molecules. Being electri-cally or magnetically polarized, polar molecules interactwith each other via long-rang anisotropic dipole-dipoleinteractions. Loading the polar molecules on a 2d opticallattice with the dipole moments perpendicular to thetrapping plane can be mapped to Eqn.1 with long-rangerepulsive interactions ∼ p /r where p is the dipole mo-ment. There are also other ways to generate long-rangeinteractions . We expect Fig.2a can be easily realized.There could also be some efficient ways to generate thering exchange interactions needed to stabilize the theVBS state in Fig.2b. VIII. ADATOM ADSORPTIONS ONSUBSTRATES ( HONEYCOMB LATTICE )
In this section, we will use the phase diagram Fig.2a todiscuss two experimental systems: (1) the reentrant ”su-perfluid” detected in a narrow region of coverages in thesecond layer of He4 ( called He/ He /graphite system) adsorbed on graphite by a previous torsional oscilla-tor experiment ( . (2) the reentrant ”fluidlike” stateof hydrogen adsorbed on Krypton-preplated graphite ( H /Kr/graphite ) near half filling which was investigatedin a recent experiment .The dual vortex approach is a Magnetic Space Group( MSG ) symmetry based approach which can be usedto classify all the possible phases and phase transitions.0However, the question if a particular phase will appearor not as a ground state depends on the specific values ofall the possible parameters in the boson Hubbard modelin Eqn.1, namely, the specific values in the real systems,it can not be addressed in this approach. There are twoways to remedy this short-coming. (1) we can compareour theoretical classifications with some known phasesobserved in experiments, then we can be more specificon our predictions on the nature of unknown phases andphase transitions. (2) As said in section VI, a microscopicapproach such as Quantum Monte-Carlo (QMC) may beneeded to supplement the dual field theoretical approach. In this section, we will take the first strategy to studythe boson Hubbard model Eqn.1 in honeycomb latticeFig.1 near q = 2 ( Fig.1 ). The honeycomb lattice in Fig.1may describe the preferred adsorption sites in the twosystems. So the He/ He /graphite and D /Kr/graphitesystems may have the same symmetry and belong to thesame universality class. A. He/ He /graphite system A superfluid is a fluid that flows through the tini-est channels or cracks without viscosity. So far, thephenomenon of superfluidity has been firmly observedin only two kinds of systems. The first system is thetwo isotopes of Helium: He and He . He T < T c = 2 . K , while He He T < T c = 2 . mK and form a superfluid . The secondsystem is Bose-Einstein condensation in ultra-cold alkaliatomic gases ( for a review, see ). Recently, intensiveresearch activities have been lavished on searching forexcitonic superfluid in excitons in electron-hole semicon-ductor bilayers . Torsional oscillator method was usedto study He films adsorbed on graphite , a large Non-Classical Rotational Inertial ( NCRI ) was detected ina narrow window of coverages c = 17 ∼ atoms/nm in the second layer ( He/ He /graphite system ). TheNCRI is a low temperature reduction in the rotationalmoment of inertia due to the superfluid component ofthe state. This important phenomenon was interpretedas reentrant ”superfluid” in this narrow window . Re-cently, also by using the torsional oscillator measurement,a PSU group lead by Chan observed a marked 1 ∼ He at ∼ . K . This was inter-preted as a new state of matter called supersolid whichhas both superfluid order and crystalline order . How-ever, so far, no NCRI in bulk solid H was detected.At the completion of the first layer, the He atomswith the coverage c ∼ atoms/nm form a triangu-lar lattice which is incommensurate with the underlyinggraphite. It is reasonable to assume the lattice structureof the preferred adsorption sites on the second layer forma honeycomb lattice ( Fig.1). When applying the Eqn.1to the He/ He /graphite system in Fig.1, b † i is the He atom creation operator. The hopping amplitude t is de- termined by the trapping potential from the incommen-surate first He layer. We assume the He atoms interactwith each other with the LJ potential. Note that the LJpotential is quite accurate in describing the long distanceattractive part, but very crude in the short distance re-pulsive part. The repulsive part in real case is expectedto be softer. The chemical potential µ determines thecoverage. The filling factor f is related to the coverage c of He with c ∼ . atoms/nm corresponding to f = 1 / He atoms occupies every two latticesites of the honeycomb lattice ( for example, all the A sublattice sites ) to form a close packed triangular latticewith d AA ∼ . A and d AB ∼ . A which is just smallerthan Lennard-Jones (LJ) parameter σ ( He ) ∼ . A .The onsite U is very big. Because d AB < σ He < d AA , V is positive, while V and further neighbor interactionsare weakly attractive and can be neglected. Note thatour theoretical value 19 . atoms/nm is only 3% awayfrom the experimental value 19 atoms/nm . Fig.2a isvery similar to the ρ s versus c ∼ f phase diagram in He/ He /graphite structure near c = 19 atoms/nm (see Fig. 1 in ). The dashed line is the experimentalpath in He/ He /graphite at T = 0. Ir shows that areentrant supersolid (SS) state is a generic state sand-wiched between the C-IC transition. The reentrant ”superfluid ” state in the second layer of the He filmsadsorbed on graphite could be a He supersolid state.Very interestingly, the data in the torsional oscillator ex-periment in do not show the characteristic form for a2d He superfluid film, instead it resemble that in char-acteristic of a possible supersolid in terms of the gradualonset temperature of the NCRI, the unusual tempera-ture dependence of T SS on the coverage. From Fig.2a,it is easy to see that it maybe difficult to reach the su-perfluid by moving along the horizontal r axis, but it isvery easy to get to the CDW-SS state by moving alongthe vertical coverage axis. In principle, there should be aKosterlitz-Thouless (KT) finite temperature phase tran-sition above which the CDW-SS becomes a coexistenceof the CDW state and normal fluids of interstitials. Theeffects of disorders on this finite temperature transitionwill be studied in and compared with Fig.2 and 3 in . B. H /Kr/graphite system There are great interests in finding superfluidity inother substances. Hydrogen molecules are relativelylight, quantum fluctuations are large at very low tem-perature. a para-H2 molecule is a boson and very similarto a He4 atom, in principle, H2 could become superfluidat low temperature as He4 does. Unfortunately, unlikeHe4, due to deeper attractive potentials, bulk H2 solidi-fies at low temperature ( T c ∼ K ), this preempts thepossible observation of the speculated superfluity. At suf-ficiently high pressure, the solid hydrogen will transforminto a metallic alkali-like crystal or an unusual two com-ponent ( protons and electrons ) quantum liquid at low1temperatures . Potential avenues to prevent p − H H H n clusters with n = 13 − Indeed, Infra-red spectroscopyalso provided evidences for superfludity in o ( D ) n and p ( H ) n clusters with n = 14 − . Another route isby the reduction of dimensionality. Extensive theoreticaland experimental work has been devoted to study H2 andD2 films in a variety of substrates. Path Integral Monte-Carlo (PIMC) simulations indicated that introduction ofcertain impurities can stabilize a 2 dimensional liquidhydrogen which undergoes a Kosterlitze-Thouless (KT)transition below 1.2 K.In a recent experiment, neutron scattering measure-ments were used to characterize all the possible phasesof D coadsorbed on graphite preplated by a monolayerof Kr called D /Kr/graphite structure . Because twodimensional graphite has the honeycomb net structure,the corresponding preferred adsorption sites form a tri-angular lattice. The precoated Kr atoms will occupy thepreferred adsorption sites and form a triangular lattice.Then the D deposited on top of the Kr monolayer will siton the preferred adsorption sites of the triangular latticeof the Kr monolayer to form a honeycomb lattice. So, thelattice geometry is very similar to Fig.1 with Kr atomssitting on the triangular lattice, while the H moleculeshopping on the honeycomb lattice. The filling factor f isrelated to the coverage c of D . In the coverage c versethe temperature T phase diagram, an unusual featureis that in a small coverage range ( 1 . < c < .
25 )at the commensurate-incommensurate (C-IC) transition,a reentrant fluid phase squeezes in between the C andIC phases down to T = 1 . D has ever been found.The filling factor f is related to the coverage c of H with c = 1 . f = 1 / H molecule occupies every two lattice sites of the honey-comb lattice with d AA ∼ . A and d AB ∼ . A inFig.1. Because d AB < σ H < d AA , V could also bepositive and very big, while V and further neighbor in-teractions are weakly attractive, the above discussions on He in subsection A can also be applied to this system.Fig.2a is very similar to the c ∼ f versus low T phasediagram in H /Kr/graphite structure near c = 1 .
