Strong-coupling expansion of multi-band interacting models: mapping onto the transverse-field J_1-J_2 Ising model
Xiaoyu Wang, Morten. H. Christensen, Erez Berg, Rafael M. Fernandes
SStrong-coupling expansion of multi-band interacting models: mapping onto thetransverse-field J - J Ising model
Xiaoyu Wang, Morten H. Christensen, Erez Berg, and Rafael M. Fernandes National High Magnetic Field Laboratory, Tallahassee, FL 32310, USA Niels Bohr Institute, University of Copenhagen, Denmark Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA (Dated: February 25, 2021)We investigate a class of two-dimensional two-band microscopic models in which the inter-bandrepulsive interactions play the dominant role. We first demonstrate three different schemes ofconstraining the ratios between the three types of inter-band interactions – density-density, spinexchange, and pair-hopping – that render the model free of the fermionic sign-problem for anyfilling and, consequently, amenable to efficient Quantum Monte Carlo simulations. We then studythe behavior of these sign-problem-free models in the strong-coupling regime. In the cases wherespin-rotational invariance is preserved or lowered to a planar symmetry, the strong-coupling groundstate is a quantum paramagnet. However, in the case where there is only a residual Ising symmetry,the strong-coupling expansion maps onto the transverse-field J - J Ising model, whose pseudospinsare associated with local inter-band magnetic order. We show that by varying the band structureparameters within a reasonable range of values, a variety of ground states and quantum criticalpoints can be accessed in the strong-coupling regime, some of which are not realized in the weak-coupling regime. We compare these results with the case of the single-band Hubbard model, whereonly intra-band repulsion is present, and whose strong-coupling behavior is captured by a simpleHeisenberg model.
I. INTRODUCTION
In systems of interacting electrons, a strong Coulombrepulsion can give rise to correlated insulating statesin which charge carriers become localized. The low-energy properties are then usually determined by emer-gent charge-neutral excitations. A prime example isthe Mott-insulating phase observed in several transitionmetal oxides, including the parent compounds of thehigh-temperature cuprate superconductors [1–3]. Thelow energy excitations are typically magnetic in nature,since the local spins usually order at low enough temper-atures inside the Mott state. More exotic correlated phe-nomena may arise in systems where the electrons haveadditional degrees of freedom besides spin. Examplesinclude the orbital-selective Mott transition [4–6] andthe Hund’s metallic state [7, 8] in multi-orbital systems,as well as ferromagnetic and Chern insulating phases ingraphene-based systems with valley degrees of freedom[9–12].From a theoretical perspective, while these materialshave important structural and chemical differences, it isuseful to consider simple models that may capture uni-versal emergent behaviors associated with these corre-lated phases [13]. In this regard, the Hubbard model [14–16] is certainly among the most studied models in con-densed matter physics, consisting of a kinetic hoppingterm (with coefficient t ) and an onsite Hubbard repulsionterm (with coefficient U ) involving electrons on a singleorbital. Upon increasing U , one generally expects a metalto Mott-insulator transition [17]. However, this is not thefull story. As Phil Anderson showed in Ref. 18, a pertur-bative calculation in the strong-coupling regime ( U (cid:29) t ) reveals that the spins of the localized charge carriers ex-perience a superexchange interaction promoted by virtualhopping processes. As a result, at half-filling, the insulat-ing state is expected to display long-range magnetic N´eelorder – unless frustration is present, in which case spinliquid phases may appear [1]. Moving away from half-filling, one obtains the rich t - J model [3]. Interestingly,in the weak-coupling regime ( U (cid:28) t ), perturbative calcu-lations generically find a N´eel state at half-filling [19, 20],which can be a metal or a Slater insulator depending onadditional hopping parameters. As a result, the magneticorder in the weak-coupling and strong-coupling regimesare the same.For intermediate coupling strengths and away fromhalf-filling, unconventional superconductivity is generallyexpected from both strong-coupling [1, 21] and weak-coupling perspectives [22–24]. Assessing this regime ofmoderate correlations, however, is theoretically challeng-ing due to its non-perturbative nature. Numerical meth-ods have played an important role in bridging the gap be-tween the strong- and weak-coupling regimes of the Hub-bard model [25]. These include density-matrix renormal-ization group methods [26–28], dynamical mean-field the-ory [17, 