Multi-channel uctuating eld approach to competing instabilities in interacting electronic systems E. Linn er1A. I. Lichtenstein2 3 4S. Biermann1 5 6 7and E. A. Stepanov1

2025-04-24 0 0 500.36KB 11 页 10玖币

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Multi-channel ﬂuctuating ﬁeld approach to competing instabilities

in interacting electronic systems

E. Linn´er,1A. I. Lichtenstein,2, 3, 4 S. Biermann,1, 5, 6, 7 and E. A. Stepanov1

1CPHT, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, F-91128 Palaiseau, France

2I. Institute of Theoretical Physics, University of Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany

3European X-Ray Free-Electron Laser Facility, Holzkoppel 4, 22869 Schenefeld, Germany

4The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany

5Coll`ege de France, 11 place Marcelin Berthelot, 75005 Paris, France

6Department of Physics, Division of Mathematical Physics,

Lund University, Professorsgatan 1, 22363 Lund, Sweden

7European Theoretical Spectroscopy Facility, 91128 Palaiseau, France

Systems with strong electronic Coulomb correlations often display rich phase diagrams exhibit-

ing diﬀerent ordered phases involving spin, charge, or orbital degrees of freedom. The theoretical

description of the interplay of the corresponding collective ﬂuctuations giving rise to this phe-

nomenology remains however a tremendous challenge. Here, we introduce a multi-channel extension

of the recently developed ﬂuctuating ﬁeld approach to competing collective ﬂuctuations in correlated

electron systems. The method is based on a variational optimization of a trial action that explicitly

contains the order parameters of the leading ﬂuctuation channels. It gives direct access to the free

energy of the system, facilitating the distinction between stable and meta-stable phases of the sys-

tem. We apply our approach to the extended Hubbard model in the weak to intermediate coupling

regime where we ﬁnd it to capture the interplay of competing charge density wave and antiferromag-

netic ﬂuctuations with qualitative agreement with more computationally expensive methods. The

multi-channel ﬂuctuation ﬁeld approach thus oﬀers a promising new route for a numerically cheap

treatment of the interplay between collective ﬂuctuations in large systems.

I. INTRODUCTION

A hallmark of materials with strong electronic

Coulomb correlations are their typically extremely rich

phase diagrams, exhibiting various kinds of ordering phe-

nomena. These result from competing instabilities in-

volving e.g. charge, spin, orbital or pairing ﬂuctuations.

The theoretical description of these collective phenom-

ena remains a challenging issue of computational com-

plexity [1,2] as well as conceptual diﬃculty, e.g. the

explicit breaking of symmetries [3,4]. In this sense, the

interplay of competing electronic ﬂuctuations constitutes

a roadblock to the understanding of the complex phase

diagrams of a wide range of material systems. Construct-

ing simpliﬁed methods to study interplaying collective

ﬂuctuations is thus of crucial importance.

The extended Hubbard model [5–8] provides a suitable

framework for investigating the interplay between collec-

tive electronic ﬂuctuations. The physics of this model

is determined by the competition between the local U

and the non-local VCoulomb interactions. A repulsive

Ustabilizes collective spin ﬂuctuations [9], which may

compete with charge ﬂuctuations driven by a strong re-

pulsive V[10,11]. The earliest considerations of the ex-

tended Hubbard model were already implicit in the initial

work of J. Hubbard in 1963 [5]. However, the ﬁrst stud-

ies of the model occurred in the 1970’s, with studies of

the strong [10,12] and weak coupling limits of the half-

ﬁlled one-dimensional (1D) chain [13,14]. Together with

an access to the intermediate coupling regime by early

numerical exact diagonalization (ED) and lattice Monte

Carlo calculations [15,16], the phase diagram of the 1D

extended Hubbard model was predicted to be composed

of regions of strong charge density wave (CDW) and an-

tiferromagnetic (AFM) ﬂuctuations, with a CDW-AFM

transition occurring in the vicinity of U= 2V. The tran-

sition was later discovered to be modiﬁed in the weak cou-

pling limit by an intermediate bond-order wave (BOW)

state [17,18].

