Traces of Electron-Phonon Coupling in One-Dimensional Cuprates Ta Tang1 2Brian Moritz2Cheng Peng2Z. X. Shen1 2 3 4and Thomas P. Devereaux2 4 5 1Department of Applied Physics Stanford University California 94305 USA.

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Traces of Electron-Phonon Coupling in One-Dimensional Cuprates

Ta Tang,1, 2 Brian Moritz,2Cheng Peng,2Z. X. Shen,1, 2, 3, 4 and Thomas P. Devereaux2, 4, 5

1Department of Applied Physics, Stanford University, California 94305, USA.

2Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory,

2575 Sand Hill Road, Menlo Park, California 94025, USA.

3Department of Physics, Stanford University, Stanford CA 94305, USA.

4Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305, USA.

5Department of Materials Science and Engineering, Stanford University, Stanford CA 94305, USA.

(Dated: October 18, 2022)

The appearance of certain spectral features in one-dimensional (1D) cuprate materials has been

attributed to a strong, extended attractive coupling between electrons. Here, using time-dependent

density matrix renormalization group methods on a Hubbard-extended Holstein model, we show that

extended electron-phonon (e-ph) coupling presents an obvious choice to produce such an attractive

interaction that reproduces the observed spectral features and doping dependence seen in angle-

resolved photoemission experiments: diminished 3kFspectral weight, prominent spectral intensity

of a holon-folding branch, and the correct holon band width. While extended e-ph coupling does not

qualitatively alter the ground state of the 1D system compared to the Hubbard model, it quantita-

tively enhances the long-range superconducting correlations and suppresses spin correlations. Such

an extended e-ph interaction may be an important missing ingredient in describing the physics of

the structurally similar two-dimensional high-temperature superconducting layered cuprates, which

may tip the balance between intertwined orders in favor of uniform d-wave superconductivity.

The origin of high-temperature superconductivity

found in layered, quasi-two-dimensional (2D) cuprates

remains a puzzle despite concerted, continuous investi-

gations over the last few decades. From the perspective

of numerical simulations, simpliﬁed models such as the

Hubbard and t-JHamiltonians have been studied ex-

tensively, which have produced rich physics relevant to

cuprates such as antiferromagnetism, stripes, and strange

metal behavior[1–3]. However, evidence that these sim-

pliﬁed models possess a uniform d-wave superconduct-

ing ground state remains elusive. Quasi-long-range su-

perconductivity has only been reported on small width

cylinders [4–13], with strong competition from coexisting

charge order. Superconducting correlations decay expo-

nentially on the hole doped side for wider clusters, in-

dicating the superconductivity is absent for parameters

thought to be relevant to hole doped cuprates.

These ﬁndings indicate that the Hubbard model

is incomplete, at least for describing the cuprates

and high-temperature superconductivity. The inclu-

sion of additional ingredients, such as phonons, which

manifest as kinks or replica bands in photoemission

measurements[14–17], may provide the crucial remedy.

However, exact numerical simulations of the 2D Hub-

bard model already are challenging (the density matrix

renormalization group (DMRG) method is limited by the

growth of entanglement entropy and determinant quan-

tum Monte Carlo (DQMC) and related methods suﬀer

from the fermion sign problem); and adding bosonic de-

grees of freedom creates an even more daunting problem.

The task may be made easier, with more numerical con-

trol, by turning to the simpler, yet structurally similar,

one-dimensional (1D) cuprates. Recent angle-resolved

photoemission spectroscopy (ARPES) experiments on

the 1D cuprate Ba2−xSrxCuO3+δ[18] provide an excel-

lent platform for testing theoretical models. Modeling in

1D has both well-established theory and numerical simu-

lations can be performed with a higher degree of control

and accuracy. The measured single-particle spectra pro-

vide a detailed proving ground for assessing the impact

of terms added to model Hamiltonians. Reference [18]

showed that the simple Hubbard model fails to repro-

duce salient details of the spectra near the Fermi surface:

a prominent holon-folding (hf)-branch emanates from kF

and quickly fades away with doping. This spectral fea-

ture, and its doping dependence, can be well reproduced

when one includes a strong nearest-neighbor attractive

interaction V∼ −tin the model Hamiltonian. A natural

near-neighbor attraction exists in the Hubbard model,

evident when downfolding to the t-Jmodel, but such a

weak attraction (∼ −J/4) cannot account for the ob-

served eﬀect. Rather, this strong attraction likely origi-

nates from extended electron-phonon (e-ph) coupling, as

discussed in recent work [18,19].

To investigate the inﬂuence of the extended e-ph cou-

pling, in this paper a time-dependent density matrix

renormalization group (tDMRG) method is employed to

study the single-particle spectral function and ground

state properties of a 1D Hubbard-extended Holstein

model. The extended e-ph coupling quantitatively re-

produces the dominant hf-branch seen in experiments,

while also correctly reproducing the holon branch band

width, matching the observed spectra. Approximating

this model using an eﬀective nearest-neighbor attraction

V fails to reproduce all of these features. Moreover,

while the extended e-ph coupling does not qualitatively

alter the ground state obtained from the Hubbard model,

which qualitatively remains a Luttinger liquid with sub-

arXiv:2210.09288v1 [cond-mat.str-el] 17 Oct 2022

FIG. 1. (a) Schematic for the one dimensional Hubbard-

extended Holstein model. On each site, the local Hilbert space

is a direct product of phonon and charge degrees of freedom.