20 (Fig. 6 in ). The dashed line is the experimental pathin H /Kr/graphite at T = 0. The so called (1 × ]commensurate phase in is the CDW state where the D atoms occupy one of the two sublattices of the un-derlying honeycomb lattice. The transition at zero tem-perature is a C-CDW to CDW-SS to IC-CDW transi-tion described by Fig.2(a). The first transition is in the z = 2 , ν = 1 / , η = 0 universality class. Because ν < H supersolid. If this is indeed the case, this may lead to the first observationof H supersolid. Torsional oscillator experiment simi-lar to that used in may be used to measure the NCRIof this H supersolid state. Obviously, this experimentcan avoid the difficulty of using D which has large co-herent neutron scattering cross section, but smaller deBoer quantum number. In fact, Kr may not be the bestspacer to observe the H supersolid state. The ideal sit-uation is to make d AB is just slightly smaller than σ H .Larger atoms like Xe, CF , SF may be more suitablespacers, because they may increase the distance d AB inthe Fig.1 just slightly smaller than σ H . We suggestthat He supersolid should also appear in, for example, He /Kr/graphite structure.The reentrant lattice SS discussed in this section isdifferent from the bulk He SS state discussed in , al-though both kinds of supersolids share many interestingcommon properties. In the former, there is a periodicsubstrate or spacer potential which breaks translationalsymmetries at the very beginning. While in the latter,the lattice results from a spontaneous translational sym-metry breaking, so the theory developed in this paper onlattices is completely different from the continuum the-ory developed in . Combined with the results in , weconclude that He supersolid can exist both in bulk andon substrate, while although H supersolid may not existin the bulk, but it may exist on wisely chosen substrates.Ultra-cold atoms supersolid could also be realized in op-tical lattices. IX. COOPER-PAIR SOLID, SUPERFLUID ANDCOOPER-PAIR SUPERSOLID IN HIGHTEMPERATURE SUPERCONDUCTORS (SQUARE LATTICE )
Very recently , by using both angle resolvedphotoemission (ARPES) and scanning tunneling mi-croscopy (STM) on high temperature superconductor La − x Ba x CuO , Valla et.al detected a quasi-particleenergy gap with d -wave symmetry even when the super-conductivity is completely suppressed at x = 1 /
8, whilehaving almost equally strong superconducting phases atboth higher and lower dopings ( Fig.5). The gap turnsout to reach maximum at x = 1 / T c → x = 1 /
8. This fact suggests that the most strongly boundCooper pairs at x = 1 / T c →
0. In this section, I propose that the formation of astripe Cooper pair supersolid may be able to explain thevery unusual phase diagram of La − x Ba x CuO at andnear doping level x = 1 / still detected an energy gap at Fermi surface with d x − y wave symmetry suggests that there are tightlybound Cooper pairs in this cuprate. We can treat theseCooper pairs as bosons, therefore many important re-sults achieved in previous sections on interacting bosons2 ∆ x501000.30.220 (K) T c FIG. 5: The dashed line is the energy gap ∆ which reachesminimum at x = 1 /
8, the solid line is the superconductingtransition temperature T c which has two domes and is zeroat x = 1 / hopping on lattices maybe applied to this cuprate. Oneimportant new feature for tightly bound Cooper pairs isthat they carry charges 2 e , therefore interact with barelong-range Coulomb interaction ( in high T c cuprates, bytaking the penetration depth λ → ∞ , one can neglectthe very weak Meissner effect ) . I propose the followingphysical picture: at exactly x = 1 /
8, the filling factorsof the bosons on the square lattice is f = − x = 7 / T c = 0. Then weneed q = 16 dual vortices to describe the stripe CDW inthe dual vortex picture. It was explicitly pointed out insection III that special care is needed to choose the cor-rect saddle point of the dual gauge field A µ to describethis stripe CDW state. Slightly away from x = 1 / ,on both sides, the groundstate is a Stripe Cooper pairsupersolid ( CP-SS ) ! When x > / x < / , onlystriped supersolid is stable in a square lattice for hard-core bosons. The quantum phase transition from thestripe Cooper pair solid to the stripe CP-SS driven bythe doping is in the same universality class as that froma Mott insulator