Slow thermalization and subdiﬀusion in U1conserving Floquet random circuits Cheryne Jonay Joaquin F. Rodriguez-Nievaand Vedika Khemani Department of Physics Stanford University Stanford CA 94305

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Slow thermalization and subdiﬀusion in U(1) conserving Floquet random circuits

Cheryne Jonay, Joaquin F. Rodriguez-Nieva,∗and Vedika Khemani

Department of Physics, Stanford University, Stanford, CA 94305

(Dated: May 23, 2023)

Random quantum circuits are paradigmatic models of minimally structured and analytically

tractable chaotic dynamics. We study a family of Floquet unitary circuits with Haar random U(1)

charge conserving dynamics; the minimal such model has nearest-neighbor gates acting on spin 1/2

qubits, and a single layer of even/odd gates repeated periodically in time. We ﬁnd that this min-

imal model is not robustly thermalizing at numerically accessible system sizes, and displays slow

subdiﬀusive dynamics for long times. We map out the thermalization dynamics in a broader param-

eter space of charge conserving circuits, and understand the origin of the slow dynamics in terms

of proximate localized and integrable regimes in parameter space. In contrast, we ﬁnd that small

extensions to the minimal model are suﬃcient to achieve robust thermalization; these include (i)

increasing the interaction range to three-site gates (ii) increasing the local Hilbert space dimension

by appending an additional unconstrained qubit to the conserved charge on each site, or (iii) using

a larger Floquet period comprised of two independent layers of gates. Our results should inform

future studies of charge conserving circuits which are relevant for a wide range of topical theoretical

questions.

I. INTRODUCTION

The dynamics of thermalization in isolated quantum

systems is a topic of fundamental interest [1–5]. Foun-

dational questions on this topic have taken special rele-

vance with the advent of quantum simulators that can

coherently evolve quantum states for long times [6–10].

Some key questions include identifying universal features

in the dynamics of thermalizing systems, such as in the

growth of entanglement entropy [11–15], the propagation

of quantum information [16,17], or the emergence of hy-

drodynamic transport [18–20]. Likewise, understanding

mechanisms by which systems evade thermalization or

thermalize slowly, say due to localization [21–23] or quan-

tum scarring [24–26] or prethermalization [27–34], is also

an active area of research.

Analytic approaches to studying the out-of-equilibrium

dynamics of interacting quantum systems remain scarce.

In this context, random quantum circuits have emerged

as a powerful class of analytically tractable models

that can capture universal features of quantum dynam-

ics [14,35–37]. Long studied in quantum information as

models of quantum computation, quantum circuits are

also the natural evolutions implemented by digital quan-

tum simulators that are the subject of much experimen-

tal investigation [6,10,38,39]. A canonical example of

a minimally structured random circuit retains only the

ingredients of unitarity and locality, such that the time

evolution of a set of a qudits on some lattice is gener-

ated by unitary gates acting on rcontiguous qudits, and

drawn independently from the Haar measure at each po-

sition and time-step. Such minimally structured circuit

models have no continuous symmetries or conservation

laws (the discrete time-evolution means even energy is

∗Equal contribution

not conserved), and exhibit thermalization to an inﬁ-

nite temperature state, with universal ballistic spread-

ing of quantum information en route to thermalization

[14,36,37].

Random circuits can also be endowed with addi-

tional structure to achieve a range of interesting dynam-

ical behaviors, with the simplest such extension being

the addition of a single U(1) conservation law of total

spin [18,19]. In the minimal implementation of this con-

servation law, each local unitary gate assumes a block di-

agonal structure, with each symmetry block within each

gate drawn independently from the Haar measure. We

will refer to such gates as “Haar random U(1) conserv-

ing" gates. With this change, the ballistic information

spreading is (provably) accompanied by diﬀusive spin dy-

namics, and the diﬀusion constant can be analytically de-

rived [18,19]. The dynamics can be made subdiﬀusive if

the U(1) conservation law is accompanied by other dipole

or multipole symmetries, or if the system is endowed with

additional kinetic constraints [40–45].

