Universality classes of thermalization for mesoscopic Floquet systems Alan Morningstar1 2David A. Huse1and Vedika Khemani2 1Department of Physics Princeton University Princeton NJ 08544 USA

2025-05-06 0 0 2.22MB 18 页 10玖币

侵权投诉

Universality classes of thermalization for mesoscopic Floquet systems

Alan Morningstar,1, 2 David A. Huse,1and Vedika Khemani2

1Department of Physics, Princeton University, Princeton, NJ 08544, USA

2Department of Physics, Stanford University, Stanford, CA 94305, USA

(Dated: May 11, 2023)

We identify several distinct phases of thermalization that describe regimes of behavior in isolated,

periodically driven (Floquet), mesoscopic quantum chaotic systems. In doing so, we also identify

a new Floquet thermal ensemble—the “ladder ensemble”—that is qualitatively distinct from the

“featureless inﬁnite-temperature” state that has long been assumed to be the appropriate maximum-

entropy equilibrium ensemble for driven systems. The phases we ﬁnd can be coarsely classiﬁed by

(i) whether or not the system irreversibly exchanges energy of order ωwith the drive, i.e., Floquet

thermalizes, and (ii) the Floquet thermal ensemble describing the ﬁnal equilibrium in systems that do

Floquet thermalize. These phases are representative of regimes of behavior in mesoscopic systems,

but they are sharply deﬁned in a particular large-system limit where the drive frequency ωscales

up with system size Nas the N→ ∞ limit is taken: we examine frequency scalings ranging from a

weakly N-dependent ω(N)∼log N, to stronger scalings ranging from ω(N)∼√Nto ω(N)∼N.

We show that the transition where Floquet thermalization breaks down happens at an extensive

drive frequency and, beyond that, systems that do not Floquet thermalize are distinguished based

on the presence or absence of rare resonances across Floquet zones. We produce a thermalization

phase diagram that is relevant for numerical studies of Floquet systems and experimental studies

on small-scale quantum simulators, both of which lack a clean separation of scales between Nand

ω. A striking prediction of our work is that, under the assumption of perfect isolation, certain

realistic quench protocols from simple pure initial states can show Floquet thermalization to a novel

type of Schrodinger-cat state that is a global superposition of states at distinct temperatures. Our

work extends and organizes the theory of Floquet thermalization, heating, and equilibrium into the

setting of mesoscopic quantum systems.

I. INTRODUCTION

Breakthrough experimental developments in building

isolated quantum systems have led to signiﬁcant recent

progress in quantum statistical mechanics. This has fu-

eled advances in our understanding of fundamental ques-

tions surrounding the process of thermalization and its

various exceptions in isolated many-body systems [1–7].

In the common case of a system governed by a time-

independent Hamiltonian, the system thermalizes if, at

late times, probability distributions of local observables

are indistinguishable from those in a relevant thermal

ensemble. The appropriate thermal ensemble is deter-

mined by the principle of entropy maximization, con-

strained by the conservation laws of the system. The

Eigenstate Thermalization Hypothesis (ETH) [1,8–13]

posits conditions for thermalization on individual eigen-

states of the dynamics, and empirically these conditions

hold in examples of thermalizing systems [14–19].

Upon the addition of a periodic drive of frequency ω,

i.e., making the system “Floquet”, the Hamiltonian and

eigenstates of the stroboscopic dynamics gain a periodic

time dependence. The drive breaks the conservation of

energy and the appropriate long-time maximum-entropy

equilibrium is assumed to be a featureless “inﬁnite tem-

perature” state [20,21]. Exceptions to this “heat death”

are possible [22,23], notably in many-body localized

(MBL) or integrable Floquet systems [24–27], in which

case the system may thermalize to a generalized peri-

odic Gibbs ensemble [28,29] and/or realize novel ordered

phases such as the discrete time-crystal [30–34] or the

anomalous Floquet insulator [35,36]. Heating can also be

suppressed for a time exponential in the drive frequency,

a transient phenomenon called Floquet prethermaliza-

tion [37–50]. All of these results on Floquet thermal-

ization and its exceptions were obtained in works aimed

at the limit where the drive frequency ωis ﬁnite and the

number of degrees of freedom in the system Nis inﬁnite.

