Bath induced phase transition in a Luttinger liquid Saptarshi Majumdar1Laura Foini2Thierry Giamarchi3and Alberto Rosso1 1Universit e Paris Saclay CNRSLPTMS 91405 Orsay France

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Bath induced phase transition in a Luttinger liquid

Saptarshi Majumdar,1Laura Foini,2Thierry Giamarchi,3and Alberto Rosso1

1Universit´e Paris Saclay, CNRS,LPTMS, 91405, Orsay, France

2IPhT, CNRS, CEA, Universit´e Paris Saclay, 91191 Gif-sur-Yvette, France

3Department of Quantum Matter Physics, University of Geneva,

24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland

(Dated: October 5, 2022)

We study an XXZ spin chain, where each spin is coupled to an independent ohmic bath of

harmonic oscillators at zero temperature. Using bosonization and numerical techniques, we show

the existence of two phases separated by an Kosterlitz-Thouless (KT) transition. At low coupling

with the bath, the chain remains in a Luttinger liquid phase with a reduced but ﬁnite spin stiﬀness,

while above a critical coupling the system is in a dissipative phase characterized by a vanishing spin

stiﬀness. We argue that the transport properties are also inhibited: the Luttinger liquid is a perfect

conductor while the dissipative phase displays ﬁnite resistivity. Our results show that the eﬀect of

the bath can be interpreted as annealed disorder inducing signatures of localization.

In quantum systems, transport properties are

strongly aﬀected by the presence of quenched dis-

order. The most spectacular eﬀect is the locali-

sation of part of the spectrum of the hamiltonian,

which is at the origin of a ﬁnite temperature metal-

insulator transition, ﬁrst predicted for free fermions

by P.W.Anderson [1]. Recently, it has been argued

that ﬁnite temperature localization can also occur in

presence of interactions via the so-called many body

localisation (MBL)[2–7]. Instead, localization is de-

troyed by contact with an external bath via Variable

Range Hopping (VRH)[8–10]. Indeed, transport be-

comes possible by the hopping of a localised electron

over a distance rdue to the emission or absorption

of phonons. Note that this process is feasible only

for acoustic phonons which are delocalized at both

donor and acceptor sites. However for systems with

single degree of freedom, such as a single spin or

a single particle, baths can induce localisation. In

particular, it has been shown that subohmic baths

with harmonic oscillators at zero temperature can

freeze the quantum dynamics of such systems [11–

14]. In our paper, we extend these works and show

the existence of a genuine bath-induced phase tran-

sition in many-body systems. To avoid processes

like VRH, we work with a bath that produces op-

tical phonons. We focus on one dimensional sys-

tem to use an approach that allows us to probe very

large systems and account for the eﬀect of the bath

in an exact way. In practice, we choose the sys-

tem to be an XXZ spin-chain with the hamiltonian

HS=PN

j=1 JzSz

jSz

j+1 +Jxy Sx

jSx

j+1 +Sy

jSy

j+1

and Jz/Jxy ∈(−1,1). This model displays a gapless

low energy spectrum and it is in a perfectly con-

FIG. 1. Schematic representation of the microscopic sys-

tem: An one-dimensional XXZ spin chain(blue color)

with each of the spin coupled to its individual dissipative

bath(red colour). The baths are described by a collection

of simple harmonic oscillators kept at zero temperature.

The parameter αis a measure of the coupling strength

between the bath and the associated spin.

ducting phase known as Luttinger Liquid (LL) [15].

Each spin jof the chain is in contact with its own

independent bath of harmonic oscillators with the

hamiltonian HB=Pjk

2mk+mkΩ2

2X2

jk. A diﬀer-

ent choice for local baths was studied in [16]. The

complete hamiltonian is given by :

H=HS+HB+HSB

HSB =

j=1

λkXjk

(1)

Note that the coupling term hj(t) = PkλkXjk is

arXiv:2210.01590v1 [cond-mat.dis-nn] 4 Oct 2022

equivalent to a time-dependent magnetic ﬁeld inter-

acting with the spins. The time-independent limit,

hj(t) = hj, corresponds to a quenched disordered

magnetic ﬁeld. Recent numerical simulations seem

to suggest that quenched disorder can induce a ﬁnite

temperature MBL transition in this model [17]. At

zero temperature instead, a localisation transition

surely occurs and can be studied using Bosonization

[18] and powerful simulation techniques [19]. Here

we replace the quenched disorder by an annealed

disorder produced by the bath and investigate the

possibility of a zero temperature localisation tran-

sition. To fully characterize the bath, we need to

specify the low-frequency behaviour of the spectral

function, deﬁned as :

J(Ω) = π

(λ2

k/mkΩk)δ(Ω −Ωk) (2)

In general, one has J(Ω) = παΩsfor Ω ∈(0,ΩD).

Here αdenotes the eﬀective coupling strength with

the bath, the cut-oﬀ ΩDis the Debye frequency and

ssets the nature of the bath. For our study we take

s= 1, which corresponds to an ohmic bath. A sim-

ilar model was already studied by Bosonisation [20]

and Monte-Carlo techniques [21] in a diﬀerent con-

text. However, its phase diagram remains contro-

versial and it is not clear how many phases appear

varying the parameter α. Here, we introduce a novel

approach, which simulates directly the bosonised ac-

tion and allows us to reach large system sizes. Our

results show a simple scenario of two phases with a

KT transition between them. Increasing α, a dis-

sipative phase with suppressed transport takes over

the LL phase.

Bosonised action: We map the chain with periodic

condition into a 1D fermionic system using Jordan-

Wigner transformation. For Jz= 0, we recover the

free-fermion problem that can be diagonised in the

momentum space q= 2πl/(Na) with aas lattice

spacing and l∈(−N/2, N/2). The Fermi momen-

tum depends on the total magnetisation Mof the

spin chain, namely qF=π(N−M)/(2Na). Two

cases should be distinguished : In the zero sector of

magnetisation, qFis commensurate with the lattice

space, while it is incommensurate for the non-zero

magnetisation sector. Here we focus on the incom-

mensurate case. Bosonisation allows us to include

both the interaction and the bath by linearizing the

spectrum around qF, which corresponds to the low-

energy physics of the system. Using the standard

techniques ([22], Sec. 1), we can map the spin-

chain problem into the following ﬁeld theory action

S=SLL +Sint, where:

SLL =1

2πK Zdxdτ 1

u∂τφ(x, τ)2+u∂xφ(x, τ)2

(3)

Sint =−α

4π2Zdxdτdτ0cos 2φ(x, τ)−φ(x, τ 0)

|τ−τ0|2

(4)

Here φ(x, τ) is a two-dimensional ﬁeld living in the

physical space x∈(0, L) and in imaginary time

τ∈(0, β), where βis the inverse temperature. Ther-

modynamic quantities of the spin chain can be ex-

pressed in terms of the correlation functions of the

ﬁeld φ. In particular, the propagator G(q, ωn) =

hφ(q, ωn)φ(−q, −ωn)ican be related to the suscep-

tibility χand spin stiﬀness ρsby the two equations:

χ= lim

q→0lim

ωn→0

π2G(q, ωn) (5)

ρs= lim

ωn→0lim

q→0

ω2

π2G(q, ωn) (6)

Here ωn= 2πn/β,n∈(−β/2, β/2) are the Mat-

subara frequencies. When α= 0, we recover the

LL action which corresponds to an isolated XXZ

spin chain and the parameters uand Kcan be di-

rectly related to Jzand Jxy either by Bethe Ansatz

or by bosonisation ([22], Sec. 1). In this phase,

GLL(q, ωn) = πK/(ω2

n/u +uq2), and hence χ=

K/(uπ) and ρs=uK/π.

The bath introduces a long-range cosine interac-

tion in τdirection only and the strength of this po-

tential is controlled by the parameter α. A pertur-

bative RG study [20] shows that for K < Kc= 0.5,

the cosine term is relevant and the LL phase is de-

stroyed, whereas for K > Kcand small α, the sys-

tem stays in LL phase but with renormalised LL

parameters Krand ur. For K&Kc, the transi-

tion is of KT type: The critical point αc(K) is still

LL with Kr=Kc= 0.5. The nature of the dis-

sipative phase is not clear : For moderate Kand

very large α, the action should be gapless and har-

monic, obtained by the quadratic expansion of the

cosine term. For KKc, a large-N argument sug-

gests the existence of a gapped disordered phase.

Monte-Carlo simulations [21] were performed on the

1D hard-core bosonic chain, which can be mapped

to free fermions (K= 1). Increasing α, they found

that χincreases and at αc, the system undergoes a

FIG. 2. Calculation of diﬀerent quantities for K= 0.75 that characterizes LL (α= 1, top row) and dissipative phase

(α= 8, bottom row). Blue and red points correspond to L=β= 384 and L=β= 128, respectively. Left : Due to

symmetry, πχ =Kr/uris equal to K/u = 0.75 for all values of αand all lengthscales. Middle : For α= 1, ωnC(ωn)

saturates to Kr/2=0.349 as ωn→0; whereas for α= 8, √ωnC(ωn) saturates to [Krπ/(8αrur)]1/2= 0.392. The

other ﬁtting constants are a1= 0.2493 and a2= 3.572. Right : For α= 1, hcos(φ)idecays as a power law, which

allows us to extract Kr= 0.694, consistent with the ﬁt of ωnC(ωn). For α= 8 it saturates to a constant, as predicted

by the variational ansatz (the ﬁt gives c1= 0.62, c2= 0.603 and c3=−0.531).

continuous second-order phase transiton with van-

ishing ρs. Below, we propose a simple scenario able

to conciliate the puzzle of contradictory results.

Methods : To make progress, we focus on the in-

teracting bosonised action. On one side, we com-

pute the correlation functions numerically by gen-

erating equilibrated conﬁgurations from the action

with the help of Langevin dynamics. On the other

hand, to understand the properties of the dissipative

phase, we improve the harmonic expansion proposed

in [20] using a variational approach where we obtain

an eﬀective quadratic action by minimizing the vari-

ational free energy ([22], Sec. 2). The propagator of

this action is given by:

G−1

var(q, ωn) = urq2

2πKr

+αr

π2|ωn|+a1|ωn|3

2+a2ω2

(7)

The macroscopic behaviour of this phase depends

only on the two parameters ur/Krand αr. The pa-

rameters a1and a2are introduced to account for

ﬁnite size eﬀects. From the analysis of our result,

we will show that by varying α, the long-distance

properties are always captured either by the LL or

by the variational propagator with renormalized pa-

rameters ur, Krand αr.

Results : In the following, we present our re-

sults for the correlation functions of the action S=

Sint +SLL with u= 1, K = 0.75 and diﬀerent α.

For our simulations, we set β=L. The ﬁrst ob-

servation is that the action of Eq.(4) is invariant

under tilt transformation([22], Sec. 5). As a conse-

quence, χis not aﬀected by the presence of Sint. We

measure Kr/urboth at low and high αas shown

in ﬁg. 4left. Note that the susceptibility corre-

sponds to q→0 limit, but due to the symme-

try, Kr/uris invariant at all length scales and all

values of α. We conclude that Kr/ur=K/u for

all values of α. In principle, we should now mea-

sure the stiﬀness, but if we focus only on the limit

q→0, it strongly ﬂuctuates. Instead we introduce a

function C(ωn) = (1/πL)Pqφ(q, ωn)

2, which for

small ωnbehaves as Kr/2ωnin the LL phase and

as pKr/8πur(αrωn/π2+a1ω3/2

n+a2ω2

n)−1/2ac-

cording to the variational prediction. From ﬁg. 4

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BathinducedphasetransitioninaLuttingerliquidSaptarshiMajumdar,1LauraFoini,2ThierryGiamarchi,3andAlbertoRosso11UniversiteParisSaclay,CNRS,LPTMS,91405,Orsay,France2IPhT,CNRS,CEA,UniversiteParisSaclay,91191Gif-sur-Yvette,France3DepartmentofQuantumMatterPhysics,UniversityofGeneva,24QuaiErnest-Ansermet...

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Bath induced phase transition in a Luttinger liquid Saptarshi Majumdar1Laura Foini2Thierry Giamarchi3and Alberto Rosso1 1Universit e Paris Saclay CNRSLPTMS 91405 Orsay France

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