Extensive studies have been conducted on the extended

Hubbard model for elucidating the interplay between

collective charge and spin ﬂuctuations [12,14–16,19–

28]. Considerable insight has been acquired for the

extended Hubbard model on a two-dimensional square

lattice at half-ﬁlling with nearest-neighbour interaction

V[20–26,29–37], which we study in the current work. It

has been found that this model displays a phase diagram

similar to the one-dimensional counterpart, besides the

apparent lack of an intermediate BOW phase. In partic-

ular, the system reveals a checker-board CDW pattern

which interplays with strong AFM ﬂuctuations in the

vicinity of a CDW-AFM transition line U= 4V[20]. In

a recent work [25] based on the dynamical cluster approx-

imation (DCA) [38–40], the competition near the transi-

tion line has been shown to induce a coexistence region

of charge- and spin-ordered states.

By the Mermin-Wagner theorem [41–43], magnetic or-

dering at ﬁnite temperatures is excluded in a broad class

of one- and two-dimensional systems, including the ex-

tended Hubbard model, due to the continuous nature of

the underlying symmetry. Thus, the regime of strong col-

lective AFM ﬂuctuations is strictly speaking not a phase.

However, in our current work the “AFM phase” will refer

to a slightly broader deﬁnition of short-range AFM or-

arXiv:2210.05540v1 [cond-mat.str-el] 11 Oct 2022

dering, which transforms to a true phase for a quasi-two-

dimensional system. In contrast, the discrete symmetry

of the CDW allows for a true phase transition. In ad-

dition, technically speaking, in the present work, we are

performing calculations for ﬁnite systems, where long-

range ﬂuctuations are eventually cut oﬀ, so neither the

AFM or CDW state are strictly speaking phases. Nev-

ertheless, in the following, we will refer to both states

as phases, since we are interested in the interplay of the

competing ﬂuctuations corresponding to these orderings.

Our conclusions should thus be understood as applying

either to ﬁnite systems replacing the notion of phase by

”state dominated by the respective ﬂuctuations” or to

a quasi-two-dimensional system in the thermodynamic

limit.

Limitations in the treatment of competing collective

ﬂuctuations arise in the currently available approaches

employed for studying quantum lattice systems. Nu-

merically exact methods, such as exact diagonalization

(ED) [1] and lattice Monte Carlo [2] have studied the

interplay between Uand V[15,16,20,21] but are re-

stricted to small system sizes and thus cannot address

long-range collective ﬂuctuations. The same problem is

also inherent in cluster extensions of the dynamical mean-

ﬁeld theory (DMFT) [44–49], such as, e.g., DCA [38–

40]. Diagrammatic methods based on the parquet ap-

proximation [50–55] allow one to account for the inter-

play between charge and spin ﬂuctuations [26] originat-

ing from the two-particle vertex functions in an unbi-

ased and powerful fashion. These vertices are incorpo-

rated with full momentum- and frequency-dependence,

and the approach is thus computationally very expen-

sive, which severally limits its applicability. Advanced

diagrammatic extensions of DMFT [56] are able to de-

scribe long-range ﬂuctuations simultaneously in diﬀerent

instability channels. In the presence of the non-local

interaction Vthis can be done within the dual boson

theory [34,36,57–59], the dynamical vertex approxi-

mation (DΓA) [60,61], the triply irreducible local ex-

pansion (TRILEX) method [62], or the dual TRILEX

(D-TRILEX) approach [27,28,63]. However, these ﬂuc-

tuations are usually treated in a ladder-like approxi-

mation, where diﬀerent instability channels aﬀect each

other only indirectly via self-consistent renormalization

of single- and two-particle quantities.

