Domain wall melting across a defect Luca Capizzi1 Stefano Scopa1 Federico Rottoli1andPasquale Calabrese12 1SISSA and INFN via Bonomea 265 34136 Trieste Italy

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Domain wall melting across a defect

Luca Capizzi1, Stefano Scopa1, Federico Rottoli1and Pasquale Calabrese1,2

1SISSA and INFN, via Bonomea 265, 34136 Trieste, Italy

2International Centre for Theoretical Physics (ICTP), I-34151, Trieste, Italy

Abstract

– We study the melting of a domain wall in a free-fermionic chain with a localised

impurity. We ﬁnd that the defect enhances quantum correlations in such a way that even the

smallest scatterer leads to a linear growth of the entanglement entropy contrasting the logarithmic

behaviour in the clean system. Exploiting the hydrodynamic approach and the quasiparticle

picture, we provide exact predictions for the evolution of the entanglement entropy for arbitrary

bipartitions. In particular, the steady production of pairs at the defect gives rise to non-local

correlations among distant points. We also characterise the subleading logarithmic corrections,

highlighting some universal features.

Introduction

– A single localised impurity or defect can

alter the global structure of a many-body quantum system,

as well known from the textbook examples of Anderson

orthogonality catastrophe [1] and the Kane-Fisher model

[2,3]. In the latter, it has been shown that for repulsive

interactions, the electrons are completely reﬂected by even

the smallest scatterer, leading to a truly insulating weak

link disconnecting the two halves. Conversely for attractive

bulk interactions, the weak link is irrelevant, i.e., it is

washed away at large scales. As a consequence free fermions

represent the most interesting system in which the defect

is marginal and there is a line of ﬁxed points characterised

by the the defect strength [4,5].

In recent years, the physics of impurities in one-

dimensional (1D) free-fermionic systems has been investi-

gated a lot through the lens of entanglement. The marginal-

ity of the defect is reﬂected into a logarithmic scaling of

the entanglement entropy with a prefactor that depends

continuously on the defect strength [6

–

20]. Overall, thanks

to all these studies nowadays we have a rather complete

understanding of the physics of defects in equilibrium free

fermionic systems. The same is deﬁnitively not true when

the free fermionic chain is driven out of equilibrium; in

fact, in spite of several works about the non-equilibrium

behaviour across one defect (see, e.g., Refs. [21

–

31]), a

complete understanding is still far because of the many

diﬀerent ways of driving a system away from equilibrium.

In this manuscript, we prepare an initial state with a

domain wall localised at the defect and we let it melt.

Without the defect, this is a protocol that has been studied

intensively in the free fermionic literature [32

–

45] and it

represents a case study for the application of hydrodynam-

ics to non-equilibrium quantum systems (recently adapted

also to interacting integrable models [46,47] with the gen-

eralised hydrodynamics formalism [48,49]). Anyhow, the

presence of the defect is expected to alter dramatically

the evolution at a qualitative level. The density, the cur-

rents, and other local quantities have been characterised in

Ref. [50] where the emergence of a local non-equilibrium

stationary state (NESS) has been rigorously established.

However, little is known for the entanglement entropy,

whose behaviour is aﬀected, as any other non-local ob-

servable, by non-local correlations generated by the defect.

Some lattice results have been derived for the domain wall

melting with defect in Ref. [21], and another important

step forward has been done by Fraenkel and Goldstein

[51,52] in a slightly diﬀerent context, but a general scheme

to describe non-local correlations is still missing. In this

regard, it is very natural to wonder whether even a small

defect alters substantially the domain wall melting. As we

shall see, this is the case.

The model and the quench protocol

– We consider

a 1D chain of free spinless fermions with 2

sites and

with nearest-neighbour hopping with a defect of strength

located at the center of the system. The Hamiltonian is

i,j=−N+1

hi,j ˆc†

iˆcj(1)

with

hi,j =−1

2(δi,j+1 +δi+1,j ),∀i, j 6= 0,1 (2)

p-1

arXiv:2210.02162v2 [cond-mat.stat-mech] 3 Feb 2023

L. Capizzi et al.

Figure 1: Illustration of the evolution of the Fermi occupation

function

n(λ)

(

x, k

) in the presence of the defect. The light-grey

area is the initial occupation

(7)

, while the colored regions

correspond to the time-evolved one.

and defect

h0,1=h1,0=−λ

2, h0,0=−h1,1=1

2p1−λ2.(3)

Here

ˆc†

ˆcj

are the creation and annihilation operators of

spinless fermions at site

, satisfying

{ˆc†

j,ˆci}

δij

. For

= 1 the Hamiltonian

(1)

reduces to a standard hopping

model. The system is initially prepared in the product

state

|ψ0i=

j=−N+1 |1ij

j=1 |0ij,(4)

where

|ζ= 0,1ij

are the eigenstates of the fermionic

number operator

ˆc†

jˆcj

with eigenvalues

= 0

1. In the spin

language, the initial state

(4)

corresponds to the domain

wall

|ψ0i

|. . . ↑↑↓↓ . . .i

. Thereafter, our discussions

can be extended to the XX spin chain with defect [8],

obtained from the model

(1)

through a Jordan-Wigner

transformation. For

t >

0, the state

(4)

is unitarily

evolved with Hamiltonian (1), |ψti=e−itˆ

H|ψ0i.

The structure of the defect

(3)

does not spoil the exact

solvability of the free fermionic model, which has spectrum

ωq

−cos

(

πq/

) (

= 1

,...,

), for any

value of

λ∈

1] [10,26]. Moreover, the eigenstates of

can be related to the eigenstates Ψ

q,2N

(

) =

sin

(

kqj

)

/√N

of (1) in the absence of defect (λ= 1) as [26]

Ψdef

q,2N(j)= Θ(−j)α+

q(λ)Ψq,2N+ Θ(j)α−

q(λ)Ψq,2N,(5)

with Θ(

) the Heaviside step function and coeﬃcients

α±

(

) = [1

(

−

q√1−λ2

]

1/2

. For

N→ ∞

, this eigen-

problem reduces to a scattering of plane waves across a

localised defect, i.e.,

Ψdef

k,∞(j)∝Θ(j)λ eikj + Θ(−j)(p1−λ2e−ikj +eikj ) (6)

with transmission probability

(

)

≡λ2

and reﬂection

probability

(

)

≡

−λ2

. These parameters do not

depend on the momentum kof the scattered particle and

so the defect

(3)

is also known as conformal defect [8,10,21].

Hydrodynamic limit

– Exact asymptotic results for

the charges proﬁles can be obtained in the hydrodynamic

Figure 2: Fermionic density as function of

x/t

for diﬀerent

values of

and

. Symbols show the numerical data obtained

with exact lattice calculations with 400 sites while the full lines

are given by Eq. (11).

limit

N→ ∞

j→ ∞

t→ ∞

at ﬁxed

j/t

. The lattice

index

and the quantised momenta

are replaced by

continuous variables for the position

ja ∈R

(

is the

lattice spacing) and for the momenta

−π≤k≤π

. In such

scaling limit, the essential information on the initial state

(4) is retained by the local fermionic occupation

n0(x, k) = (1,if x≤0 and −π≤k≤π;

0,otherwise.(7)

In the absence of defect (

= 1), the evolution of the

occupation function is given by the Euler equation

(∂t+ sin k ∂x)nt(x, k)=0,(8)

which can be simply solved as

nt(x, k) = n0(x−tsin k, k),(9)

i.e. it stays equal to 0 and 1 and only the Fermi contour

separating the two values evolves. Intuitively, this solu-

tion encodes the fact that each non-interacting particle

moves along the ballistic trajectory with constant velocity

(

) =

sin k

. Instead, for

λ6

= 1, a particle of momentum

k >

0 traveling from

x <

0 is scattered by the defect in

such a way that it is reﬂected with probability

(

) and

transmitted with probability

(

). Accordingly, the time-

evolved occupation function in the presence of the defect

takes the form [50]

n(λ)

t(x, k) = λ2Θ(x)nt(x, k)+

+ Θ(−x)(1 −λ2)nt(−x, −k) + nt(x, k),(10)

as illustrated in Fig. 1. In our notations

n(λ≡1)

Eq.

(9)

. The occupation function

(10)

gives us access to

the asymptotic proﬁles of conserved charges as elementary

integrals over the modes

, properly weighted with the

single-particle eigenvalue of the associated charge [32,33].

For instance, the particle density proﬁle for 0 < x ≤tis

n(λ)

t(x) = Zπ

−π

dk n(λ)

t(0 < x ≤t, k)

2π=λ2arccos(x/t)

π.

(11)

p-2

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DomainwallmeltingacrossadefectLucaCapizzi1,StefanoScopa1,FedericoRottoli1andPasqualeCalabrese1;21SISSAandINFN,viaBonomea265,34136Trieste,Italy2InternationalCentreforTheoreticalPhysics(ICTP),I-34151,Trieste,ItalyAbstract{Westudythemeltingofadomainwallinafree-fermionicchainwithalocalisedimpurity.Wend...

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Domain wall melting across a defect Luca Capizzi1 Stefano Scopa1 Federico Rottoli1andPasquale Calabrese12 1SISSA and INFN via Bonomea 265 34136 Trieste Italy

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