However, as we show in this paper, this limit provides

an incomplete description of thermalization in chaotic

Floquet systems. In particular, in mesoscopic systems

where Nis ﬁnite, there are other regimes of thermaliza-

tion captured by thermal ensembles that are qualitatively

distinct from a featureless inﬁnite-temperature state.

These regimes, and the crossovers between them, occur

at drive frequencies that depend on the system size N, so

to study them we allow for a drive frequency ω∝Ω(N)

that is scaled up with N. We examine frequency scalings

ranging from a weakly N-dependent Ω(N) = log N, to

stronger scalings ranging from Ω(N) = √Nto Ω(N) =

N. The distinctions between the diﬀerent regimes we ob-

tain can be made sharp in a particular large-Nlimit [51–

55], discussed later, where N→ ∞ is taken with ω/Ω(N)

held constant, i.e., a large-ωlimit is taken at the same

time.1This limit may appear non-standard when com-

pared to conventional thermodynamic limits studied in

1The width of the crossovers [in the control parameter ω/Ω(N)]

scales as Ω(N)−1and sharpens up as the limit is taken.

arXiv:2210.13444v3 [cond-mat.stat-mech] 9 May 2023

<latexit sha1_base64="qG+Wq6ZRPivR60lz1iFqqFYqq28=">AAAB7XicbVDJSgNBEO2JW4xb1KOXxiB4CjMSXG4BLx4jmAWSIfR0apI2vQzdPUIY8g9ePCji1f/x5t/YSeagiQ8KHu9VUVUvSjgz1ve/vcLa+sbmVnG7tLO7t39QPjxqGZVqCk2quNKdiBjgTELTMsuhk2ggIuLQjsa3M7/9BNowJR/sJIFQkKFkMaPEOqnVUwKGpF+u+FV/DrxKgpxUUI5Gv/zVGyiaCpCWcmJMN/ATG2ZEW0Y5TEu91EBC6JgMoeuoJAJMmM2vneIzpwxwrLQrafFc/T2REWHMRESuUxA7MsveTPzP66Y2vg4zJpPUgqSLRXHKsVV49joeMA3U8okjhGrmbsV0RDSh1gVUciEEyy+vktZFNbis1u5rlfpNHkcRnaBTdI4CdIXq6A41UBNR9Iie0St685T34r17H4vWgpfPHKM/8D5/AJBqjxs=</latexit>

featureless

inﬁnite temperature

ensemble

ladder ensemble

partial Floquet

thermalization

rare resonances

no rare

resonances

full Floquet thermalization energy localized

(a)

<latexit sha1_base64="OMqfnZ6sLw6GkmhOyKA9NdwAeq4=">AAAB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JIUZcFNy4r2Ac0IUwmt+3QmSTMTIQSu/BX3LhQxK2/4c6/cdJmoa0HBg7n3Ms9c8KUM6Ud59taWV1b39isbFW3d3b39u2Dw45KMkmhTROeyF5IFHAWQ1szzaGXSiAi5NANxzeF330AqVgS3+tJCr4gw5gNGCXaSIF97CUChiTwBNEjKXJOogjkNLBrTt2ZAS8TtyQ1VKIV2F9elNBMQKwpJ0r1XSfVfk6kZpTDtOplClJCx2QIfUNjIkD5+Sz/FJ8ZJcKDRJoXazxTf2/kRCg1EaGZLGKqRa8Q//P6mR5c+zmL00xDTOeHBhnHOsFFGThiEqjmE0MIlcxkxXREJKHaVFY1JbiLX14mnYu6e1lv3DVqzUZZRwWdoFN0jlx0hZroFrVQG1H0iJ7RK3qznqwX6936mI+uWOXOEfoD6/MHxeCWjw==</latexit>

!ladder

<latexit sha1_base64="ffWuCWOzCzSuojLcD51HfdCJvxg=">AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL4KokUtRlwY3LCvYBTQiT6aQdOo8wMxFK6MZfceNCEbd+hjv/xkmbhbYeGDicc++de0+cMqqN5307lbX1jc2t6nZtZ3dv/8A9POpqmSlMOlgyqfox0oRRQTqGGkb6qSKIx4z04slt4fceidJUigczTUnI0UjQhGJkrBS5J4HkZISigCMzVjxPkTIUsVnk1r2GNwdcJX5J6qBEO3K/gqHEGSfCYIa0HvheasK8GIcZmdWCTJMU4QkakYGlAnGiw3x+wAyeW2UIE6nsEwbO1d8dOeJaT3lsK4s99bJXiP95g8wkN2FORZoZIvDioyRj0EhYpAGHVBFs2NQShBW1u0I8RgphYzOr2RD85ZNXSfey4V81mvfNeqtZxlEFp+AMXAAfXIMWuANt0AEYzMAzeAVvzpPz4rw7H4vSilP2HIM/cD5/ALgolxo=</latexit>