Another important extension of random circuit dy-

namics is obtained by considering Floquet dynamics with

time-periodicity, so that layers of random unitary gates

are repeated in time after a period of Ttime-steps.

In generic Floquet systems, the appropriate equilibrium

state is the inﬁnite temperature state [46,47]. However,

the periodicity in time and randomness in space opens up

the possibility of many-body localization (MBL), which

is absent in circuits that are random in time. Indeed, sev-

eral families of structured random Floquet circuits have

been shown to exhibit a many-body localized regime [48–

53]. In this context, ”structured” pertains to Floquet

circuits generated by speciﬁc gate sets parameterized by

tunable interaction and disorder strengths and, impor-

tantly, with a preferred local basis for the disorder such

that the models look increasingly diagonal in this basis

as the disorder strength is increased. Increasing disorder

or decreasing interactions leads to MBL.

arXiv:2210.13429v2 [quant-ph] 19 May 2023

However, localization is generally not expected in min-

imally structured Haar random Floquet models (without

any conservation laws), in which neighboring spins are

strongly coupled with Haar random gates, and there is

no preferred local basis in which the disorder is strong. In

line with this intuition, Ref. [54] proved that local Haar

random Floquet circuits are chaotic in the limit of inﬁnite

local Hilbert space dimension qfor each qudit (speciﬁ-

cally, they proved that the spectral form factor derived

from the Floquet unitary exhibits random matrix the-

ory (RMT) behavior), while Ref. [49] numerically showed

thermalization even for spin 1/2 qubits. The derivation

of chaos in the spectral form factor was also extended

to random Floquet circuits with U(1) charge conserva-

tion symmetry [55], in which case the RMT behavior

sets in after a Thouless time governed by diﬀusive charge

transport. The U(1) Floquet calculation required ap-

pending an unconstrained qudit to a qubit whose charge

is conserved, resulting in an enlarged local Hilbert space

C2⊗Cqof dimension 2q. Once again, the limit q→ ∞

was necessary to make analytic predictions although, just

like the case without U(1) symmetry, the results were

expected to hold more generally i.e. even in the case of

q= 1.

In this work we consider thermalization dynamics in

various U(1) conserving random Floquet circuits. Sur-

prisingly, we ﬁnd that the system deviates from robust

thermalization for the minimal Haar random U(1) con-

serving Floquet model with nearest neighbor gates, spin

1/2 qubits, and no additional qudits (i.e. with q= 1).

While the random-in-time Haar random U(1) conserv-

ing circuit with q= 1 is provably diﬀusive [18,19]—and

the same behavior was expected to occur in Floquet cir-

cuits [55]—we instead ﬁnd subdiﬀusive behavior of con-

served charges for long times and slow thermalization dy-

namics for numerically accessible system sizes. We com-

pute various local and global metrics of quantum ther-

malization, such as level spacing statistics, the transport

of conserved charges, and the dynamics of entanglement.

While all metrics show trends that are consistent with

asymptotic thermalization for large enough system sizes

and times, numerically accessible sizes and times do not

show robust thermalization. By studying the parame-

terization of the gates, we argue that the slow dynamics

can be understood in terms of proximity of the nearest

neighbor Haar random U(1) conserving gates to localized

regimes in parameter space.

In contrast, robust thermalization is recovered by tak-

ing q > 1or, analogously, by increasing the range of gates

r, or the period T. Random circuits (with and with-

out symmetries) have come to serve as canonical mod-

els for numerically and experimentally studying quantum

dynamics. Our work informs future studies of charge

conserving Floquet circuits by showing that the mini-

mal such circuit is not robustly chaotic, and needs to be

extended in one of the ways mentioned above, i.e. by

increasing the range, the period, or q, to achieve robust

thermalization.

