Eigenstate thermalization hypothesis in two-dimensional XXZ model with or without SU2 symmetry Jae Dong Noh

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Eigenstate thermalization hypothesis in two-dimensional XXZ model with or without

SU(2) symmetry

Jae Dong Noh

Department of Physics, University of Seoul, Seoul 02504, Korea

(Dated: January 23, 2023)

We investigate the eigenstate thermalization properties of the spin-1/2 XXZ model in two-

dimensional rectangular lattices of size L1×L2under periodic boundary conditions. Exploiting

the symmetry property, we can perform an exact diagonalization study of the energy eigenvalues up

to system size 4×7and of the energy eigenstates up to 4×6. Numerical analysis of the Hamiltonian

eigenvalue spectrum and matrix elements of an observable in the Hamiltonian eigenstate basis sup-

ports that the two-dimensional XXZ model follows the eigenstate thermalization hypothesis. When

the spin interaction is isotropic the XXZ model Hamiltonian conserves the total spin and has SU(2)

symmetry. We show that the eigenstate thermalization hypothesis is still valid within each subspace

where the total spin is a good quantum number.

I. INTRODUCTION

The eigenstate thermalization hypothesis (ETH) ex-

plains the mechanism for thermalization of isolated quan-

tum systems [1, 2]. The ETH guarantees that a quan-

tum mechanical expectation value of a local observable

relaxes to the equilibrium ensemble averaged value and

ﬂuctuations in the steady state satisfy the ﬂuctuation

dissipation theorem (see Ref. [3] and references therein).

Numerous studies have been performed to test valid-

ity of the ETH since the early work of Ref. [4]. The

spin-1/2 XXZ model [5–14] and the quantum Ising spin

model [7, 15–18] are the paradigmatic model systems for

ETH study. The XXZ model is useful since it describes a

hardcore boson system which is relevant to experimental

ultracold atom systems [19–23]. Moreover, integrability

in these models can be tuned easily in one-dimensional

lattices. A thermal/nonthermal behavior and a crossover

between them have been studied comprehensively using

the model systems [5–7, 11, 24–30].

The ETH has been examined mostly in one-

dimensional spin systems, and there are only a few works

for two-dimensional systems [4, 15–17, 31]. In this work,

we study the eigenstate thermalization property of the

spin-1/2 XXZ model in two-dimensional rectangular lat-

tices. In comparison with the Ising spin systems [15–

17], the XXZ model is characterized by the conservation

of the magnetization in the zdirection. Furthermore,

it possesses the SU(2) symmetry when the spin interac-

tion is isotropic [32]. The SU(2) symmetry conserves the

magnetization in all directions, but the total spin oper-

ators in diﬀerent directions do not commute with each

other. Such a non-Abelian symmetry has a nontrivial

eﬀect on many-body localization [33, 34], quantum ther-

malization [32, 35, 36], and entanglement entropy [37].

This paper is organized as follows. In Sec. II, we intro-

duce the XXZ Hamiltonian with nearest and next near-

est neighbor interactions in two-dimensional rectangular

lattices. The symmetry property of the Hamiltonian is

summarized. In Secs. III and IV, we present results of

a numerical exact diagonalization study. First, we will

show in Sec. III that the ETH is valid in the XXZ model

without SU(2) symmetry. In Sec. IV, we proceed to show

that the SU(2) symmetric XXZ model also satisﬁes the

ETH in each SU(2) subsector. Our work extends the

validity of the ETH to the two-dimensional XXZ model.

II. TWO-DIMENSIONAL XXZ MODEL

We consider the spin-1/2 XXZ model on a two-

dimensional rectangular lattice. The Pauli spin σr=

(σx

r, σy

r, σz

r)resides on a lattice site rand the Hamilto-

nian is given by

H=λX

hr,r0i

h(σr,σr0) + (1 −λ)X

[r,r0]

h(σr,σr0),(1)

where hr,r0iand [r,r0]denote the pair of nearest neigh-

bor (nn) sites, connected by solid lines in Fig. 1(a), and

of next nearest neighbor (nnn) sites, connected by dotted

lines in Fig. 1(a), respectively, and h(σ,σr0)denotes the

XXZ coupling given by

h(σr,σr0) = −J

2(σx

rσx

r0+σy

rσy

r0+ ∆σz

rσz

r0).(2)

The model is deﬁned by three parameters J,∆, and λ:

λcontrols the relative strength of the nn and nnn cou-

plings, ∆is an anisotropy parameter, and Jsets the over-

all energy scale which will be kept to be 1. We assume

periodic boundary conditions, σr+L1e1=σr+L2e2=σr

where e1and e2are the unit vectors in the horizontal

and vertical directions, respectively [see Fig. 1(a)]. The

XXZ coupling with J= 1 can be rewritten as

h(σr,σr0) = −σ+

rσ−

r0+σ−

rσ+

r0+∆

2σz

rσz

r0(3)

with the raising and lowering operators σ±≡(σx±

iσy)/2. Throughout the paper, we will set ~= 1. The

total number of sites will be denoted by N=L1L2. In

this work, we only consider the lattices with even N.

arXiv:2210.14589v2 [cond-mat.stat-mech] 20 Jan 2023

0L1−1

L1−1

(a)

T1,2 X

R1,2 Sz

(Sz)′=0

(Tα)′=±1

(b)

FIG. 1. (a) Rectangular lattice of size L1×L2under pe-

riodic boundary conditions in the horizontal (e1) and verti-

cal (e2) directions. (b) Commutation relations among the

XXZ Hamiltonian and symmetry operators. Mutually com-

muting operators are connected with a solid line. A dashed

line connect operators which are commuting only within the

subspace with speciﬁc quantum numbers of the symmetry op-

erator. The Hamiltonian and S2, connected by a dashed-

dotted line, commutes only when ∆ = 1.