!partial

<latexit sha1_base64="K0L68qe7CDA2P9N8Nbf60eeoRMY=">AAAB/HicbVDLSsNAFJ3UV62vaJduBovgqiRS1GXBjcsK9gFNCJPptB06jzAzEUKov+LGhSJu/RB3/o2TNgttPTBwOOde7pkTJ4xq43nfTmVjc2t7p7pb29s/ODxyj096WqYKky6WTKpBjDRhVJCuoYaRQaII4jEj/Xh2W/j9R6I0leLBZAkJOZoIOqYYGStFbj2QnExQFHBkpornTOJ55Da8prcAXCd+SRqgRCdyv4KRxCknwmCGtB76XmLCHClDMSPzWpBqkiA8QxMytFQgTnSYL8LP4blVRnAslX3CwIX6eyNHXOuMx3ayyKhXvUL8zxumZnwT5lQkqSECLw+NUwaNhEUTcEQVwYZlliCsqM0K8RQphI3tq2ZL8Fe/vE56l03/qtm6bzXarbKOKjgFZ+AC+OAatMEd6IAuwCADz+AVvDlPzovz7nwsRytOuVMHf+B8/gB0Q5VD</latexit>

!loc

<latexit sha1_base64="pz304nZF/npx+NMDw30fh8eOppo=">AAAB+3icbVDLSsNAFJ3UV62vWJduBovgqiRS1GXBjcsK9gFNCJPppB06jzAzEUvor7hxoYhbf8Sdf+OkzUJbDwwczrmXe+bEKaPaeN63U9nY3Nreqe7W9vYPDo/c43pPy0xh0sWSSTWIkSaMCtI11DAySBVBPGakH09vC7//SJSmUjyYWUpCjsaCJhQjY6XIrQeSkzGKAo7MRPFcqHnkNrymtwBcJ35JGqBEJ3K/gpHEGSfCYIa0HvpeasIcKUMxI/NakGmSIjxFYzK0VCBOdJgvss/huVVGMJHKPmHgQv29kSOu9YzHdrKIqFe9QvzPG2YmuQlzKtLMEIGXh5KMQSNhUQQcUUWwYTNLEFbUZoV4ghTCxtZVsyX4q19eJ73Lpn/VbN23Gu1WWUcVnIIzcAF8cA3a4A50QBdg8ASewSt4c+bOi/PufCxHK065cwL+wPn8Abk8lNs=</latexit>

!nr

<latexit sha1_base64="bI3ERVa7rKDrPn7I6qQ7YmZC5fk=">AAAB8nicbVBNS8NAEN34WetX1aOXxSLUS0mkqMeCF09SwX5AEspmu2mX7mbD7kQooT/DiwdFvPprvPlv3LY5aOuDgcd7M8zMi1LBDbjut7O2vrG5tV3aKe/u7R8cVo6OO0ZlmrI2VULpXkQMEzxhbeAgWC/VjMhIsG40vp353SemDVfJI0xSFkoyTHjMKQEr+YHhMhBqWLu/6Feqbt2dA68SryBVVKDVr3wFA0UzyRKgghjje24KYU40cCrYtBxkhqWEjsmQ+ZYmRDIT5vOTp/jcKgMcK20rATxXf0/kRBozkZHtlARGZtmbif95fgbxTZjzJM2AJXSxKM4EBoVn/+MB14yCmFhCqOb2VkxHRBMKNqWyDcFbfnmVdC7r3lW98dCoNutFHCV0is5QDXnoGjXRHWqhNqJIoWf0it4ccF6cd+dj0brmFDMn6A+czx+XL5C/</latexit>

⇠log(N)