to a superfluid, therefore have exactexponents z = 2 , ν = 1 / , η = 0 ( with possible logarith-mic corrections ). The resulting stripe CP-SS has thesame lattice symmetry breaking patterns as the stripeCooper pair solid with the superfluid density scaling as ρ s ∼ | x − / | ( d + z − ν = | x − / | = | δx | with a possiblelogarithmic correction. As pointed out in section III, thesuperfluid density is anisotropic with ρ s along the stripeis larger than that normal to the stripe, but both scale inthe same way with different coefficients. From Uemurarelation, we conclude that T c ∼ | δx | ( in fact, it scales asthe smaller ρ s in the Stripe CP-SS ). This result very nat-urally explains why T c indeed looks linear in | δx | and thephase diagram is nearly symmetric near x = 1 / .The above picture only involves the sector of thetightly bound Copper pair. Of course, understanding theground state of the Cooper-pair sector is very important on its own and is also the starting point to incorporatequasi-particles and spin excitations into the pic-ture. As shown in section IV, a valence-bond supersolid( VB-SS ) can be stabilized if there is a considerable ringexchange interaction. An interesting question to addressis if valence-bond Cooper-pair supersolid can be realizedin some of these cuprates. X. CONCLUSIONS
By using the DVM, we studied superfluid, solid and su-persolid and quantum phase transitions of the extendedboson Hubbard model near half filling on bipartite op-tical lattices such as honeycomb and square lattice andmapped out the global phase diagram at T = 0 in aunified scheme. We identified boson density and bosonkinetic energy operators in terms of the dual vortex fieldsto characterize symmetry breaking patterns in the insu-lating states and supersolid states. In the DVM, startingfrom the featureless superfluid state where the the aver-age value of the dual gauge field through a dual plaquetteis taken to be uniform and equal to the boson density f = p/q , its fluctuation is coupled to q dual vortex or-der parameters and is gapless, one study all the possiblesymmetry breaking patterns by condensing the q vortexorder parameters. We first study the transition driven bythe ratio of the kinetic energy over the potential energyat the commensurate fillings f = p/q along the horizonalaxis in Fig.2-4. In the Ising limit, we found that thesaddle point of the dual gauge fields should be chosen ina self consistent way in the CDW side, this is in sharpcontrast to the more familiar f = n case where the self-consistency condition is automatically guaranteed. Dueto this change of saddle point structure of the dual gaugefields on the both sides, the SF to the CDW transition isalways a strong first order one. In the Easy-plane limit,we found that the self-consistency condition is automati-cally guaranteed, the SF to the VBS transition is a weakfirst order one. Then we study the transition driven bychemical potential slight away from the commensuratefillings f = p/q along the vertical axis in Fig.2-4. Wefound that in the insulating side, the transition at zerotemperature driven by the chemical potential must be aC-CDW ( or C-VBS ) at half filling to a narrow windowof CDW- ( VB-) supersolid, then to a IC-CDW ( IC-VBS ) transition in the Ising ( easy-plane ) limit. Thevalence bond supersolid is a novel kind of supersolid firstproposed in this paper. Although the density fluctuationat any site is very large indicating its superfluid nature,the boson kinetic energies on bonds between two sitesare given and break the lattice translational symmetryindicating its valence bound nature. The first transitionis in the same universality class as that from a Mott in-sulator to a superfluid driven by a chemical potential,therefore have exact exponents z = 2 , ν = 1 / , η = 0with a logarithmic correction. The second is a 1st ordertransition. The results achieved in this letter could guide3QMC simulations to search for all these phases and con-firm the universality class of the transitions. This VB-SSshould be stable in both hard core and sift core limit ifthere is a sufficient strong ring exchange term in Eqn.1.It will be extremely interesting to search this novel kindof supersolid in a specific microscopic model by QMCsimulation.In the second part of this paper, we study the appli-cations of the results achieved