Bipolaronic superconductivity out of a Coulomb gas J. Sous1 2C. Zhang3yM. Berciu4 5D. R. Reichman6B. V. Svistunov7 8N. V. Prokofev7and A. J. Millis9 10z 1Department of Physics Stanford University Stanford CA 93405 USA

2025-05-06 0 0 1.22MB 7 页 10玖币

Bipolaronic superconductivity out of a Coulomb gas

J. Sous,1, 2, ∗C. Zhang,3, †M. Berciu,4, 5 D. R. Reichman,6B. V. Svistunov,7, 8 N. V. Prokof’ev,7and A. J. Millis9, 10, ‡

1Department of Physics, Stanford University, Stanford, CA 93405, USA

2Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA5, USA

3State Key Laboratory of Precision Spectroscopy,

East China Normal University, Shanghai 200062, China

4Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada

5Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada

6Department of Chemistry, Columbia University, New York, New York 10027, USA

7Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA

8Wilczek Quantum Center, School of Physics and Astronomy and T. D. Lee Institute,

Shanghai Jiao Tong University, Shanghai 200240, China

9Department of Physics, Columbia University, New York, New York 10027, USA

10Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA

(Dated: December 5, 2022)

Employing unbiased sign-problem-free quantum Monte Carlo, we investigate the eﬀects of long-

range Coulomb forces on BEC of bipolarons using a model of bond phonon-modulated electron hop-

ping. In absence of long-range repulsion, this model was recently shown to give rise to small-size,

light-mass bipolarons that undergo a superﬂuid transition at high values of the critical transition

temperature Tc. We ﬁnd that Tcin our model even with the long-range Coulomb repulsion remains

much larger than that of Holstein bipolarons, and can be on the order of or greater than the typical

upper bounds on phonon-mediated Tcbased on the Migdal-Eliashberg and McMillan approxima-

tions. Our work points to a physically simple mechanism for superconductivity in the low-density

regime that may be relevant to current experiments on dilute superconductors.

Introduction. Understanding the mechanisms of su-

perconductivity in the dilute density regime is an ac-

tive theme of research, relevant to polar materials [1,2],

doped topological insulators [3,4], transition metal

dichalcogenides [5], moir´e materials [6–11] and other ma-

terials [12]. This large and growing list of ultra-low car-

rier density superconductors motivates theoretical exam-

ination of superconductivity in electron-phonon coupled

systems at very small densities where, as a matter of

principle, the Fermi liquid/Migdal-Eliashberg paradigm

must fail. Bose-Einstein condensation (BEC) of pre-

formed pairs (“bipolarons”) in principle oﬀers a robust

route to superconductivity at low densities. But, in

the low-density regime, the Coulomb repulsion is weakly

screened and thus the pairing “glue” required to bind

electron pairs into bound states must be strong enough

to overcome the Coulomb repulsion. A strong pairing

interaction is usually believed to result in heavy bound

states, implying low values of the critical transition tem-

perature Tc. These considerations [13,14] are widely be-

lieved to severely limit the maximum Tcobtained from

phonon-mediated binding of electrons into bipolarons.

We have recently shown [15] that even in the pres-

ence of a short-ranged interaction parameterized by a

large onsite Hubbard repulsion U, electrons coupled to

phonons via bond phonon-modulated electron hopping

form small-size, light-mass bipolarons [16,17] that un-

dergo a superﬂuid (“BEC”) transition at values of Tcthat

are much larger than those obtained in (Holstein) mod-

els in which the electron density is coupled to phonons or

from Migdal-Eliashberg theory of superconductivity out

of a Fermi liquid. This work did not include the long-

range part of the Coulomb interaction, so is relevant to

two-dimensional (2D) materials in which the Coulomb

repulsion is completely screened by gating or proximity

to a substrate [15]. However, in ungated 2D materials

and in three-dimensional (3D) materials in which the

Coulomb repulsion cannot be screened by an external

gate, the question of the eﬀects of long-range Coulomb re-

pulsion on bipolaronic superconductivity (and other non-

phononic BEC mechanisms [18–22]) remains open.

In this letter, we study BEC of bipolarons occuring

in a dilute, 3D Coulomb gas, showing that Tcof bond-

coupled bipolarons is still higher than that of density-

coupled (Holstein) bipolarons, and in line with the value

of Tctypically found in experiments on 3D materials be-

lieved to be close to or in the low-density regime. In a

Coulomb system, the two-electron bound state retains a

ﬁnite size even at the critical interaction strength asso-

ciated with unbinding [23], and thus the maximum Tcis

determined by a combination of binding strength, mass

and size with the constraint that the size cannot be in-

ﬁnite. To the best of our knowledge, despite decades

of debate [13,24,25], our theory is the ﬁrst quantita-

tive eﬀort that takes the presence of long-range Coulomb

interaction into account and (i) demonstrates, using an

unbiased approach, a realistic mechanism for BEC for-

mation at relatively high values of Tcand (ii) unveils the

properties of bipolarons, e.g. their mass and size, in 3D.