The XXZ Hamiltonian commutes with several symme-

try operators. First, the Hamiltonian commutes with the

magnetization operator in the zdirection

Sz=1

σz

r.(4)

The Hamiltonian also commutes with the shift operator

Tαwhich shifts a spin state by the unit distance in the

direction eαwith α= 1,2:

T−1

ασrTα=σr+eα(α= 1,2).(5)

The system has the spatial inversion symmetry so that

Hcommutes with Rαwhich maps a site r= (x, y)to

(−x, y)for α= 1 or to (x, −y)for α= 2. Finally, the

system is invariant under the spin ﬂip σz→ −σzwhich

is generated by the symmetry operator X=Qrσx

The commutation relations are summarized by a di-

agram in Fig. 1(b). (A similar diagram for the one-

dimensional system is found in Ref. [38].) Note that

[X, Sz]6= 0 and [Rα, Tα]6= 0 in general. Thus, one can-

not construct a simultaneous basis set for all the sym-

metry operators. On the other hand, one can show that

[Rα, Tα]|ψi= 0 if a state |ψiis an eigenstate of Tαof

eigenvalue (Tα)0=±1. It implies that the two opera-

tors commute within the subspace of the eigenstates of

Tαwith eigenvalues ±1, Likewise, [X, Sz] = 0 within

the subspace of the eigenstates of Szwith eigenvalue

(Sz)0= 0. In this work, we focus on the symmetry sec-

tor consisting of the eigenvalues of the symmetry oper-

ators with the eigenvalues (Tα)0= (Rα)0= (X)0= 1

and (Sz)0= 0, which will be referred to as the maximum

symmetry sector (MSS).

When the spin-spin interaction is isotropic (∆ = 1),

the Hamiltonian is invariant under spin rotation [SU(2)

symmetry]. Consequently, each component of the to-

tal spin S=1

2Prσris conserved and S2=S·S

becomes the symmetry operator commuting with the

Hamiltonian and all the other symmetry operators. The

maximum symmetry sector is then further decomposed

into subsectors characterized with the eigenvalue of S2,

(S2)0=s(s+ 1) with integer s. The SU(2) symmetry

will be investigated in detail in Sec. IV.

We have performed the exact diagonalization study.

The basis states, which are simultaneous eigenstates of

the symmetry operators appearing in Fig. 1(b) in the

MSS, can be easily constructed using the methods sum-

marized in Refs. [38–43]. The Hilbert space dimension-

alities of the MSS are D= 26,1392,15578, and 183926

when L1×L2= 4 ×3,4×5,4×6,4×7, respectively.

When L1=L2, the system has an addition symmetry

under the spatial rotation by a multiple of π/2. It will

not be addressed since we only consider the lattices with

L16=L2.

An energy eigenstate and a corresponding eigenvalue

of Hin the MSS will be denoted as |Eniand En, re-

spectively, where the quantum number n= 0,··· , D −1

is assigned in ascending order of the energy eigenvalue.

We will study the Hamiltonian spectrum and the matrix

elements of the observable,

OZ=1

α=1,2

σz

rσz

r+eα

OJ=1

α=1,2σ+

rσ−

r+eα+σ+

r+eασ−

r

OP=1

r,r0

σ+

rσ−

OF=1

σz

p1σz

p2σz

p3σz

p4,

(6)

which measure the nearest neighbor two-spins cor-

relation, nearest neighbor hopping amplitude, zero-

momentum distribution function, and the plaquette in-

teraction of four spins. The sum in OFis over all pla-

quettes and σpi(i= 1,2,3,4) refers to four spins around

a plaquette p.

III. NUMERICAL STUDY OF EIGENSTATE

THERMALIZATION HYPOTHESIS

A. Ratio of consecutive energy gaps

As a signature for the quantum chaos, we investigate

the statistics of the ratio of consecutive energy gaps [44,

45]:

rn= min En+1 −En

En−En−1

,En−En−1

En+1 −En.(7)

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Eigenstatethermalizationhypothesisintwo-dimensionalXXZmodelwithorwithoutSU(2)symmetryJaeDongNohDepartmentofPhysics,UniversityofSeoul,Seoul02504,Korea(Dated:January23,2023)Weinvestigatetheeigenstatethermalizationpropertiesofthespin-1/2XXZmodelintwo-dimensionalrectangularlatticesofsizeL1L2underperiod...

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Eigenstate thermalization hypothesis in two-dimensional XXZ model with or without SU2 symmetry Jae Dong Noh

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