Analytical results for the entanglement dynamics of disjoint blocks in the XY spin chain Gilles Parez1and Riccarda Bonsignori2

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Analytical results for the entanglement dynamics of

disjoint blocks in the XY spin chain

Gilles Parez ∗1and Riccarda Bonsignori2

1Centre de Recherches Mathématiques (CRM), Université de Montréal, P.O. Box 6128,

Centre-ville Station, Montréal (Québec), H3C 3J7, Canada

2Ruđer Bošković Institute, Bijenička cesta 54, 10000 Zagreb, Croatia

October 10, 2022

Abstract

The study of the dynamics of entanglement measures after a quench has become a very active

area of research in the last two decades, motivated by the development of experimental techniques.

However, exact results in this context are available in only very few cases. In this work, we present

the proof of the quasiparticle picture for the dynamics of entanglement entropies for two disjoint

blocks in the XY chain after a quantum quench. As a byproduct, we also prove the quasiparticle

conjecture for the mutual information in that model. Our calculations generalize those presented

in [M. Fagotti, P. Calabrese, Phys. Rev. A 78, 010306 (2008)] to the case where the correlation

matrix is a block-Toeplitz matrix, and rely on the multidimensional stationary phase approximation

in the scaling limit. We also test the quasiparticle predictions against exact numerical calculations,

and ﬁnd excellent agreement. In the case of three blocks, we show that the tripartite information

vanishes when at least two blocks are adjacent.

∗gilles.parez@umontreal.ca

arXiv:2210.03637v2 [cond-mat.stat-mech] 10 Jan 2023

Contents

1 Introduction 2

2 The XY spin chain and entanglement dynamics 4

2.1 ThequenchedXYspinchain ................................. 4

2.2 Entanglement entropies and mutual information for disjoint intervals . . . . . . . . . . . 6

3 Quasiparticle conjecture for free fermions 7

3.1 Singleinterval.......................................... 7

3.2 Twodisjointintervals...................................... 8

3.3 Threeintervals ......................................... 8

3.4 Conjecture for the Rényi entropies of disjoint blocks in the XY chain . . . . . . . . . . . 9

3.5 Numericalchecks........................................ 9

4 Proof of the quasiparticle conjecture for two disjoint blocks 11

4.1 Calculation of T2(t)....................................... 11

4.2 Calculation of T2j(t)...................................... 14

4.3 Final resummation and conclusion of the proof . . . . . . . . . . . . . . . . . . . . . . . 17

5 Conclusion 18

1 Introduction

Understanding the nonequilibrium dynamics of quantum many-body systems is a question that has

been at the heart of quantum mechanics since its early days [1]. For the last twenty years, this ﬁeld of

research has experienced a renewed interest, due to groundbreaking cold-atom and ion-trap experiments

that managed to simulate the unitary evolution of closed systems on long time scales [2–6], as well as

an intense theoretical activity [7]. A key aspect is that quantum entanglement and its dynamics out of

equilibrium play a large role in our understanding of fundamental problems, such as the equilibration

and thermalisation of isolated many-body systems [7–12] and the emergence of thermodynamics [13–16].

Moreover, several experiments managed to measure entanglement-related quantities [17–23].

The simplest protocol to drive a quantum system out of equilibrium is known as a quantum

quench [24,25]. Consider a system described by a Hamiltonian H(λ), where λis a set of external

parameters. In the quench protocol, the system is prepared in the groundstate of some initial Hamilto-

nian H(λ0), and at time t= 0 the parameters are quenched from λ0to λ1such that [H(λ0), H(λ1)] 6= 0.

For t > 0, the system evolves unitarily under the action of the post-quenched Hamiltonian H(λ1). Be-

cause the two Hamiltonians do not commute, the subsequent dynamics is non-trivial. In particular, the

investigation of entanglement dynamics after a quantum quench received considerable attention over

the last two decades [26–29].

For a quantum many-body system in a pure state |ψi, one is typically interested in the entanglement

between a subsystem Aand its complement, traditionally denoted B. The so-called Rényi entropies

are deﬁned from the reduced density matrix ρA=TrB(|ψihψ|)as

n=1

1−nlog Trρn

A.(1.1)

The limit n→1of the Rényi entropies yields the celebrated entanglement entropy [30]. It is the

von Neumann entropy of the reduced density matrix of subsystem A, namely

1≡lim

n→1SA

n=−Tr(ρAlog ρA).(1.2)

Rényi entropies quantify the entanglement between Aand B, irrespective of the geometry of A.

In many cases considered in the literature, Ais a connected spatial region, such as a segment in

a one-dimensional chain. However, the case where Aconsists of disjoints blocks has also generated

interest [31–36]. Let us consider the situation where A=A1∪A2consists of two blocks A1and A2.

From the entanglement entropies of the two blocks SA1∪A2

n, one deﬁnes the mutual information as

IA1:A2

1=SA1

1+SA2

1−SA1∪A2

1,(1.3)

as well as the Rényi mutual information

IA1:A2

n=SA1

n+SA2

n−SA1∪A2

n.(1.4)

Here, SX

n, X =A1, A2, A1∪A2, is the Rényi entropy from Eq. (1.1) where the reduced density matrix

is obtained by tracing out the degrees of freedom of the complement of Xfrom the total pure-state

density matrix ρ=|ψihψ|. The mutual information is not per se an entanglement measure between A1

and A2because it also contains classical correlations [37,38], and a proper measure of entanglement in

that context is instead the entanglement negativity [39]. However, both the mutual information and

the negativity share important properties, both in [40] and out of equilibrium [41–43], and we relegate

the study of the negativity to forthcoming investigations.

For a tripartite system A=A1∪A2∪A3, a relevant quantity that characterizes multipartite

entanglement is the tripartite information IA1:A2:A3

n. It is deﬁned as [44]

IA1:A2:A3

n=IA1:A2

n+IA1:A3

n−IA1:(A2∪A3)

n(1.5)

and measures the extensiveness of the mutual information. In particular, a negative tripartite informa-

tion indicates multipartite entanglement, and it is related to quantum chaos and scrambling [45–47]. In

the context of two-dimensional systems where A1, A2, A3are adjacent regions, the tripartite informa-

tion coincides with the celebrated topological entanglement entropy [48]. Very recently, the tripartite

information was investigated in the context monitored spin chains [49] and quantum quenches [50].

While it vanishes at all times for many quench protocols, the authors of Ref. [50] show that in some

cases its dynamics yields universal information about the system.

In integrable models, the quasiparticle picture [27–29,51,52] describes the growth of entanglement

in time after a global quench, in terms of ballistic propagation of pairs of quasiparticles of opposite

momentum, that spread entanglement and correlations through the system. In the case of disjoint

subsystems, the quasiparticle picture can be adapted to describe the dynamics of the mutual informa-

tion [41] and the entanglement negativity [53]. The quasiparticle picture also describes the growth of

symmetry-resolved entanglement [54–56], and it has recently been generalized to the case of dissipative

free fermionic and bosonic systems [57–61].

While the quasiparticle picture provides impressive quantitative results that have been checked

extensively through numerical investigations, ab initio and analytical results for the entanglement

dynamics after a quench remain scarce in the literature. In a seminal paper, Fagotti and Calabrese

computed the complete time dependence of the entanglement entropy of a single interval after a quench

in the XY chain [26]. Their derivation used the Toeplitz-matrix representation of the correlation matrix

and multidimensional phase methods. However, a similar derivation for the case of disjoint blocks is still

lacking. We also mention recent exact results for the negativity dynamics in dissipative models [60,61]

and quantum circuits [43]

In this paper, we provide the analytical derivation of the quasiparticle picture for the entanglement

entropies of two disjoint blocks in the XY chain in presence of transverse ﬁeld, generalizing the results

of Ref. [26]. As a byproduct, this allows us to prove the quasiparticle picture result for the mutual infor-

mation. Moreover, we show that the tripartite information vanishes at all times for the quench protocol

we consider, and argue more generally that the quasiparticle picture implies a vanishing tripartite in-

formation. Finally, we mention that these calculations for the entropy dynamics of disjoint blocks have

already been advertised and used in the context of symmetry-resolved entanglement measures [55,56].