More generally, our work shows that the combination

of locality, symmetry, and a small local Hilbert space

can quite generically hinder thermalization and explo-

ration of the (constrained) Hilbert space. Interestingly,

it was recently pointed out [56] that local symmetric gates

cannot produce the full group of global symmetric gates

even for random-in-time systems (regardless of the range

of interactions). This is to be contrasted with the fun-

damental result in quantum computing that any global

unitary in U(N)can be arbitrarily approximated by a

sequence of 2-local gates [57,58]. Whereas the result in

[56] does not preclude the emergence of local thermaliza-

tion, it raises intriguing questions of how truly random

the global unitaries generated by local and symmetric

evolution are.

The rest of this paper is organized as follows. We be-

gin in Section II with a description of the models consid-

ered in this work. We show how to parameterize two-site

Haar random U(1) gates in two diﬀerent ways: By di-

rectly constraining transition amplitudes in the unitary

gate in accordance with the symmetry, and in terms of

the generators of the Lie algebra of U(4) that commute

with the generator of the U(1) subgroup on two qubits.

The various Lie algebra coordinates, which we refer to

as "couplings", serve as important tuning knobs of the

models, controlling the interaction strength and real and

complex hopping amplitudes. We ﬁnd that the minimal

nearest neighbor U(1) conserving Floquet circuit fails to

saturate to the RMT prediction at numerically accessible

system sizes, and explain this using the parameterization

of U(1) gates. We present level statistics data in Section

III, charge transport in Section IV and entanglement dy-

namics in Section V. In all cases, the minimal slowly

thermalizing model is contrasted with a robustly ther-

malizing model with larger period T > 1, longer range

interactions r > 1, or enlarged local Hilbert space q > 1.

We summarize and conclude in Section VI.

II. MODELS

A. Circuit Architecture

In this subsection, we introduce the general architec-

ture of U(1) conserving Floquet circuits considered in

this work. The circuit structure will be labeled by three

parameters: the local Hilbert space dimension 2q, the

range r, and the period T, discussed in turn below. The

unitary gates comprising the circuit are drawn from var-

ious probability distributions, which are discussed in the

next subsections.

We study periodic one-dimensional chains of length L

sites, where the degree of freedom on each site is the di-

rect product of a spin-1/2 qubit and a qudit with Hilbert

space dimension q. Thus, the local Hilbert space is

C2⊗Cq, with dimension 2q. We can label the spin state

on each site ias (↑a)ior (↓b)i, where the ﬁrst label is

the spin state in the Pauli zbasis and the second label

is the qudit state. Likewise, Xj, Yj, Zjdenote Pauli ma-

trices acting on the qubit on site j, where the identity

operator acting on the qudit state is left implicit. The

time-evolution is constrained to conserve the total zcom-

ponent of the spin-1/2’s, Stot

z=PjZj, while the qudits

are not subject to any conservation laws [18].

We consider brickwork circuits with staggered layers

of range-runitary gates. The range ris the number of

contiguous qudits each individual gate acts on, so that

Uj,j+1,···,j+(r−1) acts on sites (j, j+1,··· , j+r−1). Thus,

r= 2 denotes nearest-neighbour gates while r= 3 is a

three-site gate including both nearest and next-nearest

neighbour interactions. Each gate is a (2q)r×(2q)rblock

diagonal matrix, with (r+ 1) blocks labeled by the total

zcharge of the qubits on rsites: Pj+r−1

i=jZi. The blocks

have size r

nqrwith n= 0,1, ..., r. For example, for

r= 2, the gates Uj,j+1 have the structure:

Uj,j+1 =







↑↑







q2×q2







↑↓,↓↑







2q2×2q2







↓↓







q2×q2













,(1)

comprised of (i) a (q2×q2)block acting in the (↑a)i⊗(↑

b)i+1 subspace, (ii) a (2q2×2q2)block acting in the

(↑a)i⊗(↓b)i+1,(↓a)i⊗(↑b)i+1 subspace, and (iii)

a(q2×q2)block acting in the (↓a)i⊗(↓b)i+1 subspace.