Current approaches to quantum lattice systems that

are able to capture competing collective ﬂuctuations are

too complicated for broad usage. In this work, we de-

velop a multi-channel generalisation of the ﬂuctuating

ﬁeld (FF) approach that allows us to incorporate multiple

collective ﬂuctuation channels and their interplay in a nu-

merically cheap way without explicitly breaking the sym-

metry of the model. The FF method was originally intro-

duced for the study of spin ﬂuctuations in the classical

Ising plaquettes [64] and was further developed for single-

and multi-mode treatment of collective spin ﬂuctuations

in the Hubbard model [65–67]. We employ the pro-

posed multi-channel ﬂuctuating ﬁeld (MCFF) approach

to study the interplay between CDW and AFM ﬂuctu-

ations in the extended Hubbard model on a half-ﬁlled

square lattice with a repulsive on-site Uand nearest-

neighbour Vinteractions. We show that the MCFF ap-

proach predicts results for the CDW and AFM phase

boundaries in qualitative agreement with more elaborate

numerical methods. Furthermore, it allows to model

competing collective ﬂuctuations for large system sizes

near the thermodynamic limit. In addition, the method

is able to distinguish between stable and meta-stable col-

lective ﬂuctuations. For this reason, the MCFF approach

allows us to capture the true ground state of the coex-

istence region of CDW and AFM ﬂuctuation that was

obtained in Ref. [25] on the basis of DCA calculations.

II. MODEL

For simplicity, our considerations are limited to a

single-band extended Hubbard model. However, we note

that our approach can be straightforwardly generalised

to more complex single- and multi-band quantum lat-

tice systems. The Hamiltonian of the extended Hubbard

model has the following form:

H=−tX

hi,ji,σ

ˆc†

iσ ˆcjσ +UX

ˆni↑ˆni↓+V

hi,ji,σσ0

ˆniσ ˆnjσ0.

(1)

In this expression, ˆc(†)

iσ operators correspond to annihi-

lation (creation) of electrons, where the subscripts de-

note the position iand spin projection σ∈ {↑,↓}. Our

system is modelled by the hopping tbetween nearest-

neighbor sites hi, jion a two-dimensional square lat-

tice. The Coulomb interaction between electronic densi-

ties ˆniσ = ˆc†

iσ ˆciσ contains the on-site Uand the nearest-

neighbor Vcomponents.

The extended Hubbard model (1) displays two sym-

metries of fundamental importance for our considera-

tions: a continuous SU(2) symmetry associated with

spin degrees of freedom and a discrete particle-hole sym-

metry related to charge degrees of freedom. To facil-

itate our later treatments, we include a sketch of the

ﬁnite temperature U, V phase diagram of the extended

Hubbard model on the two-dimensional square lattice in

Fig. 1. Within the sketch, we denote the regime of strong

CDW ﬂuctuations (red gradient), with asymptotics of

the CDW phase boundary highlighted, and the regime

of strong AFM ﬂuctuations (blue gradient). The CDW

phase boundary occurs along V=U/8 + cst. at weak

coupling [33], which transforms to V=U/4 at interme-

diate coupling [20], followed by V∼U+ cst. at strong

coupling [29,34,36,57,68]. At weak coupling the AFM

phase boundary starts at a critical U, which further ex-

tends to the V=U/4 phase boundary at intermediate

coupling [25]. We restrict our consideration to the weak

to intermediate coupling regime, with the strong coupling

regime being outside the scope of the current work.

FIG. 1. Sketch of the phase diagram of the quasi-two-

dimensional half-ﬁlled extended Hubbard model with repul-

sive interactions Uand Vat low but ﬁnite temperature. Be-

yond a critical local interactions, a regime of dominant an-

tiferromagnetic (AFM) ﬂuctuations is expected, while strong

non-local interactions drive the system into a charge density

wave (CDW) phase. At low Uand V, the orderings give away

for a homogeneous paramagnetic (PM) phase. The schematic

phase boundaries of the CDW phase is determined by the

asymptotic expressions V=U/8 + cst. at weak coupling [33],

V=U/4 at intermediate coupling [20,21] and V∼U+ cst.

at strong coupling [29,34,36,57,68]. At weak to intermedi-

ate coupling, the AFM regime extrapolates from a critical U

at vanishing Vto the V=U/4 phase boundary [25].

