Phase diagram of the Hubbard model on honeycomb lattice
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Phase diagram of the Hubbard model on honeycomb lattice
Abolhassan Vaezi and Xiao-Gang Wen
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Dated: August 29, 2018)In this paper we generalized the slave-particle technique to study the phase diagram of the Hub-bard model on honeycomb lattice which may contain charge fluctuations. For large U , we haveantiferromagnetic order phase. As we decrease U below U c ≃ t , the system undergoes a first orderphase transition into a gapped spin rotation invariant phase. Under a semiclassical approximationof the slave-particle approach, we find that such phase breaks the translation symmetry, the parityand the lattice rotation symmetry. However, beyond the semiclassical approximation, a Z spinliquid that does not break any lattice symmetry is also possible. Introduction.—
Hubbard model[1, 2] is believed todescribe the physics of many strongly correlated sys-tems e.g., Mott insulator[3, 4] and high temperaturesuperconductors[5–7]. It is the simplest model one canwrite capturing the strong correlation physics. So farmany theoretical[8, 9] and numerical techniques[10–12]have been developed to study this model. Among themis the slave particle[13–16] theory which was motivatedby the RVB state first introduced by P.W. Anderson[17].One of the interesting phases that have been studied andis strongly supported by the slave particle approach isthe Z spin liquid phase[18–20] which does not show anylong range order down to zero temperature. Unfortu-nately this phase has not been experimentally verifiedbut recently Meng et al [10] have studied the Hubbardmodel on honeycomb lattice at half filling using the quan-tum Monte Carlo (QMC) method and have reported theexistence of a spin liquid phase for a range of U/t . For-tunately QMC does not have sign problem on bipartitelattices at half filling so we can trust its results. Forsmall U-limit they have reported the semi-metallic phase.At U c ∼ . t they have seen a phase transition to thespin liquid with nonzero spin excitation gap (gapped spinliquid). At U c the charge gap opens up and thereforethis transition point is associated with the Mott metal-insulator transition. For a larger value of U c ∼ . t they have obtained the anti-Ferromagnetic (AF) order inwhich the charge gap is still nonzero but the spin excita-tion is the gapless Goldstone mode.In this paper, we generalized the slave-particlemethod[7, 13] to capture charge fluctuations[15, 21] tostudy the Hubbard model on honeycomb lattice. we haveobtained a similar phase diagram (see Fig. 1 and 2) asin [10] but with different numerical values for U c and U c . We obtained a superconducting phase (instead ofthe semi-metal phase) for small U/t and a AF phase forlarge
U/t . At the meanfield level, our phase between U c and U c is a spin liquid with finite charge/spin gapthat do not break any symmetry. However, the meanfieldstate is unstable. Under a semiclassical approximation,we show that phase between U c and U c is a charge/spingapped state that breaks translation and lattice rotationsymmetry but not spin rotation symmetry. On the otherhand, in the presence of the second neighbor hopping, themeanfield state may become a Z spin liquid state[18– U/t E g s & E g c i n un i t s o f t data 1 : Spin Excitation Gapdata 2 : Charge Excitation Gap FIG. 1. Spin excitation gap (blue line) and charge excitationgap (red dots)
20] that does not break any symmetry and has finitecharge/spin gaps. All phase transitions are first orderwhich agrees with experiments[4].We would like to point out that the slave-rotor methodis the other method to include charge fluctuations,[21]which give rise to a nodal spin liquid between 1 . t
U/t and givesrise to a (wrong) superconducting state.In the large
U/t limit, the Hubbard model can beapproximated by the Heisenberg model and we expectstrong AF order in it. This model has been extensivelystudied by different methods [22–28]. Here we use a dif-ferent approach to study the antiferromagnetic phase. Itis shown that the spin/chrage gapped phase has an in-stability towards antiferromagnetism.The Hubbard model is defined as the following: H = U X i n i, ↑ n i, ↓ − t X h i,j i ,σ C † jσ C iσ + h.c. (1)Here h i, j i means site j is one of the nearestneighbors of site i . We know that Hilbert space m FIG. 2. Staggered magnetization, m (( − ) i S z ( i )), as a func-tion of Ut . There is a phase transition to the antiferromagneticorder at U/t = 3. of Hubbard Hamiltonian has four sates per site. | f i i , |↑i i , |↓i i , |↑↓i i . Let’s name each state as fol-lows: | holon i i = h † i | vac i i = | i i , | spin up spinon i i = f † i, ↑ | vac i i = |↑i i , | spin down spinon i i = f † i, ↓ | vac i i = |↓i i , | doublon i i = d † i | vac i i = |↑↓i i in which | vac i is thevacuum, an unphysical state which contains no slave par-ticles even holons. Using this picture we can rewrite theelectron creation operator as: C † i,σ = f † i,σ h i + σ d † i f i, − σ =[ h i d † i ] (cid:20) f † i,σ σf i, − σ (cid:21) .It should be mentioned that the physical