<latexit sha1_base64="R56RgBwiM1nd657ue7yN7pKUHMc=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgadmVoB4DXjxJBPOAZAmzk9lkzDyWmVkhhPyDFw+KePV/vPk3TpI9aGJBQ1HVTXdXnHJmbBB8e4W19Y3NreJ2aWd3b/+gfHjUNCrThDaI4kq3Y2woZ5I2LLOctlNNsYg5bcWjm5nfeqLaMCUf7DilkcADyRJGsHVSs2uYQHe9ciXwgznQKglzUoEc9V75q9tXJBNUWsKxMZ0wSG00wdoywum01M0MTTEZ4QHtOCqxoCaazK+dojOn9FGitCtp0Vz9PTHBwpixiF2nwHZolr2Z+J/XyWxyHU2YTDNLJVksSjKOrEKz11GfaUosHzuCiWbuVkSGWGNiXUAlF0K4/PIqaV744aVfva9Wan4eRxFO4BTOIYQrqMEt1KEBBB7hGV7hzVPei/fufSxaC14+cwx/4H3+ABCgjr4=</latexit>

⇠N

FIG. 1. Distinct regimes of thermalization in mesoscopic Flo-

quet systems. (a) A sketch of the diﬀerent regimes of ther-

malization, listed in Sec. I A. The various frequencies marked

along the axis depend on the system size Nas shown, and

also on (quasi)energy, so the line shown is a cut through the

full 2D phase diagram of Fig. 2. (b) and (c): A depiction

of the (b) “featureless inﬁnite-temperature” and (c) “ladder”

ensembles. These are characterized by the probability dis-

tribution of energy deﬁned by an eﬀective Hamiltonian. In

the ladder ensemble, the system has a drive frequency ωthat

is large enough so that, while the system does Floquet ther-

malize, it maintains an approximate conservation of energy

modulo ω. The weights of the peaks follow the density of

states (DOS) at those energies, PE∝DOS(E).

many-body physics, but is standard in studies of meso-

scopic systems where interaction strengths and/or other

parameters are often taken to scale with N[51].

One motivation for studying the diﬀerent possible

regimes of thermalization in mesoscopic Floquet sys-

tems is that many settings—for instance, experiments

on near-term quantum simulators—allow controlled ac-

cess to “intermediate-scale” [56] many-body quantum

systems where there is not a clean separation of scales

between Nand ω(measured in units of a microscopic

energy scale). This is also true for numerical studies of

Floquet phenomena, which are limited to small system

sizes [57]. There is currently a major gap in the literature

in theoretically and systematically addressing thermal-

ization, Floquet heating, and equilibrium in this setting,

which we hope to bridge with this present work.

A. Summary of results

We identify a number of distinct regimes of thermal-

ization, and crossovers between them, that can occur in

mesoscopic Floquet systems. In this work, we are only

considering isolated chaotic many-body systems subject

to periodic driving, focusing on the delocalization of en-

ergy across Floquet zones; in particular, all the crossovers

we consider are between chaotic regimes with diﬀerent

degrees of energy conservation, so the physics of many-

body localization or integrability will play no role in our

discussion. Ref. [58] instead considers the fate of ﬁnite-

size driven integrable systems, with an N-dependent

driving amplitude. The diﬀerent regimes of thermaliza-

tion we ﬁnd are summarized below and in Fig. 1(a), and

explained in detail later:

•At the smallest frequencies, the system irreversibly

exchanges energy with the drive and Floquet ther-

malizes to a conventional featureless inﬁnite-

temperature state, i.e., the relevant Floquet ther-

mal ensemble is a uniform distribution over all

states [Fig. 1(b)].

•At larger frequencies, beyond ω=ωladder(N)∼

log N(or, in some physically relevant cases dis-

cussed later, ωladder(N)∼√N), the system instead

Floquet thermalizes to a ladder ensemble: In this

regime, energy conservation is not completely de-

stroyed, but downgraded to an approximate conser-

vation of energy modulo ω. Thus, energy becomes

delocalized across a “ladder” of narrow energy win-

dows that are spaced by ω. This is the relevant

maximum-entropy Floquet thermal ensemble un-

der the constraint of conservation of energy modulo

ω[Fig. 1(c)]. While in some cases this ensemble

can have the same average energy as the inﬁnite

temperature ensemble, the distribution of energy is

distinct.

•At yet larger frequencies beyond ω=ωpartial(N)∼

N, the system only partially thermalizes across the

rungs of the ladder. We call this the regime of par-

tial Floquet thermalization. In this regime, a

version of Floquet heating may still occur for many

initial states.