in the first part by theDVM to 3 very important ultra-cold atomic and con-densed matter experimental systems. The EBHM inhoneycomb and square lattices could be realized in ul-tracold atoms loaded on optical lattices. So the resultsachieved in this paper may have direct impacts on theatomic experiments in optical lattices. Then we appliedthe results to two condensed matter experimental sys-tems: (1) adatom adsorption on different substrates suchas the adsorption in the second layer of He adsorbed ongraphite and Hydrogen adsorbed on Krypton-preplatedgraphite in honeycomb lattice, we find that a reentrant He supersolid in the second layer on graphite may beresponsible for the NCRI detected in the torsional oscil-lator experiments in and a reentrant H supersolid at T = 0 maybe responsible for the reentrant liquid statesqueezed between C and IC phases detected by coherentneutron scattering. We propose that a judicious choiceof substrate may also lead to an occurrence of hydro-gen lattice supersolid. (2) superconducting phase dia-gram near x = 1 / La − x Ba x CuO in square lattice. We conclude thatthere is a stripe CDW at 1 / T c tozero. There are hole and electron doped Cooper pair su-persolids on both sides of x = 1 / x = 1 / ρ s ∼ T c ∼ | x − / | . Although theSS in lattice models is different from that in a contin-uous systems, the results achieved in this paper on lat-tice supersolids may still shed some lights on the possiblemicroscopic mechanism and phenomenological Ginsburg-Landau theory of the possible He supersolids .I thank Milton Cole for pointing out the experiment inRef. to me. I also like to thank A. Millis for pointingout Ref. to me and helpful discussions. This researchat KITP was supported in part by the NSF under GrantNo. PHY99-07949 at KITP-C by the Project of Knowl-edge Innovation Program (PKIP) of Chinese Academy ofSciences. I also thank C.P. Sun for hospitality during myvisit at KITP-C. APPENDIX A: DUALITY AT INTEGERFILLINGS f = n , THE ROLE OF THE DUALGAUGE FIELD AND THE SELF-CONSISTENCYCONDITION It is instructive to review the direct boson picture andthe duality transformation to the dual vortex picture inthis simplest case with integer filling f = n . The con- tents of this appendix are not new. However, we stressthe self-consistence check on the average value of the dualmagnetic field on both sides of SF and Mott insulator.We also explicitly spell out the physical significance ofthe fluctuations of the dual gauge field on both sides.These clarifications are very helpful to motivate the self-consistence condition on the average value of the dualmagnetic fields in the translational symmetry breakinginsulating sides, especially in the CDW side at q ≥ . Although this self-consistencecondition is automatically satisfied at f = n case, theybecome a non-trivial constraint on a self-consistent the-ory at q ≥ , theauthor found this self-consistent condition is even morenon-trivial and important in frustrated lattices such astriangular and Kagome lattices than the bipartite lat-tices discussed in this paper. As shown in the main text,in both the CDW side ( using limit ) and the valencebond ( easy plane limit ), at and slightly away from thecommensurate filling f = p/q with q ≥ φ † is always accompanied by a creation of a hole φ dueto the particle hole symmetry at f = n , so the Ginzburg-Landau theory to describe the Mott to superfluid tran-sition in terms of the boson order parameter φ is givenby the well know 2 + 1 dimensional relativistic complexscalar theory: S b = Z d rdτ [ | ∂ τ φ | + |∇ φ | + r | φ | + u | φ | + · · · ] (A1)In the Mott state r >
0, so h φ i = 0, there is a Mottgap. In the superfluid state r <
0, so h φ i 6 = 0. Due tothe symmetry breaking in the SF state, there is a gaplessgoldstone mode given by the phase fluctuation of φ .In the dual vortex picture, it is convenient to start fromthe Superfluid side and look at its low energy excitations.There are two kinds of low energy excitations. The first isjust the gapless Goldstone mode in the phase of φ whichis given by a dual gauge field fluctuation A µ . The secondis the topological vortex excitation in the phase windingof ψ . Obviously, the number of vortex ψ † is equal to thenumber of anti-vortex ψ , so the Ginzburg-Landau theoryto describe the superfluid to the Mott transition in termsof the dual vortex order parameter ψ is given by the 2 + 1dimensional scalar electrodynamics: S d = Z d rdτ [ | ( ∂ µ − iA µ ) ψ | + + r d | ψ | + u d | ψ | + · · · + 14 ( ǫ µνλ ∂ ν A λ ) ] (A2)In the superfluid state r d >