Hilbert spacecontains only four states: empty state (holon), one elec-tron (spinon) and two electrons (doublon) on each site.So we always have one and only one slave particle on eachsite. So we conclude that we should put the local con-straint n hi + n fi, ↑ + n fi, ↓ + n di = 1, to get rid of redundantstates. This is the physical constraint which should besatisfied on every site. We could also obtain this resultby noting that the electron operators are fermion andshould satisfy the anticommutation relations. From thedefinition of C † i,σ it is obvious that it is invariant underU(1) gauge transformation (We require h i and d i to re-main bosonic operators i.e. , preserve their statistics aftertransformation, otherwise we would have SU(2) gauge in-variance. However at U = ∞ we have only fermions andonly in that case we have SU(2) gauge symmetry). It is worth noting that all the slave particles carry the samecharge under the internal U(1) gauge. Since the con-straint as well as the Hubbard Hamiltonian are gaugeinvariant, so is the action of the Hubbard model.In terms of new slave particles, the Hubbard Hamilto-nian can be written as: H = X U d † i d i − t X h i,j i (cid:16) χ fi,j χ bj,i + ∆ f † i,j ∆ bi,j + h.c. (cid:17) (2)In which we have used these notations χ fi,j = P σ f † i,σ f j,σ , χ bi,j = h † i h j − d † i d j , ∆ fi,j = P σ σf − σ,i f j,σ , ∆ bi,j = d i h j + h i d j . To implement theconstraint we appeal to the path integral and the La-grange multiplier methods. S = Z Df † Df Dh † DhDd † Dd e − R dτL L = f † i,σ ∂∂τ f i,σ + d † i ∂∂τ d i + h † i ∂∂τ h i + iλ i g i + Hg i = f † i, ↑ f i, ↑ + f † i, ↓ f i, ↓ + h † i h i + d † i d i − H eff = H + i P i λ i g i . Now by using theHubbard-Stratonovic transformation we can decouplespinons from [hard-core] bosons at the mean field level.To do so we just replace χ i,j and other operators withtheir average. For translational invariant systems wecan assume: h χ i,j i = h χ i − j i and so on. From nowon χ stands for the average of χ operators and so on.Moreover h iλ i i = λ . By these assumptions we can ob-tain unknown parameters in the effective Hamiltonianfrom self-consistency equations. Now, let us focus onthe effective Hamiltonian of bosons. As long as ∆ f isnonzero, the pairing between holons and doublons isnonzero, and they form bound state. Using the Bo-goliubov transformation we can show that the ground-state wave-function of bosons is a paired state whichis completely symmetric between holons and doublons.Therefore, as long as this state represents the ground,we have D h † k,A h k,B E = D d † k,A d k,B E , and as a result: χ b = D h † i,A h j,B − d † i,A d j,B E = 0. Spinons cannot hopin this case and the system is insulator. Self-consistentequations show that χ f = 0 as well and therefore thefollowing Hamiltonians describe the low energy theory ofthis phase: H A,Bf = X k h f † k,A, ↑ f − k,B, ↓ i (cid:20) − λ − t ∆ bk − t ∆ bk + λ (cid:21) (cid:20) f k,A, ↑ f †− k,B, ↓ (cid:21) (4) H A,Bb = X k h d † k,A h − k,B i (cid:20) U − λ − t ∆ fk − t ∆ fk − λ (cid:21) (cid:20) d k,A h †− k,B (cid:21) (5)where ∆ f,b (cid:16) ~k (cid:17) = P δ ∆ f,b~δ e i~k.~δ and ~δ connects two nearest neighbors. We have similar equations for H B,Af and H B,Ab . Using the Bogoliubov transformation we candiagonalize the above Hamiltonians. The energy eigen-values for spinons are E fk = q λ + (cid:0) t ∆ bk (cid:1) . For bosonicpart we obtain: E ± ,kb = + ± U + r(cid:0) U − λ (cid:1) − (cid:16) t ∆ fk (cid:17) .At half filling, in order to excite a charge we need to an-nihilate two spinons and create a pair of holon-doublon.So we can define the charge excitation gap as the sum ofthe excitation energy of a holon and a doublon. Whenthe charge gap is nonzero then the paired holon-doublonstate is stable because exciting quasi-particles on top ofthis state costs energy. For this state, the charge gapis E gc = min E + ,kb +min E − ,kb = 2 q(cid:0) U − λ (cid:1) − (3 t ∆ f ) .Therefore, as long as U − λ > t ∆ f , charge gap isfinite and we are in the insulating phase.On the other hand, when the charge gap closes, thepaired holon-doublon state becomes unstable and freeholons and doublons proliferate. In this state, doublonsand holons condense independently (single boson conden-sation) such that h d i,A i = − h d i,B i and h h i,A i = h h i,B i ,and therefore χ b = 2 h h i,A i = 2 n h = 0, therefore spinonscan hop freely and the ground state is no longer an in-sulator. Since doublons condense at sublattice A and B with opposite signs, we show that ∆ b = 0 and as a result∆ f = 0. we relate the onset of single boson condensation, i.e. the critical point below which charge gap closes, tothe Mott transition. It should be mentioned that χ f,b as well as ∆ b,f jump at this point, so we obtain a firstorder phase transition in this way, which is consistentwith experiments. We like to point out that since d † h operator that carries 2 e electric charge, condenses in thisstate, we indeed obtain a superconducting state insteadof a semi-metallic phase. Phase diagram.—
In the following sections we discussthe three phases that we have obtained from the slaveparticle method.