29–31], and Quantum Monte Carlo (QMC) sim-ulations [32–34]. While the latter is a powerful method,due to its exact and unbiased nature, it suffers from theinfamous fermionic sign problem, which, in the case ofthe Hubbard model, can only be avoided exactly at halffilling and for bipartite lattices [35].One of the main motivations to study the Hub-bard model has undoubtedly been the cuprate high-temperature superconductors. While questions remainabout whether this single-band model can capture their a r X i v : . [ c ond - m a t . s t r- e l ] F e b FIG. 1. Phase diagram of the two-band electronic model withinter-band only repulsion, see Eq. (4). The dashed black linedenotes a metal-to-insulator transition/crossover, whereas thegreen lines denote the superconducting (SC) dome. Thedashed red line marks a first-order antiferromagnetic (AFM)phase transition. The AFM phase has a dome-like structurebounded by two putative quantum phase transitions. Theinset depicts the Fermi surface of the non-interacting Hamil-tonian. The color code refers to the charge compressibility χ c . Figure reproduced from Ref. 44. Copyright 2020 by theAmerican Physical Society. rich phenomenology [36, 37], the rise of iron-based su-perconductors, ruthenates, and other multi-band systemshas revived the interest in multi-orbital generalizationsof the Hubbard model, the so-called Hubbard-Kanamorimodels (see, for example, Ref. 38). Several works haveunearthed the unique properties of these models, includ-ing the importance of the Hund’s rule coupling in pro-moting strong-coupling behavior [7, 17] and the rich in-terplay between nesting-driven spin-density wave and un-conventional superconductivity [39]. Strong-coupling ex-pansions involving both spin and orbital degrees of free-dom have also been widely employed [40], resulting incomplex Kugel-Khomskii effective models [41]. From anumerical perspective, the fact that the Hubbard modelsuffers from the fermionic sign-problem may discouragethe use of QMC methods to investigate the more com-plicated Hubbard-Kanamori models. However, as real-ized in Ref. 34, by extending the number of bands ofcertain low-energy models that describe electrons inter-acting with bosonic excitations, it is possible to com-pletely avoid the sign problem due to the emergence ofan anti-unitary symmetry [42]. A similar reasoning wasput forward previously in Ref. 43 in the context of theHubbard-Kanamori model. Therefore, multi-band inter-acting models may provide a unique window into theregime of moderate correlations that is usually difficultto access in single-band models.In a previous work [44], together with Y. Schattner,we showed that a particular realization of the Hubbard-Kanamori model, formulated in band space rather thanin orbital space and with spin-anisotropic interactions, isamenable to be simulated with a sign-problem-free QMC method. The key point is to set the intra-band repul-sion to zero and consider only inter-band repulsion terms.While the opposite limit of large intra-band repulsion andvanishing inter-band terms should map directly onto theusual Hubbard model, the inter-band dominated regimethat we considered in Ref. 44 has been little explored.As such, it has the potential to provide important in-sights onto the properties of multi-band interacting sys-tems. Importantly, such a limit is not as artificial as itmay first look: a weak-coupling renormalization-group(RG) analysis of this model shows that the inter-bandterms grow much faster than the intra-band ones underthe RG flow, in the case of a nearly-nested Fermi sur-face [39]. Moreover, the model remains sign-problem freeeven away from half-filling and for longer-range hoppingparameters.In Fig. 1, we reproduce the temperature-interactionstrength phase diagram obtained by us in Ref. 44, viaQMC simulations for a model with nearly nested bands(see the inset). Despite the absence of intra-band repul-sion, a metal-to-insulator crossover takes place for inter-mediate coupling strengths (dashed black line). Super-conductivity (SC, green line) is also found in the metallicstate near the onset of N´eel antiferromagnetic (AFM) or-der. The main feature of this phase diagram, however,is the emergence of an AFM dome (red line). Startingfrom the weak-coupling limit, it is not surprising thatAFM order is only seen after the interaction strengthovercomes a threshold value, since the nesting betweenthe two bands is not perfect. What is more surprising isthe apparent lack of AFM order in the strong-couplinglimit, since N´eel order is a hallmark of the Mott insulat-ing state of the standard Hubbard model. In Ref. [44],we performed a strong-coupling expansion of this inter-band interacting model, and found the ground state tobe in the quantum paramagnetic phase of an effectivetransverse-field Ising model, in agreement with the QMCresults.In this