•Finally, at the largest frequencies, beyond ω=

ωloc(N)∼N, the system does not exchange energy

of order ωor more with the drive, i.e., it does not

Floquet thermalize and instead becomes energy

localized. In this regime, there exists an extensive

energy, deﬁned by a quasilocal eﬀective Hamilto-

nian, that is approximately conserved for all times.

While the extensive many-body bandwidth fur-

nishes an upper bound for ωloc(N), we ﬁnd that

energy localization sets in at a smaller scale, and

distinct non-trivial energy-localized regimes exist

that can be distinguished by the presence or ab-

sence of isolated Floquet many-body resonances in

rare states.

A few points are of note. First, complete Floquet ther-

malization occurs for frequencies less than ωpartial(N)∼

N, in the sense that the system eﬀectively exchanges en-

ergy with the drive. In most of this regime, i.e., between

the two scales ωladder(N)∼log(N) and ωpartial(N)∼N,

the ladder ensemble is the relevant description of the

ﬁnal thermal equilibrium. In contrast, the regime in

which the system thermalizes to a featureless inﬁnite-

temperature state is parametrically smaller, extending

only up to ωladder(N)∼log(N). Second, while we have

only focused on the frequency dependence of the diﬀer-

ent regimes in the discussion above [and in Fig. 1(a)],

there is also a strong dependence on (quasi)energy which

we explore below. In Fig. 2, we map out the full two-

parameter phase diagram of the diﬀerent types of ther-

malization mentioned above, and one important message

of our work is that sometimes the thermalization of Flo-

quet systems needs to be state or energy-resolved instead

of uniformly averaging over all Floquet eigenstates or ini-

tial states.

Finally, a notable consequence of our results is that un-

der a suitable quench protocol, isolated systems in pure

states can thermalize to novel Schrodinger cat states of

temperature, i.e., superpositions of states at globally dif-

ferent energy densities [Fig. 1(c)]. Although the coher-

ences of such states are notoriously fragile, the ladder-like

distribution of energy is a stable signature of this novel

form of Floquet thermalization.

The rest of this paper is organized as follows: In Sec. II

we set up our theoretical understanding of the diﬀerent

regimes of thermalization that occur in mesoscopic Flo-

quet many-body quantum systems. We support our the-

oretical reasoning with numerical evidence using a con-

crete model in Sec. III. In Sec. IV we explore some of the

prospects for studying the physics discussed in this work

experimentally, and show that indeed experimental stud-

ies seem to be accessible on some near-term platforms for

quantum simulation. Finally, we summarize and discuss

our ﬁndings in Sec. V.

II. THEORY

A. Setup and review of Floquet heating

For the purpose of discussion, we consider Nqubits

evolving under a time-periodic Hamiltonian H(t) = H0+

gω(t)V0, where gωis an O(1)-valued periodic function of

time that time-averages to zero, with period T=2π

ω.

H0is a quantum chaotic Hamiltonian that is a sum of

one- and two-body terms, and V0couples the system to

the drive, also consisting of a sum of one- and two-body

terms. A characteristic microscopic energy scale of His

set to one here. We are generally interested in behavior

at ω1, although in practice ω'ω0can be a more

accurate condition, where ω0is O(1) and depends on

the speciﬁc system. Both H0and V0are traceless, so

the energy corresponding to inﬁnite temperature is zero.

N2Ntr(H2

0), 1

N2Ntr(V2

0), and 1

N2Ntr([H0, V0]2) are all of

order one, so the only small parameters present are 1/ω

and 1/N . The stroboscopic dynamics are governed by the

Floquet unitary UF=Texp −iRT

0H(t)dtthat time-

evolves the system by one period. The Floquet unitary

deﬁnes the Floquet Hamiltonian HFvia UF≡e−iHFT.

The quasienergies θare deﬁned such that the eigenvalues

of UFare e−iθT , so θis only deﬁned modulo ωand is

strictly conserved by the dynamics, and this may be the

only such strict conservation law. The speciﬁc model we

use for later numerical demonstrations is given in Sec. III,

but our results are more general.