Superconducting phase.—
Now let us approach theMott transition point from below i.e. from supercon-ducting side. In this phase both ∆ f and ∆ b are zeroand therefore the charge excitation gap as well as thespin excitation gap vanishes. Gapless charge excitationimplies : min E bh,k +min E bd,k = U − λ − χ f = 0. Thiscondition can be satisfied up to U c = 2 λ + 6 tχ f = 2 . t .At this point the Mott transition happens. In terms ofphysical electrons, we obtain an s-wave superconductingstate with gapless charge and spin excitations. The pair-ing order parameter changes sign under parity and 60degrees rotation and transforms trivially under all othersymmetry transformations. It should be mentioned thatat small U limit, the Bose gas of holons and doublonsbecomes very dense and there is strong interaction be-tween them. So the mean-field results are unreliable inthis regime and the superconducting state is a fake re-sult. However, our method captures two important rightfeatures of the system below the phase transition, be-cause we obtain zero spin excitation energy as well aszero charge excitation energy. Charge/spin gapped phase.—
For
U > U c we have χ b = 0. So the quasi-particle weight of spinons arezero and they cannot hop since for any i and j arbi-trary sites: D f † j,σ f i,σ E = 0. Therefore this state is like asuperconductor with infinite carrier’s mass m ∼ tχ b →∞ . Now let us find U c . To do so we assume that:∆ f,b (cid:16) ~δ (cid:17) = ∆ f,b . So we have ∆ f,b (cid:16) ~k (cid:17) = ∆ f,b η ( ~k ), where η ( ~k ) = e ik y + 2 e − i ky cos √ k x and therefore the energyspectrum of spinons and bosons are q λ + | t ∆ b ( k ) | and ± U/ q ( U/ − λ ) − | t ∆ f ( k ) | respectively. From theenergy dispersion of bosons, one can read that the chargegap closes when U c = 2 λ + 6 t ∆ f . Our numerical re-sults show that near the phase transition, ∆ f ≃ . λ ≃ − . U c /t = 2 . U/t limit: ∆ f → .
53 , ∆ b ∼ tU , λ ∼ ( tU ) Ln tU and n b ∼ ∆ b ∼ ( tU ) . It is clear from the energy spec-trum of spinons that in the spin liquid phase, there is agap in their spectrum equal to: E fg = | λ | . Note that inthe spin-charge separation picture, the physics of spin isdetermined by that of spinons. Therefore the spin exci-tation gap is also E sg = | λ | .Now let us focus on the gauge theory of this phase. Inthis phase the effective action of spinons is of the follow-ing forms: H s = λ X i,σ,τ f † i,τ,σ f i,τ,σ − t X ,σ,τ ∆ b ( i, j ) σf † i,A,σ f † j,B, − σ + h.c. (6)Now if we transform operators as: f i,A,σ → e iα f i,A,σ and f i,B,σ → e − iα f i,B,σ for any arbitrary phase α , i.e. assuming a staggered global gauge transformation, thenthe effective Hamiltonian does not change. Therefore theinvariant gauge group ( IGG ) of the Hamiltonian is thestaggered U (1). The reason is that there is no hoppingterm due to the nonzero charge gap and the gauge trans-formation of two neighboring sites have opposite phases,the total phase change of the pairing term becomes zeroand therefore gauge fluctuations are described by stag-gered compact U (1) instead of compact U (1) gauge the-ory. This is equivalent to assuming positive unit chargeon sublattice A and negative unit charge on sublattice Bfor slave particles under the internal gauge transforma-tion.So, at mean field level, the charge/spin gapped phasehas a neutral spinless U (1) gapless mode as its only lowenergy excitations. However, it is well known that U (1)theory in 2+1D is confined due to instanton effects. Solet us assume that the U (1) fluctuations are weak anduse the semiclassical approach to study the U (1) confinedphase where the U (1) mode is gapped. We find[29] thatthese instanton operators, e iθ (in the dual XY model),carry a non-trivial crystal momentum. Also, under 60 de-gree lattice rotation and parity, an instanton is changedto an anti-instanton, e iθ → e − iθ . The instantons carrytrivial quantum numbers for other symmetries. However,a triple instanton operator cos(3 θ ) carries trivial quan-tum numbers for all symmetries. This allows us to con-clude that the neutral spinless U (1) mode is describedby L = g ( ∂θ ) + K cos(3 θ ). In the semiclassical limit(the small g