paper, we perform a strong-coupling expansionof the inter-band interacting model for an arbitrary banddispersion and for different types of spin-anisotropic in-teractions, thereby extending our previous results. In thecase of inter-band interactions that preserve the SU(2)spin rotational symmetry or lower it to a planar symme-try, we find a non-degenerate ground state for the singlesite problem, indicating a “trivial” quantum paramag-netic ground state in the strong-coupling regime. Onthe other hand, for the case of inter-band interactionsthat lower the SU(2) symmetry to an Ising symmetry, theground state of the single-site problem is doubly degen-erate. This gives rise to a pseudospin that corresponds tothe two polarizations of the inter-band Ising-like magne-tization. In terms of this pseudospin, the strong-couplingexpansion of our microscopic interacting model mapsonto a transverse-field Ising model with extended ex-change interactions, i.e. nearest-neighbor interactions,next-nearest-neighbor interactions, etc. In particular, weshow that, by changing the band dispersion parametersacross a reasonable range of values, one can in principleaccess the entire phase diagram of the transverse-field J - J Ising model. This includes the regime where the crit-ical behavior is that of the simple transverse-field Isingmodel – as was the case for the particular band parame-ters we considered in Ref. 44 – or that of the transversefield Ashkin-Teller (four-state clock) model. We discusshow the latter can shed new light on the emergence ofvestigial nematicity in systems that display stripe-typemagnetic order, with possible implications for the cou-pled magnetic-nematic transitions of iron-based super-conductors [45]. Overall, our work reveals the richnessof the strong-coupling regime of microscopic models inwhich inter-band interactions dominate, which can bequite different from the strong-coupling behavior of thestandard intra-band dominated Hubbard model.Our paper is organized as follows: In Section II we in-troduce the microscopic two-band model, and show howrestrictions placed on the interaction parameters lead tothe emergence of an anti-unitary symmetry, which al-lows the model to be simulated using sign-problem freeQMC. In Section III we perform a strong coupling expan-sion of the model with an inter-band spin-spin interactionof the Ising-type, and show that it is mapped onto thetransverse-field J - J Ising model. In Section IV, we dis-cuss the rich phase diagram of the transverse-field J - J Ising model, as well as the choice of microscopic tight-binding parameters to achieve each ground state. Ourconclusions are presented in Sec. V.
II. MULTI-BAND INTERACTING MODEL
We start from a two-band electronic Hamiltonian withonsite interactions only: H = H + H ,H = (cid:88) ij,µ (cid:104) ( t cij − µ c δ ij ) c † iµ c jµ + ( t dij − µ d δ ij ) d † iµ d jµ (cid:105) ,H = (cid:88) i,µν (cid:104) U c † iµ c iµ d † iν d iν − U c † iµ c iν d † iν d iµ + U (cid:16) c † iµ c † iν d iν d iµ + h.c. (cid:17) + U c † iµ c iµ c † iν c iν + U d † iµ d iµ d † iν d iν (cid:105) . (1)Here, c † i,µ ( d † i,µ ) creates an electron in band c ( d ) at site i of a square lattice with spin µ . The non-interactingelectronic dispersions ε c ( k ) and ε d ( k ) are obtained fromthe corresponding hopping parameters t c,dij . The terms µ c and µ d contain implicitly both the chemical po-tential, given by ( µ c + µ d ) /
2, and the onsite energies ± ( µ c − µ d ) / U and U ), inter-band density-density repulsion ( U ), spin exchange ( U ) and pair hopping ( U ). The model can been viewed asa projection of the usual two-orbital Hubbard-Kanamorimodel on the band basis [39], with the assumption thatthe angle dependence of the projected interaction param-eters can be neglected.Depending on the choice of band and interactionparameters, various low-energy phenomena can be ex-plored. For example, if the non-interacting band disper-sions are nearly nested (i.e. one hole-like Fermi pocketand another electron-like Fermi pocket of similar sizeand shape), density-wave order in different channels (i.e.spin, charge, loop-current) can be promoted at weak cou-pling depending on the values of the inter-band inter-actions U , U , and U [39]. Unconventional supercon-ductivity, characterized by gaps of opposite signs on thetwo bands, is a close competitor of the density-wave or-der. On the other hand, if the intra-band interactions U and U are dominant, the model reduces to two nearly-independent copies of the single-band Hubbard model.Alternatively, by constraining the inter-band interactions U , U , and U to be twice the strength of the intra-bandinteractions, the model maps onto a two-layer Hubbard-model with nearest-neighbor hopping between the layers,which has been employed to study Cooper pairing due toan incipient band [46].As discussed in