The charges of opposite spin interact with an on-site repul-

sion Uand can hop to neighboring sites. Local phonons with

a frequency ω0couple to both on-site and nearest-neighbor

charges. (b) Schematic for the dynamical LBO. We keep the

dimension of the eﬀective Hilbert space of the system and

environment blocks as m, respectively. Each site ihas dop-

timized basis. The wave function is transformed to a Dd

bare basis (D=Dch ×Dph, where Dch = 4 represents the

local charge Hilbert space dimension, and Dph is the bare

phonon basis dimension) through a D×dtransformation ma-

trix, i.e. Ti, before applying the time evolution gate of shape

D2×D2. Subsequently, a new optimal basis and transforma-

tion Tiare obtained; and the wave function is projected to

the new optimal basis before moving on to the next gate.

dominant superconducting pair-ﬁeld correlations that de-

cay as a power law with distance, the results show that

the extended e-ph coupling quantitatively enhances the

superconducting pair-ﬁeld correlations by reducing the

overall exponent, making them longer-ranged. It is sur-

mised that in two dimensions an extended e-ph coupling

may tip the balance between diﬀerent phases and help to

realize a dominant d-wave superconducting ground state.

MODELS

To produce an eﬀective nearest-neighbor attractive in-

teraction for charge, we consider an optical phonon mode,

which couples to charge density beyond the local site.

Previous estimates [19] have shown that this Hubbard-

extended Holstein model can produce an eﬀective inter-

action on par with that extracted from ARPES experi-

ments [18] for a reasonable phonon frequency and e-ph

coupling strength. For simplicity and to achieve bet-

ter numerical convergence, here, we consider only on-site

and nearest-neighbor e-ph coupling (see Fig. 1(a)). This

Hubbard-extended Holstein Hamiltonian takes the form

H=Hel +ω0X

ˆa†

iˆai

+g0X

ˆni(ˆa†

i+ ˆai) + g1X

hiji

ˆni(ˆa†

j+ ˆaj),(1)

where ˆa†

iand ˆaiare the phonon ladder operators on site

i, ˆniis the total charge number operator on site i,ω0is

the phonon frequency, g0is the on-site e-ph coupling, g1

is the nearest-neighbor e-ph coupling, and hijisums over

nearest-neighbors. Hel denotes the electronic part of the

Hamiltonian, a 1D single-band Hubbard model,

Hel =−thX

hijiσ

(ˆc†

iσ ˆcjσ +h.c.) + UX

ˆni↑ˆni↓,(2)

where ˆc†

iσ (ˆciσ ) is the charge creation (annihilation) op-

erator on site ifor spin σ, ˆniσ is the charge number op-

erator on site ifor spin σ, and Uis the on-site repulsion.

To avoid confusion with the time variable t, we use th

to denote the hopping integral. For comparison, we also

evaluate the extended-Hubbard model, which introduces

a nearest-neighbor attractive interaction,

Hv=Hel +VX

hiji

ˆniˆnj,(3)

where ˆniand ˆnjare total charge number operators on

neighboring sites.

Unless otherwise speciﬁed, we use the following pa-

rameters in our simulations: U= 8th,ω0= 0.2th,

g0= 0.3th,g1= 0.15thand V=−th. The values chosen

for Uand Vwere those that produced the best ﬁt of the

ARPES experimental spectra using cluster perturbation

theory (CPT) [20,21] for an eﬀective extended-Hubbard

model [18]; and the e-ph couplings g0and g1fall within

the range estimated in Ref. [19]. Here, we use a larger

phonon frequency than that used in Ref. [19] for better

numerical convergence, but expect that a smaller phonon

frequency would produce a stronger eﬀective attraction,

which would further enhancing the hf-branch; although,

one would need to ensure that the stronger eﬀective cou-

pling would not lead to phase separation.

We use DMRG [22,23] to obtain the ground states

of the models deﬁned in Eqs. 1,2, and 3; and we use

tDMRG [24–26] to obtain real-frequency spectra from

the Fourier transform of time-dependent correlators of

the form Dˆ

O†

i(t)ˆ

Oj(0)E. To eﬃciently deal with the inﬁ-

nite phonon Hilbert space on each site, we adopt a local

basis optimization (LBO) for the ground state [27] and

a dynamical LBO for time evolution [28], as schemati-

cally shown in Fig. 1(b). Details about the method and

numerical simulation are provided in the supplementary

material.

FIG. 2. (a) The lesser Green’s function G<

j,L/2,↑(t) for an

80-site chain at half-ﬁlling for the Hubbard model. Time is

measured in units of ~/thand ~= 1 in our calculation. We use

a time step δt = 0.04t−1

hand evolve the system for a total time

T= 20t−1

h. (b) The single-particle spectral function obtained

by Fourier transform of G<in (a), with energy and momentum

broadening of σω= 0.2thand σk= 2π/L, respectively.

SINGLE PARTICLE SPECTRAL FUNCTION

Fig. 2displays the lesser Green’s function G<

j,L/2,↑(t),

deﬁned as G<

mnσ(t) = iˆc†

mσ(t)ˆcnσ(0), and the corre-

sponding single-particle removal spectra, obtained for the

Hubbard model on an 80-site chain at half-ﬁlling. In

Fig. 2(a), following the removal of an electron from the

center of the chain, one can see that the propagator at-

tains a signiﬁcant value at the two chain ends within

a time T∼20t−1

h, which sets the maximum real-time

propagation for the simulation. Padding the Green’s

function with zeros from time Tto time 2Tlimits the

frequency resolution of a fast Fourier transform (FFT)

to ωn+1 −ωn=π/T ∼0.16 th. This provides a rather

coarse resolution, but it is nevertheless more than ade-

quate for comparison to the experimental ARPES spec-

tra from the 1D chain cuprate, which is rather broad [18].

The single-particle spectrum, which is obtained using the

tDMRG method and shown in Fig. 2(b), agrees well with

the results from cluster perturbation theory [18,20,21],

dynamical DMRG, and the Bethe ansatz [29–32]. There

are clear spinon and holon branches, demonstrating spin-

charge separation in 1D. In the following, we use a chain

of length L= 80 to compute and compare the single

particle spectral function of diﬀerent models. A small

broadening is used to give the spectra a high resolution,

at least when compared with the experiment data, to bet-

ter observe how diﬀerent models aﬀect the salient spec-

tral features.

Fig. 3(A.1-6) show the single-particle removal spectra

of the Hubbard model across a range of doping. As ob-

served in experiment, splitting between the spinon and

holon branches persists with doping. Our results cor-

respond well to previous Hubbard model results on 1D

and quasi-1D systems from dynamical DMRG and the

Bethe ansatz [30,32], and also are consistent with spectra

near the Fermi level from determinant quantum Monte

Carlo and DMRG calculations of the multi-band Hub-

bard model, which includes oxygen p-orbitals [33]. Here,

we will focus on two spectral features: the branch of the

removal spectrum emanating from kF, which disperses

downward toward π, hereafter the hf-branch; and the

3kF-branch (or more precisely 2π−3kF), which also dis-

perses downward toward π, but from 3kF. In the MDCs

obtained from Hubbard model (Figs. 3(B.1-6)), one sees

that between these two features the 3kF-peak is domi-

nant. This result is contradictory to experimental obser-

vations, where the hf-peak is dominant and the 3kF-peak

is barely visible [18].

In Figs. 3(C.1-6 and D.1-6), we conﬁrm that adding

a nearest-neighbor attractive interaction V=−then-

hances the hf-branch and produces spectra that are visi-

bly more consistent with the experimental data at lower

doping [18]. As we mentioned previously, this attractive

interaction likely originates from e-ph coupling. Here, we

also simulate the underlying e-ph Hamiltonian, with the

results shown in Figs. 3(E.1-6). Below 20% doping, one

sees an enhanced hf-branch, while the 3kF-branch has

been suppressed signiﬁcantly by the e-ph coupling (see

Figs. 3(E.1-3 and F.1-3)). In all three models, the inten-

sities in both the hf- and 3kF-branches become barely

perceptible beyond ∼20% doping. Using a larger broad-

ening to compare more closely with the experimental

spectra and to extract intensities by ﬁtting MDCs results

in a doping-dependent intensity of hf-peak that matches

well to the analyzed ARPES data (see Figs. S6 and S7

in the Supplementary Material).

One signiﬁcant diﬀerence between spectra for the ex-

tended Hubbard model and the Hubbard-extended Hol-

stein model is that the nearest-neighbor attractive in-

teraction in the extended Hubbard model signiﬁcantly

shrinks the holon bandwidth at higher doping (see

Figs. 3(C.1-6)). In Fig. 4, we plot the holon binding en-

ergy at k= 0 as a function of doping to reﬂect the change

of the holon bandwidth. By comparison, one sees that

the results from the Hubbard-extended Holstein model

are more consistent with the ARPES data, as the e-ph in-

teraction would renormalize the holon-branch only within

∼ω0of the Fermi energy.

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TracesofElectron-PhononCouplinginOne-DimensionalCupratesTaTang,1,2BrianMoritz,2ChengPeng,2Z.X.Shen,1,2,3,4andThomasP.Devereaux2,4,51DepartmentofAppliedPhysics,StanfordUniversity,California94305,USA.2StanfordInstituteforMaterialsandEnergySciences,SLACNationalAcceleratorLaboratory,2575SandHillRoad,Men...

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Traces of Electron-Phonon Coupling in One-Dimensional Cuprates Ta Tang1 2Brian Moritz2Cheng Peng2Z. X. Shen1 2 3 4and Thomas P. Devereaux2 4 5 1Department of Applied Physics Stanford University California 94305 USA.

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