This paper is organized as follows. In Sec. 2we introduce the XY model and express the related

entanglement measures in terms of the two-point correlation matrix after a quench. We discuss the

quasiparticle conjecture for the entanglement dynamics in Sec. 3, and give the analytical proof for

the case of two disjoint blocks in Sec. 4. We conclude in Sec. 5with a summary of the results and

byproducts of our proof, and discuss further research directions.

2 The XY spin chain and entanglement dynamics

In this section we review the XY spin chain and its diagonalization. We also specify the quench protocol

under consideration and recall the deﬁnition of the Rényi entropies and mutual information in terms

of the two-point correlation matrix. Finally, we review the quasiparticle picture for the entanglement

dynamics, both in the case where the subsystem consists of one or multiple blocks of contiguous sites.

2.1 The quenched XY spin chain

We consider the spin-1/2XY spin chain in a transverse magnetic ﬁeld with periodic boundary condi-

tions. The Hamiltonian is

H(γ, h) = −

j=1 1 + γ

4σx

jσx

j+1 +1−γ

4σy

jσy

j+1 +h

2σz

j,(2.1)

where γis the anisotropy parameter, his the external magnetic ﬁeld, Lis the size of the system and

σα

jare the Pauli matrices at site j. The model can be solved via the Jordan-Wigner transformation,

which maps the system into a free Fermi gas in the presence of an external potential [62].

We consider the following quench protocol. At time t= 0, the system is prepared in the groundstate

|ψ0iof the Hamiltonian H(γ0, h0)with some initial anisotropy and magnetic ﬁeld γ0and h0. The

parameters are then quenched to the values γand h, and for t > 0the time-evolved state is

|ψ(t)i=e−itH(γ,h)|ψ0i.(2.2)

In order to study the time evolution of the entanglement measures, it is useful to introduce the

Majorana operators

a2j−1= j−1

k=1

σz

k!σx

j, a2j= j−1

k=1

σz

k!σy

j,(2.3)

that satisfy

a2j−1=cj+c†

j, a2j= i(cj−c†

j),{am, an}= 2δm,n,(2.4)

where cj, c†

jare the canonical spinless fermionic operators. For a subsystem Aconsisting of `contiguous

spins, the time-dependent correlation matrix ΓA(t)is a 2`×2`matrix built from 2×2blocks [52] as

hψ(t)|a2m−1

a2m·a2n−1a2n|ψ(t)i=δm,n + i[ΓA(t)]m,n,16m, n 6`. (2.5)

More explicitly, the correlation matrix is a block-Toeplitz matrix

ΓA(t) = 





Π0Π1· · · Π`−1

Π−1Π0

.....

Π1−`· · · · · · Π0







,Πj=−fjgj

−g−jfj.(2.6)

In the large-Llimit, fjand gjread [26]

fj= i Zπ

−π

2πe−ikj sin ∆ksin(2kt),

gj=Zπ

−π

2πe−ikj e−iθk(cos ∆k+ i sin ∆kcos(2kt)) ,

(2.7)

with

2

k= (h−cos k)2+γ2sin2k, 2

0,k = (h0−cos k)2+γ2

0sin2k,

e−iθk=cos k−h−iγsin k

k

cos ∆k=hh0−cos k(h+h0) + cos2k+γγ0sin2k

k0,k

sin ∆k=−sin kγh0−γ0h−cos k(γ−γ0)

k0,k

(2.8)

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摘要：

AnalyticalresultsfortheentanglementdynamicsofdisjointblocksintheXYspinchainGillesParez*1andRiccardaBonsignori21CentredeRecherchesMathématiques(CRM),UniversitédeMontréal,P.O.Box6128,Centre-villeStation,Montréal(Québec),H3C3J7,Canada2RuerBo²kovi¢Institute,Bijeni£kacesta54,10000Zagreb,CroatiaOctober10...

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Analytical results for the entanglement dynamics of disjoint blocks in the XY spin chain Gilles Parez1and Riccarda Bonsignori2

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