the introduction, our interest here is onthe strong-coupling limit in the regime of dominant inter-band repulsion, as a counterpoint of the more widelystudied regime of dominant intra-band repulsion. There-fore, hereafter we set U = U = 0. An appealing fea-ture of the model with inter-band interactions only isthat, by properly setting the ratios between the threeinter-band interactions, U , U , and U , the Hamiltoniancan be efficiently solved numerically via sign-problem-free QMC simulations. This is because the interactionterm can then be written as an effective attractive termin the inter-band spin-channel. Introducing the notation ψ i = ( c i ↑ , c i ↓ , d i ↑ , d i ↓ ) T , we define the inter-band spin or-der parameter as: M ai ≡ (cid:80) αβ ψ † iα ( σ a ρ x ) αβ ψ iβ , where σ and ρ are Pauli matrices acting in spin and band spacerespectively, and ( α, β ) are indices of the vector spacespanned by ψ i .There are three different possibilities to do such arewriting of the interaction term. The first one corre-sponds to choosing U = 4 U , U = 2 U , and U = 6 U ,which leads to: H , Heisenberg = − U (cid:88) a = x,y,z (cid:88) i M ai M ai . (2)Here, the subscript “Heisenberg” is used to emphasizethe fact that the interaction term is invariant under spinSU(2) rotational symmetry, reminiscent of the Heisen-berg model. Other ratios between U , U , and U al-low for similar types of rewriting in terms of “XY” and“Ising” inter-band spins. Of course, in these cases theinteractions must break spin-rotational symmetry. Thisis not an unreasonable assumption, since spin-orbit cou-pling naturally breaks spin-rotational symmetry in ac-tual materials. In our treatment, such effects are treatedon a phenomenological level only. In particular, setting U = 4 U (1 − δ µν ), U = 0, and U = 4 U we obtain H , XY = − U (cid:88) a = x,y (cid:88) i M ai M ai . (3)Note that the interaction term possesses a residual U (1)(or planar) symmetry, similar to the XY model. Finally,in the case where U = 4 U δ µν , U = 2 U , and U = 2 U , H can be rewritten as H , Ising = − U (cid:88) i M zi M zi , (4)where only a Z symmetry is preserved.The key motivation to introduce these restrictions onthe inter-band repulsions is that the resulting Hamil-tonian can be simulated with QMC without the sign-problem. This can be seen by decoupling the quarticterms via a Hubbard-Stratonovich field { φ ai ( τ ) } , where τ denotes imaginary time. As a result, the partition func-tion can be represented as Z = (cid:90) D [ φ ] det { (cid:98) G − ( φ ) } exp( −S φ ) . (5)Here S φ = (cid:82) β d τ U [ φ ai ( τ )] and (cid:98) G ≡ ( ∂ τ + H + H φ ) − isthe fermionic Green’s function, with H φ = (cid:80) i φ ai ( τ ) M ai .In the single-particle Hilbert space, the quadratic Hamil-tonian ( H + H φ ) has an anti-unitary symmetry (cid:98) U = iσ y ρ z K , where K denotes complex conjugation. Since (cid:98) U = −
1, the eigenstates of the single particle spectrumare always doubly degenerate (an analogue of a Kramer’sdoublet), which guarantees a positive fermionic determi-nant [42]. In Ref. [44], we used determinantal QMC tostudy the Ising case in Eq. (4), resulting in the phasediagram of Fig. 1 discussed in the Introduction.It is interesting to note that the restriction on the ratiosbetween U , U , and U that is needed to “eliminate” thefermionic sign-problem is a much milder constraint thanthe corresponding restriction on the single-band Hubbardmodel. In the latter case, one needs to impose half-fillingand hopping parameters that preserve the bipartite na-ture of the square lattice. In the present case, on theother hand, there are no restrictions on the filling andon the type of hopping parameters. The reason is be-cause the avoidance of the fermionic sign-problem arisesfrom an anti-unitary symmetry related to the intrinsictwo-band nature of the problem, in the same spirit as inRef. 42. III. STRONG-COUPLING EXPANSION
While in our previous work [44] we performed a strong-coupling expansion of the Ising case [Eq. (4)] for a specifictight-binding parameters set, here we explore the strong-coupling regime of the three cases presented in Eqs. (2)–(4) for an arbitrary tight-binding parametrization. The
FIG. 2. Energy spectrum and level degeneracy of the single-site interacting Hamiltonian H for the Heisenberg [Eq. (2)],XY [Eq. (3)], and Ising [Eq. (4)] cases. States in the n = 2filling sector are denoted in red. procedure consists of first neglecting the kinetic termsand then solving the interacting Hamiltonian exactly ona single site. Next, we treat the kinetic terms perturba-tively, and discuss the strong-coupling physics using aneffective Hamiltonian in terms of the available degrees offreedom [47]. As we will show, the Ising case provides theonly non-trivial ground state at strong-coupling, map-ping onto the rich transverse-field J - J Ising model.
