PHYSICAL REVIEW B 99, 205145 (2019)

Competition between phase separation and spin density wave or charge density wave order:Role of long-range interactions

Bo Xiao,1 F. Hébert,2 G. Batrouni,2,3,4,5,6 and R. T. Scalettar11Department of Physics, University of California, Davis, California 95616, USA

2Université Côte d’Azur, Centre National de la Recherche Scientifique, INPHYNI, Nice, France3MajuLab, Centre National de la Recherche Scientifique–UCA-SU-NUS-NTU International Joint Research Unit, 117542 Singapore

4Centre for Quantum Technologies, National University of Singapore, 2 Science Drive 3, 117542 Singapore5Department of Physics, National University of Singapore, 2 Science Drive 3, 117542 Singapore

6Beijing Computational Science Research Center, Beijing 100193, China

(Received 30 March 2019; published 24 May 2019)

Recent studies of pairing and charge order in materials such as FeSe, SrTiO3, and 2H-NbSe2 have suggestedthat momentum dependence of the electron-phonon coupling plays an important role in their properties. Initialattempts to study Hamiltonians which either do not include or else truncate the range of Coulomb repulsion havenoted that the resulting spatial nonlocality of the electron-phonon interaction leads to a dominant tendency tophase separation. Here we present quantum Monte Carlo results for such models in which we incorporate bothon-site and intersite electron-electron interactions. We show that these can stabilize phases in which the densityis homogeneous and determine the associated phase boundaries. As a consequence, the physics of momentumdependent electron-phonon coupling can be determined outside of the trivial phase separated regime.

DOI: 10.1103/PhysRevB.99.205145

I. INTRODUCTION

The challenging nature of the computational solution ofthe quantum many-electron problem has led, to a quite con-siderable extent, to consideration of models which incor-porate only a single type of interaction. For example, theHubbard and periodic Anderson Hamiltonians focus on on-site electron-electron repulsion, the Kondo model focuseson an interaction with local spin degrees of freedom, whilethe Holstein and Su-Schrieffer-Heeger Hamiltonians considerexclusively electron-phonon interactions. This is, of course,an unfortunate situation, since the interplay between differentinteractions can lead to transitions between associated orderedphases which are of great interest. Even more importantly,this competition is present in most real materials. QuantumMonte Carlo (QMC) simulations of the Hubbard-HolsteinHamiltonian in two dimensions [1–9] have reinforced thechallenges of, as well as the interest in, including multipleinteractions. A very recent study offers great promise indeveloping a QMC approach in which electron-electron andelectron-phonon interactions can be folded together, at leastin the particular region of the phase diagram where chargeorder dominates [10].

One-dimensional systems have offered an exception to thisrule, largely because specialized algorithms exist in caseswhen the restricted geometry forbids the exchange of elec-trons [11]. Thus the extended Hubbard model (EHH), whichincludes electron-electron interactions in the form of both anon-site U and an intersite V , is now known to include notonly the expected charge density wave (CDW) correlations forU < 2V and spin density wave (SDW) correlations for U >2V but also a subtle bond ordered wave (BOW) phase ina narrow region about the line U = 2V [12–19]. These

simulations have been used to understand better the behav-ior of different materials, including organic superconductors[20], density wave materials [21], and undoped conductingpolymers [22]. Even with the applicability of these one-dimensional approaches, however, the physics remains subtle,and even controversial, owing mainly to energy gaps vanish-ing exponentially in the limit of weak coupling.

The goal of the present paper is an investigation of thephysics of a Hamiltonian which includes long-range electron-phonon coupling as well as electron-electron interactionsU and V . Motivation is given by several recent situationsin which momentum dependent electron-phonon coupling(which necessitates nonlocal coupling in real space) hasbeen suggested to be important, as we discuss below. Ourpaper builds on the considerable earlier literature of one-dimensional models with mixed interactions, which we re-view. As a first step, we focus here on half filling and theeffects of coexistence of long-range electron-phonon couplingand electron-electron interactions. However, our numericalmethod, introduced in the following section, can be appliedto this model at any fillings without a sign problem.

A further reason to focus on longer-range interactions is inthe search for one-dimensional models which exhibit super-conductivity as their dominant correlation. Despite an initialsuggestion to the contrary [23], it now seems most likely thatthe one-dimensional Hubbard-Holstein model with only on-site electron-phonon coupling does not have dominant pairingfluctuations [24,25]. Instead, an “extended Holstein” model,in which phonons residing on the midpoints between latticesites couple to the electronic charge both to the left and to theright, has been put forth as a “minimal model” in which theelectron-phonon can dominate over the more typical gappedCDW and SDW phases at half filling [26,27].