The process of Floquet heating entails a system reso-

nantly exchanging energy with the drive in quanta of size

∼ω. In this work, we will ideally consider frequencies ω

that are large compared to the microscopic energy scale

of H, which is set to 1 here (the regime when ωis com-

parable to the local energy scales leads to rapid heating,

but the ω∼O(1) boundary between these two regimes

is system dependent). In this high-frequency regime, ab-

sorbing a quanta ∼ωof energy requires a high-order

process involving O(ω) local energy moves, which occurs

at a rate that is exponentially suppressed in ω. Because

these processes can happen anywhere in the system, the

system as a whole exchanges “photons” with the drive at

a rate [37–41]

Γ∼Ne−ω/ω0,(1)

with some microscopic (order one) ω0that may, in gen-

eral, depend on the energy density.2This is the behavior

in the drive frequency range ω0/ωNω0, and, in this

regime, there is an eﬀective (“prethermal”) quasi-local

Hamiltonian Heﬀ that captures the dynamics of the sys-

tem on timescales shorter than t∼Γ−1.Heﬀ can be

obtained perturbatively, and represents the most opti-

mal quasi-local truncation of a high-frequency Magnus

expansion for HF[39]. The leading term in the expan-

sion is the time-averaged Hamiltonian, H0(see App. A).

The time-scale Γ−1sets the crossover time between

the prethermal regime with dynamics governed by Heﬀ

(which has an extensive conserved energy), and the

regime of Floquet thermalization, where the system ther-

malizes across diﬀerent Floquet zones due to a reso-

nant drive-mediated coupling between states separated

in energy by ωand therefore becomes delocalized in en-

ergy [23]. The slow thermalization across Floquet zones

2In one dimension there is a logarithmic correction such that the

heating rate is bounded by Γ ∼Ne−(ωlog ω)/ω0[59–61]. We

focus on the general case in higher dimensions for our discussion

and numerical studies below, but the results are qualitatively the

same in one dimension.

is reﬂected in the non-perturbative, non-local character of

HF. The diﬀerence between UF=e−iHFTand e−iHeff T

is the thermalization process across Floquet zones visible

on times t > Γ−1.

B. Floquet thermal ensembles and nonstandard

large-Nlimits

We now discuss the featureless inﬁnite temperature

and ladder ensembles for Floquet thermalization, the

crossover between these, and how this crossover sharp-

ens in a particular large-Nlimit.

For a system that absorbs/emits energy slowly enough,

an eigenstate of UFwith eigenvalue e−iθT will be sup-

ported on eigenstates of Heﬀ near a “ladder” of energies

that diﬀer in steps of ω, i.e., with energies

Eeﬀ =θ±nω (2)

with n∈Z. However, due to the nonzero heating rate,

each of the “rungs” of the energy ladder will have an

energy uncertainty ∼Γ∼Ne−ω/ω0(see Fig. 1(c) for a

depiction). In the commonly prioritized limit of ﬁnite ω

and N→ ∞, the rate Γ grows with Nand eventually

becomes larger than ω(when N∼ωeω/ω0). For Nwell

in excess of this, the ladder is not resolvable as the width

of the rungs exceeds the spacings between rungs, and the

energy conservation (even modulo ω) is fully lost. The

resulting equilibrium is then “inﬁnite temperature” in a

strict sense, because the relevant Floquet thermal ensem-

ble is a uniform distribution over all energy eigenstates,

as shown in Fig. 1(b).

The strict inﬁnite-temperature property of Floquet

thermalized states (and eigenstates of UF) can break

down to various degrees when Γ ω. This can occur in

systems with ﬁnite Nand ω, or in large Nsystems where

we allow ωto scale up with Nin such a way that some

behavior characteristic of ﬁnite-size systems is retained in

the limit. For example, consider ωladder(N) = ω0log N:

If ωis scaled up with Nfaster, so that ωωladder(N),

then Γ ωat large enough N. In this regime, the distri-

butions of energy in the eigenstates of UFhave signiﬁcant

weight only near a well-resolved ladder of energies with

spacing ω, as shown in Fig. 1(c). In other words, the en-

ergy deﬁned by Heﬀ is conserved modulo ωto a precision

of ∼Γ.

This “energy ladder” is a maximum-entropy Floquet

thermal ensemble that is distinct from “inﬁnite tempera-

ture”, and notably it sets in already at a frequency scale

ωladder =ω0log Nthat is only weakly dependent on N.