A. Single-site exact solutions
The energy spectra of the interacting Hamiltonians inEqs. (2)–(4) on a single site are illustrated in Figure 2.The local Fock space can be written as | η c ; η d (cid:105) where η c , η d ∈ { , ↑ , ↓ , ↑↓} . This is a 16-dimensional vectorspace. Due to number conservation, the Hamiltonianscan be analyzed within any given integer electron filling n = 0 , . . . ,
4. The linear dimension for each filling fac-tor is given by the binomial coefficient d n = C n . The n = 0 and n = 4 states have the highest energy, E = 0.All eight n = 1 and n = 3 states have equal energy, E = − U for the Heisenberg case, E = − U in the XYcase, and E = − U in the Ising case. In all three cases,the n = 2 sector contains the lowest energy states. Intotal, there are six states in the n = 2 sector, given by:Φ = (cid:0) | ↑↓(cid:105) |↑ ; ↑(cid:105) |↑ ; ↓(cid:105) |↓ ; ↑(cid:105) |↓ ; ↓(cid:105) |↑↓ ; 0 (cid:105) (cid:1) T , (6)The matrix elements of H in this sector are H , Heisenberg = − U −
60 4 0 0 0 00 0 2 2 0 00 0 2 2 0 00 0 0 0 4 0 − , (7) H , XY = − U −
40 4 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 4 0 − , (8) H , Ising = − U −
20 0 0 0 0 00 0 2 2 0 00 0 2 2 0 00 0 0 0 0 0 − . (9)The single-site energy spectrum of each case is shown inFig. 2. The states for the filling n = 2 are shown inred, whereas the states for the other filling sectors areshown in blue. Both the Heisenberg and XY Hamil-tonians have a unique ground state given by | a (cid:105) ≡ √ ( |↑↓ ; 0 (cid:105) − | ↑↓(cid:105) ), with energies − U and − U re-spectively. For the Ising Hamiltonian, on the other hand,the lowest-energy state − U is two-fold degenerate, con-sisting of both | a (cid:105) and | a (cid:105) ≡ √ ( |↑ ; ↓(cid:105) + |↓ ; ↑(cid:105) ). A gen-eral ground state can therefore be written as: | ϕ (cid:105) = α | a (cid:105) + β | a (cid:105) , (10)where | α | + | β | = 1. This degeneracy is a resultof the commutation relation [ H , Ising , M zi ] = 0, whichdoes not hold in the Heisenberg or XY cases. Because M zi | a (cid:105) = − | a (cid:105) and M zi | a (cid:105) = − | a (cid:105) , the combina-tions √ ( | a (cid:105)−| a (cid:105) ) and √ ( | a (cid:105) + | a (cid:105) ) are eigenstates of M zi with eigenvalues ±
2. These combinations can thusbe interpreted as the two possible polarizations of theinter-band Ising magnetization. Therefore, in the two-dimensional space spanned by the spinor in Eq. (10),the inter-band magnetization is represented by the τ x pseudospin.This analysis shows that, in the strong-coupling limit,the Heisenberg and XY cases display only a featurelessquantum paramagnetic state. On the other hand, theIsing case has a residual SU(2) degeneracy related tothe inter-band magnetization degree of freedom. Per-turbative interactions between the local inter-band mag-netizations, which we will study in the next subsection,may result in non-trivial ground states. Therefore, withinour model, the Ising case is the closest analogue of thesingle-band Hubbard model, whose strong-coupling limitis characterized by perturbative interactions between lo-cal (intra-band) spins. B. Effective Hamiltonian of the Ising case
We proceed to discuss the effects of the kinetic termsgiven by H . For the Heisenberg and XY cases, the many-body ground state is unique and gapped, given by | Ψ (cid:105) = Π L i =1 | a (cid:105) . Note that this is a quantum paramag-netic state with (cid:104) (cid:126)M i e i Q · r i (cid:105) = 0 for arbitrary wavevector Q . In the Ising case, however, the lowest energy state oneach site is a spinor, analogous to the physical electronspin in the one-band Hubbard model. As a result, theground state has an SU(2) L -degeneracy. Perturbationsdue to the kinetic terms can lift this degeneracy, leadingto non-trivial correlations in the Hilbert space spannedby the local spinors.The energetics of low-energy excitations can be studiedusing an effective Hamiltonian approach [47] in terms ofthe pseudospins τ µ previously introduced. Recall that,in the basis of Eq. (10), inter-band magnetic order corre-sponds to a finite expectation value for τ x . For the IsingHamiltonian we find, up to second order in peturbationtheory, H eff ,ss (cid:48) ≈ − U L δ ss (cid:48) + (cid:104) Ψ ,s | H | Ψ ,s (cid:48) (cid:105) + (cid:88) n (cid:54) =0 ,t (cid:104) Ψ ,s | H | Ψ n,t (cid:105) (cid:104) Ψ n,t | H | Ψ ,s (cid:48) (cid:105) E − E n , (11)where | Ψ ,s (cid:105) = Π L i =1 | ϕ i,s (cid:105) , and | ϕ i,s (cid:105) = α i,s | a (cid:105) + β i,s | a (cid:105) is a configuration of the local spinor. | Ψ n,t (cid:105) de-notes an excited state, having e.g. 1 electron on site i and 3 electrons on site i .The contribution from the single-particle onsite µ c,d terms to the effective Hamiltonian is given by H ( µ )eff ≈ − (cid:88) i (cid:20) ( µ c + µ d ) + 18 U ( µ c − µ d ) (cid:21) τ i − (cid:88) i U ( µ c − µ d ) τ zi , (12)Note that only the difference in the onsite energies be-tween the two bands, but not the chemical potential,gives non-trivial energetics. It corresponds to a trans-verse field, since magnetic order is given by τ x . Thisdifference between onsite energies is a tuning parameterabsent from the strong-coupling limit of one-band Hamil-tonians, such as the Hubbard model.The hopping parameter t ij generates spinor correla-tions between sites { i, j } . To second order, this leads tothe superexchange interactions: H ( t )eff ≈ (cid:88) : ij : (cid:32) − ( t cij ) + ( t dij ) U τ i τ j + t cij t dij U τ xi τ xj (cid:33) , (13)where : ij : denotes an ordered pair of sites, i.e., permut-ing the two indices does not lead to a new term in theHamiltonian. We note that the superexchange interac-tion is of an Ising-type, as expected from the fact thatthe model lacks spin-rotational symmetry. Furthermore,the signs of the exchange interactions depend on the rel-ative signs between the hopping parameters of the twobands. Hence, equal (opposite) signs favor antiferromag-netic (ferromagnetic) alignment of the inter-band magne-tization. To summarize, in the strong coupling limit, the Néel StripeVBS? T / J J / J (a) Néel Stripe (b) h / J J / J PM PM FIG. 3. (a) Schematic phase diagram at zero transversefield ( h = 0), and (b) Schematic phase diagram at zero-temperature ( T = 0) of the transverse-field J - J Ising modelfor antiferromagnetic exchange interactions. Solid (dashed)lines correspond to continuous (first order) phase boundaries.PM denotes the paramagnetic phase and VBS denotes a pos-sible string valence-bond solid. two-band electronic model with Ising interaction mapsonto the generalized transverse-field Ising model contain-ing longer-range exchange interactions: H eff ≈ (cid:88) : ij : J ij τ xi τ xj − h (cid:88) i τ zi , (14)where h i ≡ ( µ ci − µ di ) / U and J ij ≡ t cij t dij / U . Notethat we have omitted an overall shift of the ground stateenergy. IV. THE TRANSVERSE-FIELD J - J ISINGMODEL
Let us focus on the case where H has hopping param-eters defined up to next-nearest-neighbor bonds, yieldingthe band dispersions ε i ( k ) = − µ i + 2 t i (cos k x + cos k y ) +4 t i cos k x cos k y . According to Eq. (14), this leads to thesquare-lattice transverse-field J - J Ising model, with J = t c t d U ,J = t c t d U ,h = ( µ c − µ d ) U . (15) This model has been studied extensively in the recent lit-erature [48–53], and here we give a brief overview of theproposed phase diagrams of the frustrated case, where J >
0. Because the model is invariant upon chang-ing the signs of J or h , hereafter we focus on the caseof J , h >
0. For h = 0, the classical phase diagramis rather well-established, and is schematically shown inFig. 3(a) based on the Monte Carlo results of Ref. 48. For J < J /
2, the ground state is a N´eel antiferromagnet de-scribed by the order parameter ∆ N = L − (cid:80) i (cid:104) τ xi e i Q N · r i (cid:105) ,where Q N ≡ ( π, π ) is the ordering wave-vector (note thatthe ground state would be a ferromagnet if J < J > J /
2, the ground stateis a striped antiferromagnet (regardless of the sign of J ), with an order parameter ∆ S n = L − (cid:80) i (cid:104) τ xi e i Q S n · r i (cid:105) ,where Q S = ( π,
0) and Q S = (0 , π ). The transition be-tween the N´eel and the stripe phases is first-order, asexpected from the fact that they break different symme-tries.Upon increasing temperature, the N´eel to paramag-netic (PM) transition is second-order and belongs to the2D-Ising universality class – except possibly in a narrowregion near J = J /
2, where the transition may becomefirst order [48]. As for the stripe ground state, it is impor-tant to note that it is four-fold degenerate, since there aretwo stripe ordering wave-vectors and two Ising spin polar-izations. In the range 1 / < J /J (cid:47) .
