Viscous Drude weight of dual Bose and Fermi gases in one dimension Yusuke Nishida Department of Physics Tokyo Institute of Technology Ookayama Meguro Tokyo 152-8551 Japan

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Viscous Drude weight of dual Bose and Fermi gases in one dimension

Yusuke Nishida

Department of Physics, Tokyo Institute of Technology, Ookayama, Meguro, Tokyo 152-8551, Japan

(Dated: October 2022)

We continue to study frequency-dependent complex bulk viscosities of one-dimensional Bose and

Fermi gases with contact interactions, which exhibit the weak-strong duality according to our recent

work. Here we show that they are contributed to by Drude peaks divergent at zero frequency as

typical for transport coeﬃcients of quantum integrable systems in one dimension. In particular, their

Drude weights are evaluated based on the Kubo formula in the high-temperature limit at arbitrary

coupling as well as in the weak-coupling and strong-coupling limits at arbitrary temperature, where

systematic expansions in terms of small parameters are available. In all three limits, the Drude

peaks are found at higher orders compared to the ﬁnite regular parts.

I. INTRODUCTION

A nonvanishing Drude weight indicates a divergent

transport coeﬃcient at zero frequency and serves as di-

agnostics of whether the transport is ballistic or diﬀu-

sive [1]. A simple example of ballistic transports is pro-

vided by a mass transport for ﬂuids with translational in-

variance, where the mass current (i.e., momentum) can-

not dissipate due to its conservation law [2]. On the

other hand, an energy transport is typically diﬀusive for

interacting systems because the energy current is noncon-

served. However, quantum integrable systems in one di-

mension have been found so exceptional that their Drude

weights remain nonvanishing for various transports even

when corresponding currents are nonconserved [3, 4].

This is because a macroscopic number of conservation

laws allows the currents to have some overlaps with con-

served quantities. Since then, anomalous conductivities

of quantum integrable systems in one dimension have

been subjected to active study from both theoretical and

experimental perspectives [5–8].

In spite of such active study, little attention has been

paid so far to another transport coeﬃcient possible in one

dimension, that is, the bulk viscosity [9, 10]. Recently,

we showed in Ref. [11] that the frequency-dependent com-

plex bulk viscosity of a Bose gas with a contact interac-

tion known as the Lieb-Liniger model [12, 13] is iden-

tical to that of a dual Fermi gas known as the Cheon-

Shigehara model [14] at the same scattering length a. In

particular, it is the weak-strong duality, where one sys-

tem at weak coupling corresponds to the other system at

strong coupling so that the bulk viscosity in the strong-

coupling regime can be accessed with the perturbation

theory of the dual system. Their bulk viscosities were

then computed in the high-temperature, weak-coupling,

and strong-coupling limits, where systematic expansions

in terms of small parameters are available, and found to

be ﬁnite with no Drude peaks at their leading orders [11].

The purpose of this paper is to go one step further

beyond the leading orders and show that the frequency-

dependent complex bulk viscosities of one-dimensional

Bose and Fermi gases with contact interactions indeed

have the structure of

ζ(ω) = ζreg(ω) + iD

ω+i0+,(1)

where the ﬁrst term on the right-hand side is the ﬁnite

regular part and the second term is the Drude peak di-

vergent at zero frequency. In particular, we will evaluate

the Drude weights Dbased on the Kubo formula in the

high-temperature limit at arbitrary coupling as well as

in the weak-coupling limit at arbitrary temperature for

bosons in Sec. III and for fermions in Sec. IV. The two

results in the high-temperature limit are also useful to

conﬁrm the Bose-Fermi duality explicitly, whereas those

in the weak-coupling limit are applicable to the Fermi

and Bose gases in the strong-coupling limit, as indicated

in Fig. 1.

We will set ~=kB= 1 throughout this paper and

the bosonic and fermionic frequencies in the Matsub-

ara formalism are denoted by p0= 2πn/β and p0

2π(n+ 1/2)/β, respectively, for n∈Zand β= 1/T .

Also, an integration over wave number or momentum

is denoted by Rp≡R∞

−∞ dp/2πfor the sake of brevity,

∞

high temperature

strong BG ↔weak FG

weak BG ↔strong FG

−1

FIG. 1. Bulk viscosities of Bose and Fermi gases are eval-

uated in the high-temperature limit as well as in the weak-

coupling limit, which corresponds to a→ −∞ for bosons

(BG) and a→ −0 for fermions (FG) [11]. The system is

thermodynamically unstable at a > 0.

arXiv:2210.09712v2 [cond-mat.quant-gas] 21 Dec 2022

whereas the same deﬁnition as in Ref. [15] is employed

for the response function [see Eq. (10) therein].