Model. We consider the bond-Peierls [26] (also known

as bond-Su-Schrieﬀer-Heeger [27]) electron-phonon cou-

pling on a 3D cubic lattice. In this model the electronic

arXiv:2210.14236v2 [cond-mat.supr-con] 1 Dec 2022

0.80 0.85 0.90 0.95 1.00 1.05

0.00

0.05

0.10

0.15

0.20

0.25

Tc/Ω

U/t = 8

V=U/10 t/Ω=1

t/Ω=2

≈

FIG. 1. Estimates of Tcof the bond-Peierls (bP) bipolaronic

superconductor (ﬁlled squares, blue lines) for diﬀerent t/Ω at

U= 8t,V(r > 0) = V /r with V=U/10 as a function of

λcomputed according to Eq. (3) from QMC simulations of

the bipolaron eﬀective mass m

BP and its mean squared-radius

BP. We contrast this to superconductivity of Holstein (H)

bipolarons (open circles, orange line) at t/Ω = 2 for the same

values of U/t and V. Here we use λ=α2/3Ωtfor bond-Peierls

bipolarons, λ= 0.85α2/6Ωtfor Holstein bipolarons (we use a

factor of 0.85 so that the two sets of data can be presented

on the same scale of λ). The doubly wavy symbol indicates

absence of bipolarons in the Holstein model for λ.0.91 where

a crossover from BEC to BCS may occur.

hopping between two sites is modulated by a single oscil-

lator centered on the bond connecting the two sites [15].

The Hamiltonian is

H=ˆ

He+ˆ

Hph +ˆ

Ve-ph.(1)

Here, the lattice Coulomb model for electrons with

spin σ∈ {↑,↓} is ˆ

He=−tPhi,ji,σ ˆc†

i,σ ˆcj,σ + h.c.+

UPiˆni,↑ˆni,↓+(1/2) Pi6=jVi,j ˆniˆnjwith nearest-neighbor

(NN) hopping t, onsite Hubbard repulsion U, NN repul-

sion Vand longer-range repulsion Vij =V a

|ri−rj|, where

ˆni= ˆni,↑+ ˆni,↓with ˆni,σ = ˆc†

i,σ ˆci,σ at site ri, and ais the

lattice constant (NN distance). The notation hi, jirefers

to NN sites. Writing the Coulomb repulsion as e2/r

(where the dielectric constant encodes short-ranged po-

larization eﬀects arising from degrees of freedom not ex-

plicitly included in the model) we estimate the onsite

U=e2/aorb with aorb the unit cell orbital size, the NN

V=e2/a ≈Uaorb/a, and further-neighbor (distance

r > a)Vr>a =V a/r. We henceforth set the lattice con-

stant a= 1. We model distortions of the bonds connect-

ing sites iand jas Einstein oscillators with Hamiltonian

Hph =Phi,ji1

2Kˆ

i,j +ˆ

i,j /2M= Ω Phi,jiˆ

b†

i,jˆ

bi,j with

oscillator frequency (~= 1) Ω = pK/M. The interac-

tion between electrons and phonons takes the form

Ve-ph =α√2MΩX

hi,ji,σ ˆc†

i,σ ˆcj,σ + h.c.ˆ

Xi,j (2)

describing the modulation of electron hopping by an os-

cillator ˆ

Xi,j :=1

√2MΩˆ

b†

i,j +ˆ

bi,j associated with the

FIG. 2. Estimates of Tcof the bond-Peierls bipolaronic su-

perconductor for t/Ω = 2 as a function of λ=α2/(3Ωt), for

a. diﬀerent strengths of the Coulomb repulsion Vat ﬁxed

onsite U/t = 8, and for b. diﬀerent strengths of the onsite U

at ﬁxed long-range repulsion V=U/10, computed according

to Eq. (3) from QMC simulations of the bipolaron eﬀective

mass m

BP and mean squared-radius R2

BP.

bond connecting sites iand jwith coupling coeﬃcient

α√2MΩ. We set M= 1. The relevant parameters are a

dimensionless coupling constant λ= ((α√2Ω)2/K)/6t=

α2/(3Ωt), the ratio of the typical polaronic energy scale

to the free electron energy scale, and an adiabaticity pa-

rameter t/Ω. It is important to note that a typical phys-

ical origin for this phonon-modulated hopping is from

interference of diﬀerent hopping pathways [15]. This im-

plies that the magnitude of the coupling term (Eq. (2))

is independent of the bare hopping, which means that

the model remains valid in the strong-coupling regime

even when the electronic hopping changes sign. This is

diﬀerent from other models of phonon-modulated hop-

ping [16,28–32] in which an equation of the general form

of Eq. (2) applies only in the small displacement regime

where the net change in hopping amplitude is small rel-

ative to the bare hopping [33].

Method. Using a numerically exact sign-problem-free

quantum Monte Carlo (QMC) approach based on a path-

integral formulation of the electronic sector combined

with either a real-space diagrammatic or a Fock-space

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BipolaronicsuperconductivityoutofaCoulombgasJ.Sous,1,2,C.Zhang,3,yM.Berciu,4,5D.R.Reichman,6B.V.Svistunov,7,8N.V.Prokof'ev,7andA.J.Millis9,10,z1DepartmentofPhysics,StanfordUniversity,Stanford,CA93405,USA2StanfordInstituteforTheoreticalPhysics,StanfordUniversity,Stanford,CA5,USA3StateKeyLaboratoryof...

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Bipolaronic superconductivity out of a Coulomb gas J. Sous1 2C. Zhang3yM. Berciu4 5D. R. Reichman6B. V. Svistunov7 8N. V. Prokofev7and A. J. Millis9 10z 1Department of Physics Stanford University Stanford CA 93405 USA.pdf

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Bipolaronic superconductivity out of a Coulomb gas J. Sous1 2C. Zhang3yM. Berciu4 5D. R. Reichman6B. V. Svistunov7 8N. V. Prokofev7and A. J. Millis9 10z 1Department of Physics Stanford University Stanford CA 93405 USA

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