67, the stripe-PMtransition is first-order. However, for J /J (cid:39) .
67, thestripe-PM transition is second-order and described by the2D four-state clock model [48]. A special property of thismodel is that it only has weak universality, in the sensethat only the anomalous critical exponent η (and con-sequently the δ exponent) is universal, while the othercritical exponents are non-universal [54]. In the particu-lar case of the classical J - J Ising model, it was shownin Ref. 48 that the non-universal critical exponents asa function of the ratio J /J map onto the critical ex-ponents of another model that belongs to the four-stateclock weak universality: the Ashkin-Teller model. In par-ticular, for J /J ≈ .
67, the critical exponents are thoseof the four-state Potts model, whereas for J /J → ∞ ,they are those of the 2D Ising model.The quantum phase diagram ( T = 0) of the model re-mains widely debated, although some properties seemto be consistent across different methods [49–53]. InFig. 3(b), we show a schematic candidate phase diagrambased on these works. As in the classical case, the N´eelstate is realized for J /J < / J /J > /
2. For zero transverse field, thetransition at J = J / h increases, however, the situ-ation is less clear. For instance, Ref. 50 reported theonset of a string valence-bond solid for finite transversefields. The quantum phase transition from the N´eel tothe PM state is believed to be second-order and in the 3DIsing universality class [53] – although some methods re-port a regime of first-order transition near J /J = 1 / µ c t c t c µ d t d t d J /J h/J G.S. ( U (cid:29) t )4(a) t t − t t / t t − t t t . t . t − t . t t / / t . t . t − t . t t / / t t t t t . t t t . t t /
16 0 StripeTABLE I. Parameters of the dispersions shown in Fig. 4 anddiscussed in the text, and their respective ground states (G.S.)at strong coupling. play a tricritical point separating a first-order transitionline from a second-order transition line – similarly to theclassical case [52]. Interestingly, the position of the quan-tum tricritical point seems to be closer to the degener-acy point than the classical tricritical point [52, 53], i.e.( J /J ) q < ( J /J ) cl ≈ .
67, as illustrated schematicallyin Fig. 3(a) and (b) by the red dots. The nature of thesecond-order quantum stripe-PM transition remains un-clear. Because the classical transition is described by thefour-state clock model, it is natural to expect that thequantum transition should be described by the quantumversion of the same model. Based on the recent results ofRef. 55 on the quantum q -state clock model, this tran-sition is expected to belong to the 3D XY universalityclass. However, Ref. 53 found that the quantum stripe-PM transition has non-universal critical exponents.This rich landscape of possible ground states of thetwo-band model with dominant inter-band interactionsin the strong-coupling regime contrasts with the simpleN´eel state obtained for the single-band Hubbard model.We now show that the ( h/J , J /J ) phase diagram ofFig. 3(b) can in principle be traversed with reason-able band structure parameters, contrasting the strong-coupling and weak-coupling behaviors of the model.A commonly studied situation is when the two bandsgive rise to a hole-like and an electron-like Fermi pocket,as illustrated in Fig. 4 (a)-(d). In (a), the c -band cre-ates a hole-like Fermi pocket centered at Γ in the Bril-louin zone, whereas the d -band gives rise to an electron-like Fermi pocket of identical size centered at Q N . Thisis achieved by a dominant nearest-neighbor hopping, aswell as opposite onsite energies, µ c = − µ d , of magnitudescomparable to the bandwidth. As shown in Table I, thetight-binding parametrization of Fig. 4(a) favors a N´eelground state in the strong-coupling regime. Upon in-creasing ( µ c − µ d ), which corresponds to shrinking thesizes of the hole-like and electron-like pockets, the valueof the effective random field h/J increases and moves thesystem towards the quantum paramagnetic ground state.The corresponding Fermi surface is shown in Fig. 4 (b).Therefore, without changing the electronic occupation,it is possible to induce a quantum phase transition fromthe N´eel phase to the paramagnetic phase. Physically,changing the band offset may be achieved via pressure.It is also interesting to compare the strong-couplingground state with the weak-coupling one. The - π π - π π - π π - π π - π π - π π - π π - π π (a) (b) - π π - π π - π π - π π - π π - π π - π π - π π (e) (f) - π π - π π - π π - π π - π π - π π - π π - π π (c) (d) FIG. 4. (a-d) Two types of non-interacting Fermi surfacesfeaturing one hole-like pocket centered at Γ and one electron-like pocket centered at either (a,b) Q N = ( π, π ) or (c,d) Q S = ( π, / (0 , π ). (e-f) Two identical copies of Fermi sur-faces with a larger hopping parameter for (e) the nearest-neighbor hopping and (f) the next-nearest-neighbor hopping,respectively. The band parameters are given in Table I. parametrization in Fig. 4 (a) satisfies the perfect nestingcondition ( ε c k = − ε d k + Q N ), which leads to a spin-densitywave order, even for infinitesimal small U , described bythe same N´eel order parameter. Varying ( µ c − µ d ) doesnot spoil the perfect nesting condition, and therefore isnot expected to drive a transition to a paramagneticphase, in contrast to the strong-coupling limit.By including sizable next-nearest-neighbor hopping,i.e. increasing t /t , the d -band electron-like Fermi pock-ets become centered at Q S = ( π,