limit), (cid:10) e iθ (cid:11) = 0 and we obtain a phase thatbreaks the translation, the parity and the 60 ◦ rotationsymmetries, but not spin rotation symmetry.We like to point out that in the presence of secondneighbor hopping in the Hubbard model the charge/spingapped phase can be spin liquid that do not break trans-lation, parity, 60 degree lattice rotation, and spin rota-tion symmetries. It is because we can break the stag-gered compact U (1) gauge symmetry down to a Z oneby Anderson-Higgs mechanism. If we add second neigh-bor hopping to the Hubbard model, within slave par-ticle approach, this term generates pairing terms of theform f † i,τ,σ f † j,τ, − σ , i.e. it induces the same sublattice pair-ing and the Hamiltonian is no longer invariant under thestaggered global U (1) gauge transformation. In this casethe staggered compact U (1) gauge symmetry is brokendown to a Z gauge symmetry. The U (1) gauge fluctua-tions are gapped and thus our mean filed state is stableand we can trust our meanfield results. Therefore weobtain a spin liquid phase. Antiferromagnetic phase.—
In this part we show thatthe charge/spin gapped phase is unstable towards an-tiferromagnetic order above U c = 3 t . To obtain Neelorder in the t-J model we simply assume that D ~S z,A E = − D ~S z,B E = m . But how can one implement this ideain the Hubbard model within slave particle approach?In the Neel order phase, translation symmetry is bro-ken and there is an asymmetric situation between sub-lattice A and B. For example we can obtain a antifer-romagnetic phase by assuming ∆ ,f = h f j,B, ↓ f i,A, ↑ i 6 = h f j,A, ↓ f i,B, ↑ i = ∆ ,f . This assumption simply meansthat the chance of finding a spin-up spinon on sublat-tice A and another spin-down spinon on sublattice Bis more than finding the opposite one, so this methodintroduces staggered sublattice magnetization and leadsto the Neel order. If there is a Neel order in thesystem then the chance of creating one holon-doublonpair from annihilating a spin-up spinon on sublattice Aand a spin-down spinon on sublattice B is more thanthe other process. Therefore the excitation energy ofspinons for up-spin on A and down-spin on B is E f ( k ) = p λ + | t ∆ ,b ( k ) | while for down-spin on A and up-spinon B is E f ( k ) = p λ + | t ∆ ,b ( k ) | . On the other hand,since we are not interested in CDW, we need a sym-metric situation between sublattices A and B for thecharge sector. So the energy excitation of bosons is E b ( k ) = q ( U/ − λ ) − | t ∆ f ( k ) | . Using these assump-tions we lead to the following self-consistency equations:∆ ,f = tN s X k | η ( k ) | ∆ ,b E ,f ( k ) (7)∆ ,f = tN s X k | η ( k ) | ∆ ,b E ,f ( k ) (8)∆ f = ∆ ,f + ∆ ,f (9)∆ ,b ∆ ,f + ∆ ,b ∆ ,f ∆ ,f + ∆ ,f = tN s X k | η ( k ) | ∆ f E b ( k ) (10)By solving the above equations we find that above U c = 3 t , m = 0. So we conclude that for U > U c we obtain AF order. It is interesting that in this phase,the gap of spinons is very small and negligible (for ex-ample at U=4, it is − × − ). So in this phase we canassume that spinons are massless quasi-particles.In conclusion, we have used a generalized slave parti-cle method to derive the phase diagram of the Hubbardmodel at half filling on the honeycomb lattice. Withinthe mean field approximation we can decouple fermionsfrom bosons to achieve the effective Hamiltonian that de-scribes the low energy physics of the system. The physicsof the Mott transition is discussed and it turns out to bea first order phase transition. It is shown that the phasetransition occurs when the charge gap opens up. Abovethe critical point, within meanfield theory we obtain aspin liquid phase. But after including gauge fluctuationsof the emergent spin liquid and investigating the instan-ton effect, we argue that this phase is unstable and wefinally obtain a spin/charge gapped phase that breaksthe translation symmetry. For large U limit, a new ap-proach to study antiferromagnetic phase within the slaveparticle picture has been developed. It is shown thatthe gapped spin liquid phase has an instability towardsantiferromagnetism. Acknowledgement.—
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