2469-9950/2019/99(20)/205145(9) 205145-1 ©2019 American Physical Society

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XIAO, HÉBERT, BATROUNI, AND SCALETTAR PHYSICAL REVIEW B 99, 205145 (2019)

We begin by writing down the full Hamiltonian, and thenconsider the various limiting cases before investigating thephysics when all terms are present. Our model is

Ĥ = K̂ + Ĥph + P̂u + P̂v + P̂ep,K̂ = −t

∑

l,σ

( ĉ†l,σ ĉl+1, σ + ĉ†l+1, σ ĉl,σ ),

Ĥph = 12

∑

l

(ω2X̂ 2l + P̂2l

),

P̂u = U∑

l

n̂l,↑ n̂l,↓,

P̂v = V∑

l

( n̂l,↑ + n̂l,↓)(n̂l+1, ↑ + n̂l+1, ↓),

P̂ep =∑

l,r

λ(r)X̂l (n̂l+r, ↑ + n̂l+r,↓). (1)

Here Ĥ is composed of a kinetic energy (K̂) describing thehopping of fermions along a one-dimensional chain of N sites;an on-site repulsion U between fermions of opposite spin σ(P̂u); an intersite repulsion V between fermions on adjacentsites (P̂v); and, finally, an electron-phonon coupling (P̂ep). Allenergies will be measured in units of t = 1. We choose theFröhlich form,

λ(r) = λ0 e−r/ξ

(1 + r2)3/2 , (2)

which modulates the long-range interaction between electronsand phonons with a screening length ξ (measured in unitsof the lattice constant a = 1). In the ξ → 0 limit, the localelectron-phonon coupling of the Holstein model is recovered.In the ξ → ∞ limit, the interaction is reduced to a latticeversion of the Fröhlich model. Throughout this paper we willconsider the half-filled situation where ρ = (N↑ + N↓)/N = 1with Nσ the numbers of electrons of spin σ . Although weare generalizing the range of the electron-phonon interactionfrom the ξ = 0 Holstein limit, we will continue to consideronly local phonon degrees of freedom, i.e., neglecting intersiteinteractions between the phonons which would give rise todispersive phonon modes [28].

The (“pure”) Hubbard Hamiltonian (HH) K̂ + P̂u is ex-actly soluble in one dimension via the Bethe ansatz [29]. Forall nonzero values of U , the ground state has spin order, withpower-law decaying spin-spin correlations. The Bethe ansatzsolution already interjects a cautionary note for numericalwork: The gap which immediately opens for U > 0 is expo-nentially small.

The physics of the ground state of the EHH K̂ + P̂u + P̂vhas already been noted above: CDW with true long-rangeorder (owing to the discrete symmetry of the order parameter)dominates for U < 2V , with weaker, power-law decayingSDW correlations at U > 2V when the symmetry is contin-uous. The BOW phase consists of a pattern in which thehopping ĉ†l,σ ĉl+1, σ + ĉ†l+1, σ ĉl,σ serves as a staggered orderparameter, alternating in amplitude for l odd and even. Asfor the CDW phase, the discrete nature of the breaking ofthe translational symmetry in the BOW gives true long-rangeorder [12].

The Holstein Hamiltonian K̂ + Ĥph + P̂ep neglects theelectron-electron interaction terms, and also sets ξ = 0 so thatonly the on-site piece of P̂ep survives. Similar to the HH, an(exponentially small) gap opens immediately for any λ0 > 0for small phonon frequencies. The existence of a gap for largeω is still controversial, despite an impressively large set ofnumerical and analytic investigations [30–39].

The Holstein extended Hubbard (HEH) Hamiltonian in-cludes all the terms K̂ + Ĥph + P̂u + P̂v + P̂ep, again settingξ = 0. The key added feature here is the possible existenceof a gapless metallic phase. This occurs when the effec-tive electron-electron attraction Ueff = −λ20/ω2 mediated byintegrating out the phonons balances the on-site repulsionU . Contradictory results concerning this gapless phase exist[23,24,35,38,40–46].

Finally, some work has been done on extensions of theHEH which allow for nonlocal electron-phonon coupling.As noted earlier, Bonča and Trugman [26] and later Tamet al. [27] considered a situation in which a phonon modeexists on the bond between two sites, coupling to the sumof the fermionic density on the two end points. This paperwas based on the observation that, for the commensuratedensities under consideration here, diagonal long-rangeorders, i.e., SDW and CDW, almost invariably dominate oversuperconductivity. The bond phonon mode will clearly favorthe formation of pairs on adjacent sites, opening the doorto the possibility of off-diagonal long-range order. Indeed,singlet superconductivity was shown to occur in the smallU,V portion of the phase diagram.