Likewise, for r= 3, the gates Uj,j+1,j+2 have the struc-

ture:

Uj,j+1,j+2 =







↑↑↑







q2×q2







↑↑↓,↑↓↑,↓↑↑







3q2×3q2







↓↓↑,↓↑↓,↑↓↓







3q2×3q2







↓↓↓







q2×q2













,(2)

comprised of (i) a (q2×q2)block acting in the (↑a)i⊗(↑

b)i+1 ⊗(↑c)i+2 subspace, (ii) a (3q2×3q2)block acting

in the (↑a)i⊗(↑b)i+1 ⊗(↓c)i+2,(↑a)i⊗(↓b)i+1 ⊗

(↑c)i+2,(↓a)i⊗(↑b)i+1 ⊗(↑c)i+2 subspace, (iii) a

(3q2×3q2)block acting in the (↓a)i⊗(↓b)i+1 ⊗(↑

c)i+2,(↓a)i⊗(↑b)i+1 ⊗(↓c)i+2,(↑a)i⊗(↓b)i+1 ⊗(↓

c)i+2 subspace, and (iv) a (q2×q2)block acting in the

(↓a)i⊗(↓b)i+1 ⊗(↓c)i+2 subspace.

The circuit architecture has a periodic brickwork lay-

out with variable period T∈Z. The circuit implements

discrete time evolution, and advancing by one unit of

time comprises the application of a “layer” comprised of

rstaggered sub-layers. Each sub-layer is displaced by

one lattice site with respect to the prior sublayer (See

Fig. 1). For example, for r= 2, advancing by one unit of

time entails applying one layer of even and odd gates:

U(t+ 1, t) = Y

U2j+1,2j+2(t)

| {z }

Uodd (t)≡U1(t)

U2i,2i+1(t)

| {z }

Ueven(t)≡U0(t)

(3)

Likewise, r= 3 requires applying three staggered sub-

layers of gates starting from the (0,1,2),(1,2,3) and

(2,3,4) bonds respectively:

U(t+ 1, t) = Y

U3k+2,3k+3,3k+4(t)

| {z }

U2(t)

U3j+1,3j+2,3j+3(t)

| {z }

U1(t)

U3i,3i+1,3i+2(t)

| {z }

U0(t)

(4)

More generally, for range rgates,

U(t+ 1, t) =

r−1

α=0

Uα(t)

Uα(t) = Y

Urj+α,rj+α+1,···,rj+α+r−1(5)

For a circuit with periodicity T, the gates in the ﬁrst T

layers are chosen independently, and layers repeat after

Ttime-steps: U(t+T+ 1, t +T) = U(t+ 1, t). The

Floquet unitary is deﬁned as the time-evolution operator

for period T:

UF(T) =

T−1

t=0

U(t+ 1, t),(6)

and U(t=nT, 0) = UF(T)n.

B. Haar Random U(1) Conserving Gates

A central focus of our study is on the most random

U(1) Floquet dynamics, where each symmetry block in

each unitary gate in the ﬁrst Tlayers is sampled ran-

domly and independently from the Haar distribution.

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SlowthermalizationandsubdiusioninU(1)conservingFloquetrandomcircuitsCheryneJonay,JoaquinF.Rodriguez-Nieva,andVedikaKhemaniDepartmentofPhysics,StanfordUniversity,Stanford,CA94305(Dated:May23,2023)Randomquantumcircuitsareparadigmaticmodelsofminimallystructuredandanalyticallytractablechaoticdynamics....

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Slow thermalization and subdiﬀusion in U1conserving Floquet random circuits Cheryne Jonay Joaquin F. Rodriguez-Nievaand Vedika Khemani Department of Physics Stanford University Stanford CA 94305

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