If we consider the rescaled frequency ν=ω

ωladder(N)=

ω0log N, then the crossover from the featureless inﬁnite

temperature ensemble to the ladder ensemble happens

near ν= 1. This crossover in the rescaled variable ν

becomes sharp in the large-Nlimit as can be seen from

the behavior of Γ

ω=Ne−ω/ω0

ω=N1−ν

νω0log(N)near ν= 1: it

diverges with Nif ν < 1 and approaches zero if ν > 1.

The ladder ensemble sets in at ω∼log(N) and ex-

tends to parametrically larger frequency scalings, ω=

ωpartial(N)∼N. First consider ω∼Nα, with 0 < α < 1.

As long as α < 1, consecutive rungs on the ladder have

diﬀerent energies but the same energy density as N→ ∞,

i.e., the spacing in energy density between the rungs

tends to zero. Each Floquet quasienergy θcorresponds to

populating a ladder of energies Eeﬀ mod ω=θthat spans

across all energy densities. In particular, the ladder con-

tains a subset of rungs that have the same energy (and

entropy) density as inﬁnite temperature in the N→ ∞

limit, E∞

eﬀ /N =1

N2NTr[Heﬀ ] = 0. Thus the ﬁnal equilib-

rium is one where the average energy density corresponds

to inﬁnite temperature, but the distribution of energy is

markedly distinct from the uniform distribution and in-

stead concentrated near a ladder of well-spaced energies.

Finally, we have the case of ω∝N(α= 1). In this

case, the frequency ωis extensive and corresponds to

transitions between diﬀerent energy densities e; thus we

denote ∆e≡ω

Nin the rest of this paper. Since ωis exten-

sive in this case, it is not generally true that the system

thermalizes to the same average energy density as inﬁ-

nite temperature in the N→ ∞ limit, even when it does

equilibrate across Floquet zones (Floquet thermalizes).

This is because the ﬁnal energy distribution resides on a

ladder of diﬀerent energy densities, and the inﬁnite tem-

perature energy density (e= 0) is not generally one of

them (see Fig. 5for a demonstration).

This brings us to an important point: Floquet ther-

malization (also referred to as Floquet heating) refers to

reaching the appropriate equilibrium ensemble with en-

ergies distributed either according to the uniform inﬁnite

temperature ensemble or the appropriate ladder ensem-

ble. The ladder ensemble is always a distinct ensem-

ble from inﬁnite temperature, and need not even have

the same average energy density as inﬁnite temperature.

Thus, Floquet thermalization does not imply that the sys-

tem thermalizes to inﬁnite temperature, even on average.

Some comments are in order before concluding this

section. When deﬁning the ladder ensemble, we consid-

ered energies deﬁned according to Heﬀ , the most optimal

quasi-local truncation of HF. In this case, the energy

deﬁned by Heﬀ is conserved modulo ωto a precision of

∼Γ set by the heating rate. However, if Heﬀ is not

chosen optimally—for instance, if energy is deﬁned with

respect to the leading term H0—then the precision is

lower and the width of the rungs is accordingly broader,

as discussed in App. A. Likewise, if we consider thermal-

ization of generic initial states (instead of eigenstates of

UF), then the energy uncertainty of the initial state also

contributes to the broadening of the rungs. A typical

product initial state has energy uncertainty ∝√N, and

hence requires ω∼Nαwith α > 1

2to resolve the rungs of

the ladder for large enough N. In this case, the crossover

from the featureless inﬁnite temperature ensemble to the

ladder ensemble happens at ωladder(N)∼√Nand it does

not sharpen up in the large-Nlimit.

In sum, in this section we’ve discussed diﬀerent Flo-

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

UniversalityclassesofthermalizationformesoscopicFloquetsystemsAlanMorningstar,1,2DavidA.Huse,1andVedikaKhemani21DepartmentofPhysics,PrincetonUniversity,Princeton,NJ08544,USA2DepartmentofPhysics,StanfordUniversity,Stanford,CA94305,USA(Dated:May11,2023)Weidentifyseveraldistinctphasesofthermalizationth...

展开>> 收起<<

Universality classes of thermalization for mesoscopic Floquet systems Alan Morningstar1 2David A. Huse1and Vedika Khemani2 1Department of Physics Princeton University Princeton NJ 08544 USA.pdf

共18页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Universality classes of thermalization for mesoscopic Floquet systems Alan Morningstar1 2David A. Huse1and Vedika Khemani2 1Department of Physics Princeton University Princeton NJ 08544 USA

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: