Generalized Thermalization in Quantum-Chaotic Quadratic Hamiltonians

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Patrycja Lyd˙zba,1Marcin Mierzejewski,1Marcos Rigol,2and Lev Vidmar3, 4

1Institute of Theoretical Physics, Wroclaw University of Science and Technology, 50-370 Wroc law, Poland

2Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA

3Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia

4Department of Physics, Faculty of Mathematics and Physics,

University of Ljubljana, SI-1000 Ljubljana, Slovenia

Thermalization (generalized thermalization) in nonintegrable (integrable) quantum systems re-

quires two ingredients: equilibration and agreement with the predictions of the Gibbs (generalized

Gibbs) ensemble. We prove that observables that exhibit eigenstate thermalization in single-particle

sector equilibrate in many-body sectors of quantum-chaotic quadratic models. Remarkably, the same

observables do not exhibit eigenstate thermalization in many-body sectors (we establish that there

are exponentially many outliers). Hence, the generalized Gibbs ensemble is generally needed to

describe their expectation values after equilibration, and it is characterized by Lagrange multipliers

that are smooth functions of single-particle energies.

Introduction.—Over the past 15 years, we have im-

proved signiﬁcantly our understanding of quantum dy-

namics in isolated many-body quantum systems [1–4].

A paradigmatic setup for these studies is the quantum

quench, in which a sudden change of a tuning parameter

pushes the system far from equilibrium. Following quan-

tum quenches, observables in nonintegrable systems have

been found to equilibrate to the predictions of the Gibbs

ensemble (GE) [1,5], while in integrable systems they

have been found to equilibrate to the predictions of the

generalized Gibbs ensemble (GGE) [6,7]. The validity

of the GGE has been tested in many theoretical studies

of integrable models that are mappable onto quadratic

ones [6–16] and integrable models that are not mappable

onto quadratic ones [17–28] (see Ref. [29] for reviews).

The GGE is also a starting point for the recently intro-

duced [30,31] and experimentally tested [32,33] theory

of generalized hydrodynamics.

Quadratic fermionic models, which are central to un-

derstanding a wide range of phenomena in condensed

matter physics, can be thought as being a special (nonin-

teracting) class of integrable models. Their Hamiltonians

consist of bilinear forms of creation and annihilation op-

erators. The inﬁnite-time averages of one-body observ-

ables after quenches in these models are always described

by GGEs [7,34–36]. However, there are one-body ob-

servables that generically fail to equilibrate because the

one-body density matrix evolves unitarily [37], i.e., gen-

eralized thermalization fails to occur. Such equilibration

failures have been discussed in the context of localization

in real [13,34–36] and momentum [36,37] space. Equi-

libration in quadratic models has been argued to occur

for local observables in the absence of real-space localiza-

tion. In particular, it has been shown to occur for initial

states that are ground states of local Hamiltonians [38],

as well as for initial states that exhibit suﬃciently rapidly

decaying correlations in real space [39–41].

In this Letter, we show that there is a broad class of

quadratic fermionic models for which generalized ther-

malization is ensured by the properties of the Hamilto-

nian. Hence, it is robust and resembles thermalization

(generalized thermalization), which occurs in interacting

nonintegrable (interacting integrable) models. The class

in question is that of quantum-chaotic quadratic (QCQ)

Hamiltonians, namely, quadratic Hamiltonians that ex-

hibit single-particle quantum chaos [42,43]. Paradig-

matic examples of local QCQ models are the three-

dimensional (3D) Anderson model in the delocalized

regime [42,44] and chaotic tight-binding billiards [45],

while their nonlocal counterparts include variants of the

quadratic Sachdev-Ye-Kitaev (SYK2) model [46,47] and

the power-law random banded matrix model in the de-

localized regime [46]. The single-particle sector of those

models exhibits random-matrix-like statistics of the en-

ergy levels [44,48,49], as well as single-particle eigenstate

thermalization [43], i.e., the matrix elements of properly

normalized one-body observables ˆ

O[50] in the single-

particle energy eigenkets are described by the eigenstate

thermalization hypothesis ansatz [1,51]

⟨α|ˆ

O|β⟩=O(¯ϵ)δαβ +ρ(¯ϵ)−1/2FO(¯ϵ, ω)RO

αβ ,(1)

where ¯ϵ= (ϵα+ϵβ)/2, ω=ϵβ−ϵα,O(¯ϵ) and FO(¯ϵ, ω)

are smooth functions of their arguments, while ρ(¯ϵ) =

δN/δϵ|¯ϵis the single-particle density of states (typically

proportional to the volume) at energy ¯ϵ. The distribu-

tion of matrix elements is described by the random vari-

able RO

αβ, which has zero mean and unit variance. The

many-body energy eigenstates, on the other hand, exhibit

eigenstate entanglement properties typical of Gaussian

states [42,46,47,52,53]; see also [54].

We prove that single-particle eigenstate thermaliza-

tion ensures equilibration in many-body sectors of QCQ

Hamiltonians, and we also prove that eigenstate ther-

malization does not occur in those sectors. We then

show that the GGE is needed to describe observables

after equilibration, and that it is characterized by the

Lagrange multipliers that are smooth functions of the

single-particle energies. The latter is also a consequence

of single-particle eigenstate thermalization. Our analyt-

ical results are tested numerically in QCQ Hamiltonians

and contrasted with results obtained for quadratic mod-

els that are not quantum chaotic.

arXiv:2210.00016v2 [cond-mat.stat-mech] 9 Aug 2023

Quantum quench and equilibration.—We consider a

quantum quench setup; the system is prepared in an ini-

tial many-body pure state |Ψ0⟩and evolves unitarily un-

der a quadratic Hamiltonian ˆ

H=PV

i,j=1 hij ˆc†

iˆcj, where

ˆc†

i(ˆci) creates (annihilates) a spinless fermion at site i,

and Vdenotes the number of lattice sites. In what fol-

lows, we use uppercase (lowercase) Greek letters to de-

note quantum states in the many-body (single-particle)

Hilbert space. One can diagonalize ˆ

Hvia a unitary trans-

formation of the creation and annihilation operators,

H=Pαϵαˆ

f†

αˆ

fα. The single-particle energy eigenstates,

with eigenenergies ϵα, can be written as |α⟩ ≡ ˆ

f†

α|∅⟩.

The many-body energy eigenstates, with eigenenergies

EΩ=P{α}ϵα, can be written as |Ω⟩=Q{α}ˆ

f†

α|∅⟩,

where {α}is the set of Noccupied |α⟩for any given lat-

tice ﬁlling ¯n=N/V . Any initial many-body pure state

can be written as |Ψ0⟩=PΩ⟨Ω|Ψ0⟩|Ω⟩.

Our focus is on one-body observables with rank

O(1) [55], such as site and quasimomentum occupations,

which are experimentally relevant and have the follow-

ing form ˆ

O=Pαβ Oαβ ˆ

f†

αˆ

fβwith Oαβ =⟨α|ˆ

O|β⟩. Their

time evolution can be written in the many-body basis as

⟨ˆ

O(t)⟩=PΩ⟨Ω|e−iˆ

Ht ˆρ0eiˆ

Ht ˆ

O|Ω⟩, where ˆρ0=|Ψ0⟩⟨Ψ0|

is the density matrix of the initial state. In quadratic

models, we can write it using the single-particle basis as

⟨ˆ

O(t)⟩=X

α⟨α|e−iˆ

Ht ˆ

Reiˆ

Ht ˆ

O|α⟩=

α,β=1

RαβOβαeiωβαt,

(2)

where ˆ

R=Pαβ Rαβ ˆ

f†

αˆ

fβis the one-body density matrix

of the initial state, with Rαβ =⟨Ψ0|ˆ

f†

βˆ

fα|Ψ0⟩[56,57],

and ωβα =ϵβ−ϵα.

The inﬁnite time average of ⟨ˆ

O(t)⟩, for a nondegener-

ated single-particle spectrum, is given by

⟨ˆ

O(t)⟩ ≡ lim

τ→∞

τZτ

0⟨ˆ

O(t)⟩dt =X

OααRαα.(3)

The density matrix in the GGE is deﬁned as ˆρGGE =

ZGGE e−Pαλαˆ

Iαwith ZGGE = Tr[e−Pαλαˆ

Iα], the con-

stants of motion being ˆ

Iα=ˆ

f†

αˆ

fα, and the Lagrange

multipliers ﬁxed such that Rαα = Tr[ˆρGGE ˆ

Iα]. There-

fore, the inﬁnite-time average of ⟨ˆ

O(t)⟩is reproduced by

the GGE prediction [7,35,36]

⟨ˆ

O(t)⟩=X

OααTr[ˆρGGE ˆ

Iα] = Tr[ˆρGGE ˆ

O],(4)

where we have used that ˆρGGE is diagonal in

the single-particle energy eigenbasis, so that

Tr[ˆρGGE Pαβ Oαβ ˆ

f†

αˆ

fβ] = Tr[ˆρGGE PαOαα ˆ

f†

αˆ

fα].

Given that the inﬁnite-time averages are guaranteed

to be described by the GGE, all one needs for general-

ized thermalization to occur is the temporal ﬂuctuations

about the inﬁnite-time average to vanish in the thermo-

dynamic limit. The temporal ﬂuctuations can be charac-

terized by the variance [1]

σ2

t=⟨ˆ

O(t)⟩2− ⟨ ˆ

O(t)⟩2.(5)

Recall that the standard derivation of the upper bound

for σ2

t, which is based on the time evolution written in the

many-body basis, requires that there are no gap degen-

eracies in the many-body spectrum [1]. This condition

need not be fulﬁlled in quadratic models. The deriva-

tion that we provide below, which is based on Eq. (2),

requires the absence of gap degeneracies in the single-

particle spectrum. The latter is satisﬁed by QCQ Hamil-

tonians. Speciﬁcally, one can write

⟨ˆ

O(t)⟩2=X

α,β,ω,ρ

OβαOρωRαβRωρei(ϵβ−ϵα+ϵρ−ϵω)t,(6)

which simpliﬁes to (see Ref. [58])

⟨ˆ

O(t)⟩2=X

α̸=β|Oαβ|2|Rαβ |2+⟨ˆ

O(t)⟩2.(7)

We can therefore deﬁne the upper bound for the variance

σ2

α̸=β=1 |Oαβ|2|Rαβ |2≤max |Oαβ |2

α=1

(R2)αα .

(8)

Since the eigenvalues of ˆ

Rbelong to the interval [0,1],

one can replace R2→Rin Eq. (8), and we obtain

σ2

t≤max V|Oαβ|21

α=1

Rαα = max V|Oαβ |2¯n ,

(9)

where we used that PαRαα =⟨ψ0|Pαˆ

f†

αˆ

fα|ψ0⟩=N.

Because the properly normalized one-body observables

with rank O(1) [50,55] can be written as ˆ

O ≃ ˆ

O√V[43],

single-particle eigenstate thermalization in QCQ models

results in max V|Oαβ|2∝1/V . Hence, the equilibra-

tion of these one-body observables is guaranteed in the

thermodynamic limit. Notice that the polynomial scal-

ing of the upper bound for σ2

twith the system size is

independent of the details of the quantum quench, like

the energy of the initial state |Ψ0⟩or the ﬁlling factor

¯n. The above analysis can be extended to one-body op-

erators that have rank O(V). Furthermore, in Ref. [58]

we show that equilibration also occurs for q-body ob-

servables (q= 2,3, ...) that are products of one-body ob-

servables, all of which exhibit single-particle eigenstate

thermalization. Remarkably, our analysis applies to ar-

bitrary initial states [58].

Numerical tests of equilibration.—We consider local

Hamiltonians that can be written as

H1=−X

⟨i,j⟩

ˆc†

iˆcj+

i=1

εiˆc†

iˆci.(10)

The ﬁrst term describes hoppings between nearest neigh-

bor sites, and εiis the on-site potential. We focus on

the 3D Anderson model on a cubic lattice with periodic

boundary conditions, for which εi= (W/2)riwith ri

being a random number drawn from a uniform distri-

bution in the interval [−1,1] [59]. We study dynamics

in the two regimes of this model (which has a transi-

tion at Wc≈16.5 [60,61]), at the W= 5 (delocal-

ized, QCQ [42]), and W= 25 (localized) points. For

the preparation of initial states in quantum quenches,

which are always taken to be ground states in this work,

we introduce a 3D superlattice model with εi=±W

in Eq. (10), where the sign alternates between nearest

neighbor sites. This 3D superlattice model allows us to

create highly nonthermal distributions of momenta in the

initial state (in the spirit of the quantum Newton’s cra-

dle experiment [62]). We complement our analysis with

a quadratic model that is not quantum chaotic, i.e., 1D

noninteracting fermions in a homogeneous lattice with

open boundary conditions [εi= 0 in Eq. (10)]. To pre-

pare the initial states for the quenches, we use the Aubry-

Andr´e model [εi=−Λ cos(2πσi) with σ= (√5−1)/2]

in Eq. (10).

We also consider a paradigmatic nonlocal QCQ model,

the SYK2 model in the Dirac fermion formulation [63],

H2=

i,j=1

[(1 −γ)aij +γbij ] ˆc†

iˆcj,(11)

where the diagonal (oﬀ-diagonal) elements of the ma-

trices aand bare real normally distributed random

numbers with zero mean and 2/V (1/V ) variance, while

γ∈[0,1]. The choice of an unconventional form of the

SYK2 Hamiltonian (as a sum of two one-body operators)

allows us to distinguish between weak and strong quan-

tum quenches, as explained in Ref. [58].

In Fig. 1, we show results of numerical tests of equi-

libration for two observables, the occupation of a lattice

site, ˆn1= ˆc†

1ˆc1, and the occupation of the zero quasimo-

mentum mode, ˆm0=1

VPij ˆc†

iˆcj. Speciﬁcally, we plot

the time evolution of ⟨ˆ

O(t)⟩−⟨ ˆ

O⟩GGE in Figs. 1(a)–1(d),

while the temporal ﬂuctuations σtas functions of Vare

shown in Fig. 1(e). For the quench from the 3D super-

lattice model at W= 1 to the 3D Anderson model at

W= 5, see Figs. 1(a) and 1(b), the temporal ﬂuctua-

tions σtof both observables decrease with increasing sys-

tem size, and a scaling σt∝V−ζwith ζ≈0.5 is observed

in Fig. 1(e). An exponent ζ= 0.5 is expected because

V|Oαβ|2∝1/V for those observables [43]. In contrast,

for the quench from the 3D Anderson model at W0= 30

to the same model (with a diﬀerent disorder realization)

at W= 25 in Fig. 1(c) [Fig. 1(d)], the temporal ﬂuctu-

ations σtdo not decrease (do decrease) with increasing

system size for ˆn1( ˆm0), and a scaling σt∝V−ζwith

ζ≈0 (ζ≈0.5) is observed in Fig. 1(e). This a conse-

quence of the fact that ˆm0, but not ˆn1, exhibits signa-

tures of single-particle eigenstate thermalization in the

101103105

- 0 . 1

0 . 1

101103105

- 0 . 0 3

0 . 0 3

101103105

- 0 . 3

0 . 3

101103105

- 0 . 1

0 . 1

102103104

10- 3

10- 2

10- 1

FIG. 1. (a)–(d) Time evolution of ⟨ˆ

O(t)⟩−⟨ ˆ

O⟩GGE after quan-

tum quenches in 3D models. The numerical results for system

with V= 63,83,143,and 183are marked with black, red,

blue, and green, respectively. We show results for two (solid

and dashed) quench realizations for each V. (a), (b) Quenches

from the 3D superlattice model at W= 1 and ¯n= 1/4 to the

3D Anderson model at W= 5. (c), (d) Quenches from the

3D Anderson model at W0= 30 and ¯n= 1/2 to the same

model (with a diﬀerent disorder realization) at W= 25. Two

operators are considered (a), (c) ˆn1and (b), (d) ˆm0. (e)

Temporal ﬂuctuations σtcalculated within the time interval

t∈[102,105] and averaged over 20 quench realizations. The

lines show the outcome of two parameter ﬁts κ/V ζ. We get

ζ∈[0.46,0.5] for (a), (b), and (d).

localized regime of the 3D Anderson model [43]. The re-

sults for ˆn1show that equilibration is not guaranteed for

quadratic Hamiltonians that are not quantum chaotic.

Qualitatively similar results to those for W= 25 in the

3D Anderson model were reported in the presence of real-

space localization in the 1D Anderson model [34,35], and

in the 1D Aubry-Andr´e model [13,36].

Stationary state.—Since eigenstate thermalization oc-

curs in single-particle eigenstates of QCQ models, it is

natural to wonder whether it also occurs in the many-

body eigenstates of those models. If this is the case,

the predictions of the GGE will be identical to those of

the GE in the thermodynamic limit, ⟨ˆ

O⟩GGE =⟨ˆ

O⟩GE,

where ⟨ˆ

O⟩GGE = Tr[ˆρGGE ˆ

O] and ⟨ˆ

O⟩GE = Tr[ˆρGE ˆ

O],

with ˆρGE =1

ZGE e−Pα(ϵα−µ)/(kBT)ˆ

f†

αˆ

fα, and ZGE =

Tr[e−Pα(ϵα−µ)/(kBT)ˆ

f†

αˆ

fα]. kB,T, and µare the Boltz-

mann constant, the temperature, and the chemical po-

tential, respectively.

We address this question in the context of the quenches

to the 3D Anderson model with W= 5. We focus on

ˆm0. The ﬁnite-size scaling of the diﬀerence |∆⟨ˆm0⟩| =

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GeneralizedThermalizationinQuantum-ChaoticQuadraticHamiltoniansPatrycjaLyd˙zba,1MarcinMierzejewski,1MarcosRigol,2andLevVidmar3,41InstituteofTheoreticalPhysics,WroclawUniversityofScienceandTechnology,50-370Wroclaw,Poland2DepartmentofPhysics,ThePennsylvaniaStateUniversity,UniversityPark,Pennsylvania16...

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Generalized Thermalization in Quantum-Chaotic Quadratic Hamiltonians

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