0) and Q S = (0 , π ),as shown in Fig. 4(c). Here, a stripe state is preferred atstrong-coupling, as shown in Table I. Similarly to the pre-vious case, by increasing the onsite energy difference – seeFig. 4(d) – the system moves towards a quantum para-magnetic state due to the increase in the transverse fieldvalue. In the weak-coupling regime, the stripe magnetictransition temperature is suppressed upon making thenesting conditions poorer. Moreover, as discussed in Ref.56, in the weak-coupling regime, strong deviations fromperfect nesting can change the magnetic ground statefrom the stripe phase to a charge-spin density-wave – i.e.a collinear double- Q state consisting of a linear combina-tion of the order parameters ∆ S and ∆ S . Therefore, themagnetic ground states in the weak- and strong-couplingregimes may be different.There is an important difference between the N´eel stateand the stripe state. The fourfold degeneracy of the lat-ter corresponds to two distinct Ising symmetries: onerelated to the polarization of the Ising-magnetic orderand one related to the tetragonal symmetry of the lat-tice – since there are two possible stripe directions thatlower the tetragonal symmetry to orthorhombic in dif-ferent ways. The latter is thus associated with nematicorder [45, 57], described in terms of the composite or-der parameter ∆ − ∆ [58]. Such a nematic phaseis called a vestigial order of the underlying stripe phase[59, 60]. In the weak-coupling regime, the quantum ne-matic phase transition is generally expected to be first-order and simultaneous to the stripe one [61]. However,in the strong-coupling regime of the model, both orderparameters can onset simultaneously at a single quantumcritical point, which likely belongs to the 3D XY univer-sality class. This result has important implications forthe possibility of nematic and magnetic quantum critical-ity being realized in iron-based superconductors [45, 62–64], whose band structure contains hole pockets and elec-tron pockets separated by the wave-vectors Q S n .The above examples demonstrate that the magneticground states at weak and at strong coupling can breakthe same symmetry, therefore allowing for a smooth con-nection at moderate coupling strengths. However, thisis by no means necessary. To illustrate this point, weconsider a simple example of identical tight-binding pa-rameters, µ c = µ d and t c = t d , as illustrated inFigs. 4(e) and (f) . In this case, the transverse field van-ishes, and the ground state in the strong-coupling regimeis either a N´eel state or a stripe state, depending on theratio t /t (see Table I). This is in stark contrast to whatone would expect from a weak-coupling approach, sincethis case features two identical Fermi surfaces, which arefar from satisfying any nesting condition. V. CONCLUSIONS
In this work, we generalized the strong-coupling ex-pansion of the single-band Hubbard model, as pioneered by Phil Anderson [18], to a two-band electronic modelwith dominant inter-band repulsion. While the for-mer maps onto the Heisenberg model, the latter mapsonto the transverse-field Ising model with extended ex-change interactions. In particular, the onsite energy dif-ference between the two electronic bands gives rise toa transverse field, while the Ising superexchange inter-actions arise from virtual hopping processes involvingnearest-neighbors and next-nearest-neighbors. Impor-tantly, by fixing the ratio between the inter-band re-pulsive interactions, this electronic model can be sim-ulated using sign-problem free QMC for arbitrary elec-tronic filling and hopping parameters. In contrast, theHubbard model, where intra-band repulsion dominates,is only sign-problem-free at half-filling and for bipartitekinetic Hamiltonians.We also showed how the rich ( h/J , J /J ) phase di-agram of the transverse-field J - J model can be probedby appropriately tuning the microscopic band parametersover a reasonable range of values. This opens an interest-ing avenue for future sign-problem-free QMC studies toexplore the impact of different types of strong-couplingground states on the emergence of superconductivity andother emergent phenomena at intermediate interactionstrengths. ACKNOWLEDGMENTS
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