II. DRUDE WEIGHT

According to the linear-response theory [16–18], the

complex bulk viscosity at frequency ωis microscopically

provided by

ζ(ω) = RΠΠ(w)−RΠΠ(i0+)

iw (2)

with the substitution of w∈C→ω+i0+on the right-

hand side [15]. Here RΠΠ(w) is the response function of

the modiﬁed stress operator at zero wave number,

Π≡ˆπ−∂p

∂NE

N − ∂p

∂EN

H,(3)

where ˆπ,ˆ

N, and ˆ

Hare the stress operator, the number

density operator, and the Hamiltonian density, respec-

tively, with p=hˆπi,N=hˆ

Ni, and E=hˆ

Hi. Although

the above Kubo formula may look diﬀerent from that

employed in Ref. [11], they are actually equivalent but

the present form will prove to be convenient for the sake

of evaluating the Drude weight.

The Drude peak appears from the type of diagrams

depicted in Fig. 2 for the stress-stress response function,

which reads

RΠΠ(ik0) = ±1

βX

p0Zp

G(ik0+ip0, p)G(ip0, p)

×[Γ(ik0+ip0, ip0;p)]2.(4)

Here G(ip0, p) = 1/(ip0−εp) with εp=p2/2m−µ

is the single-particle propagator, Γ(ik0+ip0, ip0;p) is

the vertex function to be speciﬁed below, and the up-

per (lower) sign corresponds to bosons (fermions under

p0→p0

0). The Matsubara frequency summation is re-

placed with the complex contour integration over ip0→ν

and its integration contour is deformed into four lines

along Im(ν) = ±0+,−k0±0+[19, 20]. Then the analytic

continuation of ik0→ω+i0+leads to

RCC (ω+i0+) = ZR\{0}

dν

2πi fB,F (ν)Zp

×G+(ν+ω, p)G+(ν, p)Γ(ν+ω+i0+, ν +i0+;p)2

−G+(ν+ω, p)G−(ν, p)Γ(ν+ω+i0+, ν −i0+;p)2

+G+(ν, p)G−(ν−ω, p)Γ(ν+i0+, ν −ω−i0+;p)2

−G−(ν, p)G−(ν−ω, p)Γ(ν−i0+, ν −ω−i0+;p)2,

(5)

where fB,F (ε)=1/(eβε ∓1) are the Bose-Einstein and

Fermi-Dirac distribution functions and G±(ν, p)≡G(ν±

i0+, p) are the retarded and advanced propagators. An

ΓΓ

FIG. 2. Diagrammatic representation of the stress-stress

response function in Eq. (4). The single line represents the

single-particle propagator, whereas the circle (Γ) is the vertex

function.

important fact is that the product of retarded and ad-

vanced propagators with the same wave number has the

singularity of

G+(ν+ω, p)G−(ν, p) = 2πi δ(ν−εp)

ω+i0++O(ω0) (6)

at zero frequency. Consequently, Eq. (5) substituted into

Eq. (2) gives rise to the Drude peak in Eq. (1) with its

weight provided by

D=βZp

fB,F (εp) [1 ±fB,F (εp)] [Γ+−(p)]2,(7)

where Γ+−(p)≡Γ(εp+i0+, εp−i0+;p) is the on-shell

vertex function. Here we note that the two subtracted

terms on the right-hand side of Eq. (3) have the role of

imposing

fB,F (εp) [1 ±fB,F (εp)] Γ+−(p) = 0,(8a)

fB,F (εp) [1 ±fB,F (εp)] Γ+−(p)εp= 0,(8b)

which are essential to deﬁne the static bulk viscosity in

the Boltzmann equation [21–23].

It may be recalled that higher-order diagrams involving

npairs of counterpropagating propagators have stronger

singularities ∼Dτ/(ωτ)nat zero frequency and the re-

summation of them leads to ﬁnite transport coeﬃcients

of Dτ in two and three dimensions [19, 23–26]. However,

one dimension is exceptional because the two-body inter-

action does not contribute to a ﬁnite relaxation time τ

under the energy and momentum conservations.1More

technically, such stronger singularities disappear due to

cancellation among the self-energy and vertex (Maki-

Thompson and Aslamazov-Larkin) corrections [20] (see

the Appendix therein) so that the higher-order diagrams

are to provide at most corrections to D. Therefore, the

Drude peak remains in one dimension and we now study

its weights for the Lieb-Liniger model and the Cheon-

Shigehara model.

1On the other hand, the three-body interaction ∼g3in one di-

mension does contribute to the ﬁnite relaxation time τ∼g−2

3so

as to reduce the Drude peak to iD/(ω+i/τ ) [20].

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ViscousDrudeweightofdualBoseandFermigasesinonedimensionYusukeNishidaDepartmentofPhysics,TokyoInstituteofTechnology,Ookayama,Meguro,Tokyo152-8551,Japan(Dated:October2022)Wecontinuetostudyfrequency-dependentcomplexbulkviscositiesofone-dimensionalBoseandFermigaseswithcontactinteractions,whichexhibitthe...

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Viscous Drude weight of dual Bose and Fermi gases in one dimension Yusuke Nishida Department of Physics Tokyo Institute of Technology Ookayama Meguro Tokyo 152-8551 Japan

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