Local Hilbert space fragmentation and weak thermalization in Bose-Hubbard diamond necklaces Eloi Nicolau1Anselmo M. Marques2Jordi Mompart1Ver onica Ahunger1and Ricardo G. Dias2

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Local Hilbert space fragmentation and weak thermalization

in Bose-Hubbard diamond necklaces

Eloi Nicolau,1Anselmo M. Marques,2Jordi Mompart,1Ver`onica Ahuﬁnger,1and Ricardo G. Dias2

1Departament de F´ısica, Universitat Aut`onoma de Barcelona, E-08193 Bellaterra, Spain.

2Department of Physics and i3N, University of Aveiro, 3810-193 Aveiro, Portugal.

We study Bose-Hubbard models in a family of diamond necklace lattices with ncentral sites. The

single-particle spectrum of these models presents compact localized states (CLSs) that occupy the up

and down sites of each diamond. By performing an appropriate basis rotation, the fragmentation of

the many-boson Hilbert space becomes apparent in the adjacency graph of the Hamiltonian, showing

disconnected sub-sectors with a wide range of dimensions. The models present a conserved quantity

related to the occupation of the single-particle CLSs that uniquely identiﬁes the diﬀerent sub-sectors

of the many-boson Hilbert space. Due to the fragmentation of the Hilbert space, the distribution of

entanglement entropies of the system presents a nested-dome structure. We ﬁnd weak thermalization

through sub-sector-restricted entanglement evolution and a wide range of entanglement entropy

scalings from area-law to logarithmic growth. Additionally, we observe how the distinguishability

between the diﬀerent domes increases with the number of central sites and we explain the mechanism

behind this fact by analyzing the graph structure of the Hamiltonian.

I. INTRODUCTION

The Eigenstate Thermalization Hypothesis (ETH) pre-

dicts how an excited state of a many-body closed quan-

tum system should thermalize [1–3]. Although most sys-

tems obey this hypothesis, numerous examples of non-

ergodic systems have been found. Perhaps the most

prominent example is integrable systems, where the num-

ber of conserved quantities equals or exceeds the degrees

of freedom of the system, thus exactly determining all

the eigenstates [4]. In many-body localized systems [5],

the interplay between disorder and interactions gives rise

to emergent integrability, which also leads to a strong

violation of the ETH. More recently, it was shown that

the ETH can also be weakly violated by a vanishing sub-

set of non-thermal eigenstates, dubbed Quantum Many

Body Scars (QMBS). They were initially found in one-

dimensional Rydberg arrays [6] with the underlying PXP

model [7,8], and were also discovered in parallel in the

AKLT model [9,10]. Since these initial works, QMBS

have been found in several systems where there is either

a tower of scarred eigenstates [10–23] or an isolated scar

[24–32].

A broader phenomenon that also leads to weak ther-

malization is Hilbert space fragmentation, also known as

Hilbert space shattering or Krylov fracture [33]. The

Hilbert space presents exponentially many dynamically

disconnnected sectors that prevent the system from ther-

malizing completely. Remarkably, this mechanism can

lead both to a weak or a strong violation of the ETH.

This eﬀect can arise in a wide variety of systems, such

as dipole moment or center-of-mass conserving systems

[34–38], the 1D t-Jzmodel [39], the t-Vand t-V1-V2

models [40,41], and models with dipolar interactions

[42]. All the above examples exhibit fragmentation of

the Hilbert space in the product state basis [43], i.e.,

classical fragmentation. Quantum fragmentation, which

occurs in an entangled basis, has been recently shown to

arise in Temperley-Lieb spin chains [43] and in quantum

East models [44]. However, it has yet to be determined if

quantum fragmentation leads to diﬀerent phenomenology

than its classical analogue.

The fragmentation in the above examples has recently

been referred to as standard Hilbert space fragmenta-

tion, to distinguish it from local Hilbert space fragmen-

tation [45], that arises in models with [21,27,32,46,47]

or without [48] frustration and in ﬂat band models [49].

While standard fragmentation is due to the presence of

non-local conserved quantities, locally fragmented sys-

tems present strictly local conservation laws.

In this work, we report on a family of Bose-Hubbard di-

amond necklaces [50] that exhibit quantum local Hilbert

space fragmentation. Here, the presence of a single-

particle ﬂat band composed of compact localized states

(CLSs) gives rise to the fragmentation of the Hilbert

space when introducing on-site interactions. As a conse-

quence of this fragmentation, one ﬁnds a nested distribu-

tion of entanglement entropies, sector-restricted thermal-

ization, and a broad range of sub-sectors of the Hamil-

tonian that range from frozen sub-sectors following area-

law to non-integrable sub-sectors with logarithmic scal-

ing.

The article is structured as follows: in Section II, we

introduce the system and we describe the basis rotation

that reveals the fragmentation of the Hilbert space in Sec.

II A. In Sec. II B, we analyze the conserved quantity that

characterizes the sub-sectors of the Hamiltonian, discuss

the adjacency graphs of the fragmented Hamiltonian, and

demonstrate that the system is strongly fragmented. The

numerical results are discussed in Sec. III, which include

the distribution of entanglement entropies, the entangle-

ment evolution and scaling, the level spacing analysis and

a comparison between the diﬀerent models of the dia-

mond necklace family. Finally, we summarize our con-

clusions in Sec. IV.

arXiv:2210.02429v2 [cond-mat.stat-mech] 4 Jan 2023

CLS

(a)

C1,k C2,k Cn,k

CLS

J√2J

C1,k C2,k Cn,k Sk

(b)

FIG. 1. (a) Diagram of the one-dimensional diamond necklace

model with ncentral sites. All couplings have a strength J

and the unit cell is shadowed in gray. In the second unit cell

we represent the CLS with the site amplitude being the radius

of the circle and the phase being the color (zero, red; π, blue).