Motivated by the observation that the long-range electron-phonon interaction supports light polaron and bipolaronphysics observed in the cuprates, Hohenadler et al. [47] sim-ulated the fermion-boson model with the Fröhlich interaction.They found that the extended interactions suppress the Peierlsinstability and balance CDW and s-wave pairing. With similarmotivation, for fullerenes and manganites, Spencer et al. [48]studied the ground-state properties of the screened Fröhlichpolaron in weak-, strong-, and intermediate-coupling regions,as well as various screening lengths.

Following this review, we conclude this introduction withthe background and motivation for investigation of the fullHEH with the Fröhlich form of the electron-phonon coupling[Eq. (1)]. In the dilute limit, continuous-time quantum MonteCarlo (CTQMC) has been used to study the effect of varyingthe range ξ on polaron and bipolaron formation [49]. Theinterest in studying general momentum dependent couplingconstants is driven by a number of factors. First, a momentumdependent λ(q) is implicated in several experimental situa-tions of considerable current interest, including the origin ofthe tenfold increase in the superconducting Tc of FeSe mono-layers [50], the disparity in the values of the electron-phononcoupling in SrTiO3 inferred from tunneling below Tc andangle-resolved photoemission in the normal state [51], andthe “extended phonon collapse” in 2H-NbSe2 [52]. Second,there are qualitative issues to be addressed, e.g., how the rangeof the electron-phonon interaction ξ affects the competitionbetween metallic and Peierls/CDW phases at half filling. Hererecent CTQMC studies in one dimension have shown that asξ increases from zero the metallic phase is stabilized and, forsufficiently large λ, phase separation (PS) can also occur [47].

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Finally, it has been argued that recent improvements in theenergy resolution of resonant inelastic x-ray scattering haveopened the possibility of an experimental determination ofthe electron-phonon coupling across the full Brillouin zone[53]. This observation carries the exciting implication thatmaterials-specific forms for λ(q) can be incorporated intoQMC simulations of appropriate model Hamiltonians.

Unfortunately, one cannot immediately study the simplecase of a longer-range electron-phonon interaction in isola-tion. The reason is that, when a fermion distorts the lattice forξ finite, it does so at a collection of sites in its vicinity. Thisdistortion attracts additional fermions, further increasing thedistortion. The resulting cascade leads to a very high tendencyto phase separation. The Holstein Hamiltonian (ξ = 0) neatlyevades this collapse since the Pauli principle caps the fermioncount on a displaced site. Likewise, the Bonča and Trugmanmodel [26] of bond phonons restricts the electron-phononcoupling to a pair of sites. The solution to this dilemma inthe general case is clear—the inclusion of nonzero electron-electron repulsion. A recent study which incorporates on-siteU but retains V = 0 [54] revealed continued phase separationover much of the phase diagram, with SDW occurring onlyif ξ and λ0 were rather small. We show here that V cansignificantly stabilize the CDW and SDW phases againstcollapse of the density.

II. COMPUTATIONAL METHODOLOGY

Our technical approach is a straightforward generalizationof the world-line quantum Monte Carlo (WLQMC) methodfor lattice fermions of Hirsch et al. [11]. In this method, a pathintegral is written for the partition function by discretizing theinverse temperature β into intervals of length �τ = β/L:

Z = Tr[e−βĤ]= Tr[e−�τĤ1 e−�τĤ2 · · · e−�τĤ1 e−�τĤ2 ]. (3)

On the odd imaginary time intervals, half of the Hamiltonianof Eq. (1) acts,

Ĥ1 = K̂1 + 12

(Ĥph + P̂u + P̂v + P̂ep),

K̂1 = −t∑

l∈odd, σ(ĉ†l,σ ĉl+1, σ + ĉ†l+1, σ ĉl,σ ), (4)

while on the even intervals the other half acts:

Ĥ2 = K̂2 + 12

(Ĥph + P̂u + P̂v + P̂ep),

K̂2 = −t∑

l∈even, σ(ĉ†l,σ ĉl+1, σ + ĉ†l+1, σ ĉl,σ ). (5)

The division of Ĥ into two pieces necessitates the presence of2L time slices.

Complete sets of occupation number and phonon coordi-nate states are inserted between each incremental time evolu-tion operator. The matrix elements are then evaluated, replac-ing all operators by space and imaginary time dependent cnumbers corresponding to the eigenvalues of the intermediatestates. The fermion occupation number states and phononcoordinate states are sampled stochastically by introducing

local changes and accepting/rejecting according to the ratioof matrix elements.

The utility of the “checkerboard decomposition” [11] isthat on each imaginary time slice the matrix element of theincremental time evolution operator factorizes into a productof independent two site problems. The matrix elements arethus simple to evaluate. The Monte Carlo moves are chosen topreserve local conservation laws of the particle count on eachtwo site plaquette.

The strengths of the WLQMC approach include a linearscaling in spatial system size N and imaginary time β (incontrast to the N3 scaling of auxiliary field QMC). This isa consequence of the locality of the values of the matrixelements. More significantly, there is no fermion sign problem[55,56] at any filling as long as the hopping occurs exclusivelybetween sites which are adjacent in the list of occupationnumbers labeling the state.

In this paper we will be interested in the ground-state prop-erties of Eq. (1), which we access by choosing β sufficientlylarge. Our typical choice is β ∼ N . We have checked that wehave reached the low T limit, with the caveat noted earlierthat the exponential scaling of the gap precludes this at weakcoupling. In what follows, we note when that concern affectsour conclusions significantly.

The realization Eqs. (3)–(5) of WLQMC works in thecanonical ensemble, although it is straightforward to connectto grand-canonical ensemble results by doing simulations onadjacent particle number sectors and extracting the chemicalpotential from a finite difference of ground-state energies ofsystems of different particle number [57].

There are some weaknesses to the method. Most notably,it is challenging to evaluate expectation values of operatorswhich “break” the world lines. Thus access to single- and two-particle Green’s functions is restricted, unlike other methodslike the “worm” [58] and stochastic Green’s-function (SGF)[59] approaches. In addition, WLQMC often suffers fromlong autocorrelation times which are associated with the localnature of the moves, and the tendency for world lines tohave significantly extended spatial patterns. Both of these areaddressed in the worm and SGF methods, at, however, thecost of somewhat greater algorithmic complexity. Here, ind = 1 due to the speed of the approach, one can tolerate longautocorrelations simply by doing many updates.

In order to determine the phase diagram of Eq. (1),we monitor first a set of local observables—the fermionkinetic energy, phonon kinetic energy, and fermion doubleoccupancy:

Kel = −t 〈(ĉ†l,σ ĉl+1, σ + ĉ†l+1, σ ĉl,σ )〉,Kph = 12

〈P̂2l

〉, D = 〈n̂l,↑ n̂l,↓〉. (6)

In the absence of a broken CDW or BOW symmetry, these areindependent of lattice site l since H is translation invariant.Second, we evaluate the CDW and SDW structure factors,

SCDW = 1N

∑

l, j

〈(n̂l,↑ + n̂l,↓)(n̂ j,↑ + n̂ j,↓)〉(−1) j+l ,

SSDW = 1N

∑

l, j

〈(n̂l,↑ − n̂l,↓)(n̂ j,↑ − n̂ j,↓)〉(−1) j+l , (7)

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XIAO, HÉBERT, BATROUNI, AND SCALETTAR PHYSICAL REVIEW B 99, 205145 (2019)

to get insight into long-range order. These structure factorsare independent of N in a disordered phase, but grow withN in an ordered or quasiordered phase, as the associatedreal-space correlation functions remain nonzero at largeseparations | j − l|. Although the quantities of Eq. (6) arelocal and hence in principle not appropriate to determiningthe appearance of nonanalyticities associated with transitions,we shall see that they nevertheless show sharp signatures atthe phase boundaries.

We also investigate the nonlocal observables including thecharge and spin susceptibilities which involve an additionalintegration over imaginary time:

χcharge = �τN

∑

τ,l, j

〈 n̂l (τ ) n̂l (0)〉(−1) j+l ,

n̂l (τ ) = ( n̂l,↑ + n̂l,↓ )(τ )= eτĤ ( n̂l,↑ + n̂l,↓ )(0) e−τĤ,

χspin = �τN

∑

τ,l, j

〈 m̂l (τ ) m̂l (0)〉(−1) j+l ,

m̂l (τ ) = ( n̂l,↑ − n̂l,↓ )(τ )= eτĤ ( n̂l,↑ − n̂l,↓ )(0) e−τĤ. (8)

The susceptibility provides a clearer signal of the SDW tran-sition, which is more subtle than the CDW transition owing tothe continuous nature of the symmetry involved.

III. PHASE DIAGRAMS AT FIXED V

In this section we present the phase diagrams in the U -ξplane at three fixed values of intersite electron-electron repul-sion V , focusing on the stabilization of the ordered phases byV against phase separation. We begin by showing, in Fig. 1(a),the phase diagram for vanishing intersite repulsion V = 0 andwith ω = 0.5, λ0 = 1.0. CDW correlations, driven solely bythe electron-phonon interaction (since V = 0), dominate atsmall U and ξ , but are replaced by SDW as U increases, orby PS as ξ increases. Figure 1(a) emphasizes the fragility ofCDW order to PS: An electron-phonon interaction range asshort as a few tenths of a lattice spacing is sufficient to drivethe system into a regime where all electrons clump together.Although the errors bars make it somewhat uncertain, thedata in the range 0.2 � ξ � 0.35 indicate the possibility ofpenetration of a thin PS wedge at the SDW-CDW boundary.

Figure 1(b) provides direct visualization of total electrondensity on each lattice site. The alternating pattern in theCDW phase region reflects true long-range charge order inthe ground state. Quantum fluctuations due to the hopping treduce the magnitudes of the largest and the smallest densitiesfrom perfect double occupation (ni = 2) and empty (ni = 0).For fixed screening length ξ = 0.2, Fig. 1(b) shows the CDWphase is most stable at small U , with the charge oscillationsdecreasing in size as U grows, ultimately vanishing at U = 7in the SDW. The figure also displays a subtle difference inthe double occupancy D between the CDW and PS regions.Both have half their sites nearly empty and half nearly doublyoccupied, but in the CDW case the ni values are fartherfrom their extreme limits ni = 0, 2. The reason is that the

FIG. 1. (a) Phase diagram of the HEH in the U -ξ plane at fixedV = 0 and ω = 0.5, λ0 = 1.0. The lattice size is N = 32 and inversetemperature β = 26. The procedure for determining the positionsof the phase boundaries is described in subsequent figures anddiscussion, and in the right-hand panel. (b) Snapshots of electrondensity (ni = ni,↑ + ni,↓) profiles. For ξ = 0.5,U = 3, the particlesare clumped in one region of the lattice—the system exhibits PS. Forξ = 0.2,U = 7, the total density is uniform, as expected for an SDWwhere spin density oscillates, but charge density is constant. Thethree data sets at ξ = 0.2 with U = 1, 2, 3 are in the CDW phase.The order parameter (size of charge oscillation) decreases as U growsand the phase boundary to the SDW is approached.

alternation of empty and doubly occupied sites in the CDWallows for a much greater degree of quantum fluctuations dueto the hopping t than can occur in the PS state where the Pauliprinciple blocks fermion mobility.

As noted in the introduction, mitigating the strong ten-dency to PS at V = 0 is what motivates our paper here. Theresults of Fig. 1 are consistent with those obtained at V = 0 inRef. [54], which uses the alternate stochastic Green’s-function[59] method.

We now turn to nonzero V . Our discussion will detailhow Fig. 1, and subsequent phase diagrams, are obtainedthrough the analysis of the evolution of the observables ofEqs. (6)–(8).

Figure 2(a) shows the kinetic energy of the electrons, Kel,for a range of values of ξ as a function of the on-site Ufor fixed V = 0.5, ω = 0.5, and λ0 = 1.0. For ξ � 0.8, Kelis small until U exceeds a critical value. This change isassociated with a transition from a small U PS state, wherethe clumped electrons are unable to move (except at theboundary of the occupied region) due to Pauli blocking, to alarge U SDW state where alternating up and down occupationallows considerable intersite hopping and hence large (inmagnitude) Kel [see Fig. 3(a)]. At strong U we expect thehopping to go as t2/U (second-order perturbation induced byvirtual hopping), and indeed this falloff fits the large U datareasonably well. The value of the on-site repulsion U whichis needed to eliminate PS is reduced as ξ is lowered, as isexpected since the tendency for particles to clump is reducedas their interaction range is shortened.

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-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0=0.05=0.2=0.5=0.7=0.8=1.0

0

0.1

0.2

0.3

0.4

0.5

=0.05=0.2=0.5=0.7=0.8=1.0

0 1 2 3 4 5 6 7U

0

200

400

600

800

char

ge(

) =0.05=0.2=0.5=0.7=0.8=1.0

0 1 2 3 4 5 6 7U

0

20

40

60

80

100

spin

()

=0.05=0.2=0.5=0.7=0.8=1.0

(a) (b)

(c) (d)

FIG. 2. (a) Electron kinetic energy as a function of on-site re-pulsion U for fixed intersite repulsion, V = 0.5, and different valuesof electron-phonon interaction screening length ξ . (b) Double occu-pancy as a function of U . A sharp drop in D and a sharp increase inthe magnitude of Kel indicate the entry into the SDW phase from thelarge ξ PS region as U increases. Structure at smaller ξ is associatedwith the CDW-SDW boundary [see text and Fig. 3(a)]. (c) Chargesusceptibility as a function of U . χcharge is big in the weak U CDWphase, and drops at the SDW boundary. (d) The spin susceptibilityshows the opposite trend: χspin is small at weak U , but increasesas the SDW phase is entered at large U . The lattice size is N = 32and inverse temperature β = 26. The phonon frequency is fixed atω = 0.5 and electron-phonon coupling strength λ = 1.0.

In the small ξ (Holstein) limit PS does not occur. Never-theless Kel and D show appropriate signals of the CDW-SDWtransition: Kel is largest in magnitude at the boundary wherethe two insulators most closely compete, and D becomessmall as the SDW is entered. The more rounded feature of Kelat intermediate ξ is one indication of the possible intrusion ofPS between CDW and SDW [again, see Fig. 3(a)].

The double occupancy D is given in Fig. 2(b). To interpretit, we first note that D is not a good discriminator betweenPS and CDW, because, in both, one has (subject to quantumfluctuations) roughly half the sites doubly occupied and halfempty, and hence D ∼ 0.5. In the SDW phase, on the otherhand, D ∼ 0 since U precludes double occupancy. Thus themost evident feature of Fig. 2(b) is the sharp drop in D as Uincreases, which occurs for all ξ . These occur at values consis-tent with the increase in the magnitude of Kel of Fig. 2(a). Dis not strictly zero in the SDW phase because of the quantum(charge) fluctuations induced by t . These gradually go downat U increases.

Despite the fact that D ∼ 0.5 in both the CDW and PSstates, Fig. 2(b) still shows a subtle signature of the CDW-PStransition with increasing ξ at fixed U : D initially falls asξ increases from ξ = 0.2 to 0.7, but then rises again fromξ = 0.7 to 1.0. The somewhat smaller values of D in the CDWresult from fluctuations which are more likely when doublyoccupied sites are surrounded by empty sites than in the PSstate where they are adjacent to each other. This signal of theCDW-PS boundary of Fig. 3(a), although weak, is quite clear,and lines up well with the small U features in Kel in Fig. 2(a).

FIG. 3. (a) Phase diagram of the HEH model in the U -ξ planefor V = 0.5. Compared to Fig. 1(a) the CDW-PS line is pushedout from ξ ≈ 0.38 at V = 0 to ξ ≈ 0.8 here. In the Holstein limit,CDW correlations are stabilized by an amount 2V against SDW.The uncertainty in the precise nature of the phase where the CDW,SDW, and PS regions meet is indicated by the shaded region. Seetext. (b) Finite-size effect on the normalized charge structure factorScharge(π )/N at ξ = 0.1. The magnitude of Scharge(π ) is proportionalto the size of lattice, indicating the existence of true long-rangeorder in the CDW phase. Therefore, the normalized charge structurefactor Scharge(π )/N is N independent in the CDW phase. Scharge(π )/Nchanges abruptly at the same U/t value for all N . (c) Electron kineticenergy Kel (per site) for different lattice sizes at ξ = 0.1. The positionof the cusp is N independent.

The spin susceptibility in Fig. 2(d) reinforces the infer-ences made from the local observables and confirms the largeU phase is SDW. For each value of ξ , a sharp increase inχspin occurs as U increases. The critical value is not verysensitive to ξ , changing from Uc ∼ 4.8 at ξ ∼ 0.2 to Uc ∼ 4at ξ ∼ 0.8. This is reflected in the nearly horizontal characterof the phase boundary of Fig. 3(a). In the complete absenceof any thermal or quantum fluctuations we have χspin = Nβfrom Eq. (8). The values of Fig. 2(d) are an order of magni-tude smaller: Quantum fluctuation reduces correlations signif-icantly, especially in one dimension and for the case of a con-tinuous order parameter which has power-law correlations atT = 0.

Finally, the charge susceptibility is shown in Fig. 2(c). Forξ � 0.7, χcharge is large at small U , indicating the presence ofCDW order until PS and then SDW occur as U increases. Forlarger ξ there is no small U CDW phase. These observationsare consistent with the measurement of the local observablesand spin susceptibility of Figs. 2(a), 2(b), and 2(d).

The full phase diagram at V = 0.5 is given in Fig. 3(a). ThePS regime has been pushed out to ξ ∼ 0.8, in contrast to ξ ∼0.38 in Fig. 1(a). The SDW phase boundary is remarkablyflat: Uc ∼ 4–5 regardless of the nature of the phase (CDW orPS) beneath it. For small ξ the CDW region is also increasedin size vertically upward in going from V = 0 to 0.5. Thisupward shift has been discussed by Hirsch and Fradkin [30]in the Holstein limit of local electron-phonon interaction:Nonzero V combines with the tendency to form CDW order

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XIAO, HÉBERT, BATROUNI, AND SCALETTAR PHYSICAL REVIEW B 99, 205145 (2019)

FIG. 4. Phase diagram of the HEH model in the U -ξ plane forV = 1.0 (solid lines). The dashed and dotted lines indicate the phaseboundary (PB) of the CDW region V = 0 and 0.5, with the SDWboundaries of Figs. 1(a) and 3(a) suppressed for clarity. As theintersite V increases, the charge density wave is stabilized to longer-range electron-phonon interaction range ξ . At ξ = 0 the CDW-SDWphase boundary moves upward approximately by 2V [30]. As withthe preceding figure, the uncertainty in the precise nature of thephase where the CDW, SDW, and PS regions meet is indicated bythe shaded region.

driven by the electron-phonon interaction, producing a 2Vchange in the position of the CDW-SDW boundary. Thisestimate is in good quantitative agreement with what we findin comparing Figs. 1(a) and 3(a). In Figs. 3(b) and 3(c), weshow the finite-size effects on the normalized charge structurefactor Scharge(π )/N and Kel at ξ = 0.1. We have verified thatthe finite-size effects for this parameter set are typical ofthose throughout phase space, allowing us to locate the phaseboundaries accurately.

Just as the precise nature of the boundary between CDWand SDW in the extended Hubbard model was challenging touncover, with the original picture of a direct transition (with achange from first to second order at the tricritical point) beingreplaced by an intervening BOW [12], in our studies the natureof a narrow region where the CDW-SDW boundary meets PSis ambiguous. The technical issue is the difficulty in locatingthe precise positions of the CDW and SDW transitions giventhe rounding effects of finite-size lattices. We have indicatedthis uncertainty, which is greater as we turn on V , by a shadedregion in Fig. 3(a).

We will not show the detailed evolution of observables forlarger V = 1, since their basic structure is the same as for V =0.5, just discussed. Figure 4 is the resulting phase diagram. PShas been pushed to the region ξ � 1.2. For smaller ξ , whilePS is absent, the increasing range of the e-p coupling tiltsthe CDW-SDW in favor of spin order. In the Holstein limit,ξ = 0, the electron-phonon and nearest-neighbor electron-electron interactions work in concert to promote charge order.It is clear that as ξ increases the electron-phonon interactionno longer favors double occupation on the same site over

0

0.5

1

1.5

V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20

1

2

3

3.5

V

Spin Density Wave (SDW)

Charge Density Wave (CDW)

Phase Separation (PS)

Charge Density Wave (CDW)

(b)

(a)

FIG. 5. (a) Phase diagram of the HEH model in the V -ξ planefor U = 3.0. The on-site repulsion is too weak to produce SDW.(b) Phase diagram of the HEH model in the V -ξ plane for U = 8.0.Here the on-site repulsion results in the robust SDW phase and thecombination of U and V eliminates all phase separation.

occupation of adjacent sites, leading to the downward slopeof the CDW boundary.

One might expect that as ξ → ∞ the electron-phononinteraction becomes independent of the fermion positions,and hence that the CDW-SDW boundary should approachthe usual U = 2V position. The CDW boundary is indeeddecreasing as ξ grows, but for V = 1 phase separation stilloccurs before the V = 2U values. For large ξ , the fact that theboundary of phase transition from CDW or PS state to SDWstate decreases compared to the Holstein limit suggests thatthe effective attraction, Ueff, is smaller in the Fröhlich modelthan in the Holstein model.

IV. PHASE DIAGRAMS AT FIXED U

We now show a set of complementary phase diagrams inthe V -ξ plane at fixed U . Since their derivation lies in the samedetailed analysis of χcharge, χspin, as well as Kel and D, as thatof the preceding section, we focus mostly on the final phasediagrams.

We consider first small (U = 3) and large (U = 8) on-siterepulsion, which have the phase diagrams shown in Figs. 5(a)and 5(b). For U = 3 the on-site repulsion is small enough thatno SDW region appears. The intersite V and electron-phononinteraction range compete to give either CDW or PS. For U =8, PS is replaced by SDW. We can estimate the transition pointin the Holstein limit: V = 12 (U − Ueff ) = 12 (U − λ2/ω2) =2.0, which agrees well with the U = 8 phase diagram at ξ =0. Despite its simplicity, Fig. 5(b) carries a central message ofthis paper: Reasonable choices of V and U suppress the PS,which was noted in the first attempts to simulate momentumdependent electron-phonon couplings.

The behavior at U = 4 is considerably more complexand hence worth discussing in more detail. Figure 6(a) in-dicates a small electron kinetic energy until V exceeds athreshold value. Careful examination of the line shapes for

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COMPETITION BETWEEN PHASE SEPARATION AND SPIN … PHYSICAL REVIEW B 99, 205145 (2019)

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

=0.1=0.2=0.3=0.5=0.7=0.9=1.0=1.1

0.1

0.2

0.3

0.4

0.5

=0.1=0.2=0.3=0.5=0.7=0.9=1.0=1.1

0 1 2 3V

0

200

400

600

800

char

ge(

)

=0.1=0.2=0.3=0.5=0.7=0.9=1.0=1.1

0 1 2 3V

0

5

10

15

20

25

30

spin

()

=0.1=0.2=0.3=0.5=0.7=0.9=1.0=1.1

(a) (b)

)d()c(

FIG. 6. For the HEH model at U = 4 and several values of ξwe show as functions of V the (a) electron kinetic energy, (b) thedouble occupancy, (c) the charge susceptibility, (d) the spin suscepti-bility. The number of sites N = 32 and inverse temperature β = 26.Phonon frequency ω = 0.5 and electron-phonon coupling strengthλ0 = 1.0.

0.8 � ξ � 1.1 reveals that there are two regimes of largeKel. As V is increased initially, Kel increases abruptly inmagnitude, and then continues a much slower increase. Ata second critical V a kink appears, there is a change in thesign of the slope, and the magnitude of Kel begins to fall.In combination with the behavior of other observables (seebelow), we interpret this to indicate a PS to SDW to CDWevolution with increasing V for these values of ξ . At larger ξthere is an abrupt jump in Kel, but then immediately a decreasein magnitude from the newly large values. For ξ � 0.7, theabsence of small Kel indicates that there is no phase separationregion for these values of ξ . These correspond to single PS toCDW and SDW to CDW transitions, respectively.

The double occupancy curves [Fig. 6(b)] look rather sim-ilar to the electron kinetic energy. Here, as discussed earlier,low values of D are associated with SDW, while large valuescan be either PS or CDW. The existence of two separatetransitions, quite evident in Kel, is less clear in D. However,close inspection of the ξ = 0.9, 1.0, and 1.1 curves shows thatD increases gradually with V after entering the SDW, but thenabruptly changes slope for yet larger V . This is consistent withchanges from PS to SDW to CDW for these ξ .

Figures 6(c) and 6(d) give the evolution of the charge andspin susceptibilities, respectively. Consistent with the datafrom the local observables, χspin is big only in an intermediaterange of V for ξ � 0.7, below which lies PS and above whichlies the CDW. For smaller ξ , the SDW phase extends allthe way to V = 1.0. For all ξ , χcharge grows large above acritical V . The shapes or the curves are not markedly differentfor ξ � 0.7 where the CDW emerges from the SDW, or forξ � 0.7 where it emerges from a PS region.

Putting these plots together, we infer the U = 4 phasediagram of Fig. 7, which has CDW physics dominant atlarge V , but two possibilities, SDW and PS, for the natureof the ground state at small V . As with the constant V = 0.5phase diagram of Fig. 3(a) one thus encounters all three

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

V

FIG. 7. Phase diagram of the HEH model in the V -ξ plane forU = 4.0 inferred from Fig. 6. The on-site repulsion is now bigenough to give SDW sandwiched in between PS at large ξ and CDWat large V .

ground states as one tunes the value of the electron-electroninteractions.

V. CONCLUSIONS

Recent attempts to include momentum dependent (finiterange) electron-phonon coupling without electron-electroninteractions have concluded that the tendency to phase sep-aration dominates the physics. This is mitigated to someextent [54] by an on-site repulsion U , nevertheless PS isalready dominant beyond relatively small ξ � 0.5. This putssignificant restrictions on the nature of λ(q) which wouldexhibit spatially homogeneous densities—most of the weightwould have to be at momenta q > 2π/ξ , i.e., well outside thefirst Brillouin zone.

Here we have introduced a nearest-neighbor Coulombinteraction V to the extended Holstein-Hubbard Hamiltonianand obtained the resulting phase diagrams at half filling.We have shown even relatively small values of V ∼ λ0 canstabilize SDW and CDW phases and eliminate (long period)density inhomogeneity. Figure 5(b) is a key result: PS istotally suppressed and the effects of ξ can be discerned in acontext of global thermodynamic stability.

Important questions remain open. First, we have notattempted to explore possible metallic phases. These areknown to be extremely challenging to address with QMCeven in much more simple models, since the gaps can van-ish exponentially at weak coupling. Second, we have notdoped the system. This paper has exclusively focused onhalf filling. CDW and SDW phases tend to be optimized atcommensurate filling, but in d = 1 the continued presenceof Fermi-surface nesting can give rise to order at k < π .We generally expect superconductivity to be aided by somedegree of doping; it is already known to exist in modelswhere bond phonons couple to densities on both neighboringsites [26,27]. Much of the existing experimental data on thematerials which motivate the study of the model, includingsome organic superconductors and density wave systems,

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XIAO, HÉBERT, BATROUNI, AND SCALETTAR PHYSICAL REVIEW B 99, 205145 (2019)

are away from half filling. Because the sign problem doesnot occur at any filling for our model in one dimension,it will be interesting to apply our method to study dopedsystems. As discussed here at half filling, a longer-rangeelectron-phonon interaction will play an important role in thecompetition between phase separation and superconductivity,and we expect this to be the case in the doped system aswell.

ACKNOWLEDGMENTS

B.X. and R.T.S. were supported by Department of EnergyGrant No. DE-SC0014671. F.H. and G.G.B. were supportedby the French government, through the UCAJEDI Invest-ments in the Future project managed by the National ResearchAgency Project No. ANR-15-IDEX-01, and by Beijing Com-putational Science Research Center.

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