The MCFF approach to be introduced in the next sec-

tion is based on a variational principle conveniently for-

mulated within the action formalism. Thus, it is suitable

to rewrite the extended Hubbard model (1) in the form

of the action:

S=−1

βN X

k,ν,σ

c∗

kνσ G−1

kνckνσ +U

βN X

q,ω

ρqω↑ρ−q,−ω↓

2βN X

q,ω,σσ0

Vqρqωσρ−q,−ωσ0,(2)

with the inverse temperature βand number of sites N.

Grassmann variables c(∗)correspond to the annihilation

(creation) of electrons, where the subscripts denote the

momentum kand fermionic Matsubara frequency ν. The

inverse of the bare (non-interacting) Green’s function

is deﬁned as G−1

kν=iν +µ−k, where µis the chemi-

cal potential and k=−2t(cos kx+ cos ky) is the disper-

sion relation for the nearest-neighbor hopping on a two-

dimensional square lattice. For convenience, the interac-

tion parts of the action (2) are written in terms of the

shifted densities ρqωσ =nqωσ − hnqωσiδq,0δω,0, where q

and ωare the momentum and bosonic Matsubara fre-

quency indices, respectively. This choice of shift will

be argued for in our later derivation. In our consider-

ations the momentum-space representation for the non-

local interaction is following Vq= 2V(cos qx+ cos qy) as

it is limited to only a nearest-neighbour interaction.

III. MULTI-CHANNEL FLUCTUATING FIELD

METHOD

In this section we derive a multi-channel generalisation

of the ﬂuctuating ﬁeld method that was originally intro-

duced to address the ﬂuctuations in a single (magnetic)

channel [64–67]. We derive the MCFF method by utiliz-

ing a variational approach formulated in Ref. 65, which

allows to incorporate the leading instabilities of the col-

lective ﬂuctuations.

A. Deﬁnition of trial action

We deﬁne a MC-FF trial action

S∗=−1

βN X

k,ν,σ

c∗

kνσ G−1

kνckνσ

Q,ς "φς

Qρς

−Q−1

βN

Jς

φς

Qφς

−Q#,(3)

that explicitly considers sets of scalar charge (ς=c) and

vector spin (ς=s∈ {x, y, z}) ﬁelds φς

Qcoupled to the

operators ρς

Q=nς

Q− hnς

QiδQ,0associated with the re-

spective classical (ω= 0) order parameters of interest.

Here

nς

Q=1

βN X

k,ν,σσ0

c∗

k+Q,νσ σς

σσ0ckνσ0,(4)

where Qis the ordering wave vector, σcis the identity

and σsis the Pauli spin matrices. The interaction part

of the trial action (3) contains a set of stiﬀness constants

Jς

Qthat will be determined.

B. Integrating out fermionic degrees of freedom

The trial action (3) has a Gaussian form with respect

to the Grassmann variables c(∗)and classical ﬁelds φς.

This allows one to obtain an eﬀective action for either

fermionic or classical degrees of freedom by analytically

integrating out the other degrees of freedom. Integrating

out the fermionic degrees of freedom, the eﬀective action

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Multi-channeluctuatingeldapproachtocompetinginstabilitiesininteractingelectronicsystemsE.Linner,1A.I.Lichtenstein,2,3,4S.Biermann,1,5,6,7andE.A.Stepanov11CPHT,CNRS,EcolePolytechnique,InstitutPolytechniquedeParis,F-91128Palaiseau,France2I.InstituteofTheoreticalPhysics,UniversityofHamburg,Jungiusstr...

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Multi-channel uctuating eld approach to competing instabilities in interacting electronic systems E. Linn er1A. I. Lichtenstein2 3 4S. Biermann1 5 6 7and E. A. Stepanov1

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