0, so h ψ i = 0, there is agapless fluctuation given by the dual gauge field A mu .Integrating out the gauge field fluctuation A µ will leadto a logarithmic interaction between the vortices. In the4Mott insulating state r d <
0, so h ψ i 6 = 0. Due to the”symmetry” breaking in the Mott insulating state, thedual gauge field A µ acquires a mass due to Higgs mech-anism, so there is a Mott gap in the Mott phase. Thevortex action Eqn.A2 is dual to the boson action Eqn.A1.Both actions lead to equivalent description of the SF toMott transition which is in the 3 DXY universality classand the same excitation spectra in both phases. Indeed,this universality class has been confirmed by QMC inthe first reference in from both the direct boson actionEqn.A1 and the dual vortex action Eqn.A2.In the direct boson picture, a vortex is a singularityin the boson wavefunction, so a boson wavefunction ac-quires a 2 π phase when in encircles a vortex. In the dualvortex picture, a boson is a singularity in the vortex wave-function, so a vortex wavefunction acquires a 2 π phasewhen in encircles a boson. So the average strength of thedual magnetic field of the gauge field A µ through a dualplaquette is equal to the boson density f = n , because2 πn is equivalent to 0, so the average value can be simplytaken to be zero. It is important to stress that the aver-age density of bosons is the same in both the SF and theMott insulating side, namely, it takes the integer n onboth sides, so the average strength of the dual magneticfield can be taken as zero on both side, then the fluctua-tions in the dual gauge field reflects the fluctuations of theboson density. As said in the last paragraph, the gaugefield is gapless in the SF phase, so the boson density fluc-tuation in the SF is very large, this is expected, becausethe SF is a phase ordered state, so it has a large den-sity fluctuation. However, in the Mott phase, the gaugefiled is massive, so the density fluctuation is suppressedin the Mott phase. This is expected also, because theMott phase is a density ordered phase, so has very littledensity fluctuations. So by looking at the behaviors of the dual gauge field on both sides, one can distinguishthe properties of the two phases. In short, although theaverage value of the dual gauge field is the same on boththe SF and the Mott insulating side, its fluctuations arecompletely different which, in turn, is determined by theaverage value of the vortex order parameter ψ in Eqn.A2.When slightly away from f = n which is at a in-commensurate density, there is no particle-hole symme-try anymore, so there should be a first order imaginarytime derivative, Eqn.A1 becomes: S ic − b = Z d rdτ [ φ † ∂ τ φ + |∇ φ | + r | φ | + u | φ | + · · · ](A3)where we have dropped the second derivative term | ∂ τ φ | which is less important than the linear derivative term.If there is only a onsite interaction in Eqn.1, then awayfrom f = 1, the system is always in the superfluid state.The transition from the Mott to the SF transition drivenby the chemical potential has the critical exponents z =2 , ν = 1 / , η = 0 with a logarithmic correction.In the dual vortex picture, the linear time derivativeterm corresponds to adding a small mean dual magneticfield δf = f − n to Eqn.A2: S ic − d = Z d rdτ [ | ( ∂ µ − iA µ ) ψ | + + r d | ψ | + u d | ψ | + · · · + 14 ( ǫ µνλ ∂ ν A λ − πδf δ µτ ) ] (A4)This action is essentially the same as the Ginzburg-Landau model for a superconductor in a external mag-netic field if we identify the τ direction as the ˆ z directionalong which the magnetic field δf is applied. For a typeII superconductor, its phase diagram is shown in Fig.2a. M. P. A. Fisher, P. B. Weichman, G. Grinstein and D. S.Fisher; Phys. Rev. B 40, 546 (1989). M. P. A. Fisher and G. Grinstein, Phys. Rev. Lett. 60, 208(1988). Jinwu Ye, Phys. Rev. B 58, 9450-9459 (1998). C. Dasgupta and B. I. Halperin , Phys. Rev. Lett. 47, 1556-1560 (1981), David R. Nelson, Phys. Rev. Lett. 60, 1973-1976 (1988); Matthew P. A. Fisher and D. H. Lee, Phys.Rev. B 39, 2756-2759 (1989). L. Balents, et.al
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