Charge transport information scrambling and quantum operator-coherence in a many-body system with U1 symmetry Lakshya Agarwal1Subhayan Sahu2 3and Shenglong Xu1

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Charge transport, information scrambling and quantum operator-coherence in a

many-body system with U(1) symmetry

Lakshya Agarwal,1Subhayan Sahu,2, 3 and Shenglong Xu1

1Department of Physics & Astronomy, Texas A&M University, College Station, Texas 77843, USA

2Condensed Matter Theory Center and Department of Physics,

University of Maryland, College Park, MD 20742, USA

3Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

In this work, we derive an exact hydrodynamical description for the coupled, charge and operator

dynamics, in a quantum many-body system with U(1) symmetry. Using an emergent symmetry

in the complex Brownian SYK model with charge conservation, we map the operator dynamics in

the model to the imaginary-time dynamics of an SU(4) spin-chain. We utilize the emergent SU(4)

description to demonstrate that the U(1) symmetry causes quantum-coherence to persist even after

disorder-averaging, in sharp contrast to models without symmetries. In line with this property, we

write down a ