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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
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XIAO, HÉBERT, BATROUNI, AND SCALETTAR PHYSICAL REVIEW B 99,
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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)
205145-3
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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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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,
205145-7
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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.
[1] E. Berger, P. Valášek, and W. von der Linden, Phys. Rev. B
52,4806 (1995).
[2] F. F. Assaad and T. C. Lang, Phys. Rev. B 76,
035116(2007).
[3] E. A. Nowadnick, S. Johnston, B. Moritz, R. T. Scalettar,
andT. P. Devereaux, Phys. Rev. Lett. 109, 246404 (2012).
[4] S. Johnston, E. A. Nowadnick, Y. F. Kung, B. Moritz, R.
T.Scalettar, and T. P. Devereaux, Phys. Rev. B 87, 235133
(2013).
[5] S. Yamazaki, S. Hoshino, and Y. Kuramoto, JPS Conf. Proc.
3,016021 (2014).
[6] A. Macridin, B. Moritz, M. Jarrell, and T. Maier, J.
Phys.:Condens. Matter 24, 475603 (2012).
[7] C. B. Mendl, E. A. Nowadnick, E. W. Huang, S. Johnston,
B.Moritz, and T. P. Devereaux, Phys. Rev. B 96, 205141 (2017).
[8] A. Ghosh, S. Kar, and S. Yarlagadda1a, Eur. Phys. J. B 91,
205(2018).
[9] S. Karakuzu, L. F. Tocchio, S. Sorella, and F. Becca, Phys.
Rev.B 96, 205145 (2017).
[10] S. Karakuzu, K. Seki, and S. Sorella, Phys. Rev. B
98,201108(R) (2018).
[11] J. E. Hirsch, R. L. Sugar, D. J. Scalapino, and R.
Blankenbecler,Phys. Rev. B 26, 5033 (1982).
[12] P. Sengupta, A. W. Sandvik, and D. K. Campbell, Phys. Rev.
B65, 155113 (2002).
[13] M. Nakamura, J. Phys. Soc. Jpn. 68, 3123 (1999).[14] M.
Nakamura, Phys. Rev. B 61, 16377 (2000).[15] A. W. Sandvik, L.
Balents, and D. K. Campbell, Phys. Rev. Lett.
92, 236401 (2004).[16] M. Tsuchiizu and A. Furusaki, Phys. Rev.
Lett. 88, 056402
(2002).[17] M. Tsuchiizu and A. Furusaki, Phys. Rev. B 69,
035103 (2004).[18] Y. Z. Zhang, Phys. Rev. Lett. 92, 246404
(2004).[19] K-M. Tam, S-W. Tsai, and D. K. Campbell, Phys. Rev.
Lett. 96,
036408 (2006).[20] A. J. Berlinsky, Rep. Prog. Phys. 42, 1243
(1979).[21] G. Grüner, Density Waves in Solids (Taylor &
Francis, London,
1994).[22] C. Bourbonnais and D. Jérome, in Physics of Organic
Super-
conductors and Conductors, edited by A. G. Lebed
(Springer,Berlin, 2008).
[23] R. T. Clay and R. P. Hardikar, Phys. Rev. Lett. 95,
096401(2005).
[24] R. P. Hardikar and R. T. Clay, Phys. Rev. B 75, 245103
(2007).[25] Ka-Ming Tam, S.-W. Tsai, D. K. Campbell, and A. H.
Castro
Neto, Phys. Rev. B 75, 161103(R) (2007).[26] J. Bonča and S. A.
Trugman, Phys. Rev. B 64, 094507 (2001).[27] K. M. Tam, S. W. Tsai,
and D. K. Campbell, Phys. Rev. B 89,
014513 (2014).
[28] N. C. Costa, T. Blommel, W.-T. Chiu, G. G. Batrouni, and R.
T.Scalettar, Phys. Rev. Lett. 120, 187003 (2018).
[29] E. H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445
(1968).[30] J. E. Hirsch and E. Fradkin, Phys. Rev. B 27, 4302
(1983).[31] L. G. Caron and C. Bourbonnais, Phys. Rev. B 29, 4230
(1984).[32] D. Schmeltzer, J. Phys. C 20, 3131 (1987).[33] C.
Bourbonnais and L. G. Caron, J. Phys. (France) 50, 2751
(1989).[34] C. Q. Wu, Q. F. Huang, and X. Sun, Phys. Rev. B 52,
R15683
(1995).[35] Y. Takada, J. Phys. Soc. Jpn. 65, 1544 (1996).[36]
T. Hotta and Y. Takada, Physica B 230-232, 1037 (1997).[37] E.
Jeckelmann, C. Zhang, and S. R. White, Phys. Rev. B 60,
7950 (1999).[38] Y. Takada and A. Chatterjee, Phys. Rev. B 67,
081102(R)
(2003).[39] J. Zhao and K. Ueda, J. Phys. Soc. Jpn. 79, 074602
(2010).[40] H. Fehske, A. P. Kampf, M. Sekania, and G. Wellein,
Eur. Phys.
J. B 31, 11 (2003).[41] H. Fehske, G. Wellein, G. Hager, A.
Weisse, and A. R. Bishop,
Phys. Rev. B 69, 165115 (2004).[42] H. Fehske, G. Hager, and E.
Jeckelmann, Europhys. Lett. 84,
57001 (2008).[43] M. Tezuka, R. Arita, and H. Aoki, Phys. Rev.
Lett. 95, 226401
(2005).[44] M. Tezuka, R. Arita, and H. Aoki, Phys. Rev. B 76,
155114
(2007).[45] S. Ejima and H. Fehske, J. Phys.: Conf. Ser. 200,
012031
(2010)[46] A. Chatterjee, Adv. Condens. Matter Phys. 2010,
350787
(2010).[47] M. Hohenadler, F. F. Assaad, and H. Fehske, Phys.
Rev. Lett.
109, 116407 (2012).[48] P. E. Spencer, J. H. Samson, P. E.
Kornilovitch, and A. S.
Alexandrov, Phys. Rev. B 71, 184310 (2005).[49] J. P. Hague and
P. E. Kornilovitch, Phys. Rev. B 80, 054301
(2009).[50] Y. Wang, K. Nakatsukasa, L. Rademaker, T. Berlijn,
and S.
Johnston, Supercond. Sci. Technol. 29, 054009 (2016).[51] A. G.
Swartz, H. Inoue, T. A. Merz, Y. Hikita, S. Raghu, T. P.
Devereaux, S. Johnston, and H. Y. Hwang, arXiv:1608.05621.[52]
F. Weber, S. Rosenkranz, J.-P. Castellan, R. Osborn, R. Hott,
R.
Heid, K.-P. Bohnen, T. Egami, A. H. Said, and D. Reznik,
Phys.Rev. Lett. 107, 107403 (2011).
[53] T. P. Devereaux, A. M. Shvaika, K. Wu, K. Wohlfeld, C.
J.Jia, Y. Wang, B. Moritz, L. Chaix, W.-S. Lee, Z.-X. Shen,G.
Ghiringhelli, and L. Braicovich, Phys. Rev. X 6, 041019(2016).
205145-8
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-
COMPETITION BETWEEN PHASE SEPARATION AND SPIN … PHYSICAL REVIEW
B 99, 205145 (2019)
[54] F. Hébert, B. Xiao, V. G. Rousseau, R. T. Scalettar, and G.
G.Batrouni, Phys. Rev. B 99, 075108 (2019).
[55] E. Y. Loh, J. E. Gubernatis, R. T. Scalettar, S. R. White,
D. J.Scalapino, and R. L. Sugar, Phys. Rev. B 41, 9301 (1990).
[56] M. Troyer and U.-J. Wiese, Phys. Rev. Lett. 94, 170201
(2005).
[57] G. G. Batrouni, R. T. Scalettar, and G. T. Zimanyi, Phys.
Rev.Lett. 65, 1765 (1990).
[58] N. V. Prokofév, B. V. Svistunov, and I. S. Tupitsyn, Phys.
Lett.A 238, 253 (1998).
[59] V. G. Rousseau, Phys. Rev. E 77, 056705 (2008).
205145-9
https://doi.org/10.1103/PhysRevB.99.075108https://doi.org/10.1103/PhysRevB.99.075108https://doi.org/10.1103/PhysRevB.99.075108https://doi.org/10.1103/PhysRevB.99.075108https://doi.org/10.1103/PhysRevB.41.9301https://doi.org/10.1103/PhysRevB.41.9301https://doi.org/10.1103/PhysRevB.41.9301https://doi.org/10.1103/PhysRevB.41.9301https://doi.org/10.1103/PhysRevLett.94.170201https://doi.org/10.1103/PhysRevLett.94.170201https://doi.org/10.1103/PhysRevLett.94.170201https://doi.org/10.1103/PhysRevLett.94.170201https://doi.org/10.1103/PhysRevLett.65.1765https://doi.org/10.1103/PhysRevLett.65.1765https://doi.org/10.1103/PhysRevLett.65.1765https://doi.org/10.1103/PhysRevLett.65.1765https://doi.org/10.1016/S0375-9601(97)00957-2https://doi.org/10.1016/S0375-9601(97)00957-2https://doi.org/10.1016/S0375-9601(97)00957-2https://doi.org/10.1016/S0375-9601(97)00957-2https://doi.org/10.1103/PhysRevE.77.056705https://doi.org/10.1103/PhysRevE.77.056705https://doi.org/10.1103/PhysRevE.77.056705https://doi.org/10.1103/PhysRevE.77.056705