(b) Diagram of the rotated model with the renormalized cou-

plings, √2J, denoted by a dashed line. The uncoupled states

represent the CLSs, |Aki.

II. PHYSICAL SYSTEM

We study a system of interacting bosons loaded onto

a one-dimensional lattice of diamond necklaces with n

central (i.e. spinal) sites [see Fig. 1(a)]. Each unit

cell kis composed of the sites C1,k ···Cn,k,Ukand

Dk(with k= 1, ..., Nc), and all the couplings have the

same magnitude J. The Hamiltonian of this system is

Hn=ˆ

n+ˆ

Hint

n, where the single-particle Hamiltonian

reads

n=JX

k"ˆc†

n,k(ˆuk+ˆ

dk) + (ˆu†

k+ˆ

d†

k)ˆc1,k+1+

n−1

j=1

(ˆc†

j,k ˆcj+1,k)#+ H.C.,

(1)

where ˆcj,k is the annihilation operator of the state |Cj,ki

at the central site j= 1, ..., n in each unit cell k, and

ˆukand ˆ

dkare the annihilation operators of the states

|Ukiand |Dkiat the up and down sites of each diamond,

respectively. In particular, the n= 2 case corresponds

to a type of orthogonal dimer chain [51–61] with absent

vertical couplings. The interaction Hamiltonian reads

Hint

n=U

k=1 "ˆnu,k(ˆnu,k −1) + ˆnd,k(ˆnd,k −1)

j=1

ˆnj,k(ˆnj,k −1)#=ˆ

Hint

n,diam. +ˆ

Hint

n,cent.,

(2)

where we distinguish the terms of the up and down sites

of each diamond, ˆ

Hint

n,diam., and the central sites, ˆ

Hint

n,cent..

ˆnu,k = ˆu†

kˆuk, ˆnd,k =ˆ

d†

kˆ

dkand ˆnj,k = ˆc†

j,k ˆcj,k are the

number operators at the up, down and central sites, re-

spectively.

An interesting characteristic of this family of Hamil-

tonians is that each diamond presents a single-particle

compact localized state (CLS) that only populates the

sites Ukand Dk, (|Uki−|Dki)/√2, [see Fig. 1(a)]. Due

to the presence of the CLS in each diamond of the lattice,

all models of this family exhibit a single-particle spec-

trum with a zero-energy ﬂat band. We are interested in

the many-body states where some of the particles occupy

a CLS, and how the existence of these states modiﬁes

the thermalization properties of the whole system. The

numerical results that we present in Section III can be

better interpreted by performing a basis rotation and an-

alyzing the symmetries of the system, which we discuss

in the next subsection.

A. Basis rotation

Consider the symmetric and antisymmetric superposi-

tions of the up and down states of each diamond,

|Ski=1

√2(|Uki+|Dki),|Aki=1

√2(|Uki−|Dki),

(3)

where ˆs†

kand ˆa†

kare the respective creation operators

and |Akiis the CLS in unit cell k. By using these states

to perform a basis rotation on the single-particle Hamil-

tonian, in Eq. (1), only the couplings associated to the

diamonds are altered,

H00

n=X

k"√2Jˆc†

n,k ˆsk+ ˆs†

kˆc1,k+1+

n−1

j=1

(ˆc†

j,k ˆcj+1,k)#+ H.C.

(4)

One obtains a linear chain that includes the symmetric

states, |Ski, and the central states |Cj,ki, with renor-

malized couplings corresponding to the diamonds, √2J.

Additionally, the CLSs in each unit cell, |Aki, become

decoupled, see Fig. 1(b). In analogy with the transfor-

mation of ˆ

n, only the interaction term of the up and

down sites of each diamond, ˆ

Hint

n,diam. in Eq. (2), is altered

by the basis rotation,

Hint0

n,diam. =U

k=1 4ˆs†

kˆa†

kˆskˆak+X

σ=a,s ˆσ†

kˆσ†

kˆσkˆσk

+ ˆa†

kˆa†

kˆskˆsk+ ˆs†

kˆs†

kˆakˆak,

(5)

where ˆσk(ˆσ= ˆs, ˆa) are the annihilation operators of |Ski

and |Aki, respectively. The ﬁrst term corresponds to a

nearest-neighbor interaction that arises when there is at

least one particle in |Skiand one in |Aki, akin to the

inter-circulation interaction term appearing in Hubbard

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LocalHilbertspacefragmentationandweakthermalizationinBose-HubbarddiamondnecklacesEloiNicolau,1AnselmoM.Marques,2JordiMompart,1VeronicaAhunger,1andRicardoG.Dias21DepartamentdeFsica,UniversitatAutonomadeBarcelona,E-08193Bellaterra,Spain.2DepartmentofPhysicsandi3N,UniversityofAveiro,3810-193Aveiro...

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Local Hilbert space fragmentation and weak thermalization in Bose-Hubbard diamond necklaces Eloi Nicolau1Anselmo M. Marques2Jordi Mompart1Ver onica Ahunger1and Ricardo G. Dias2

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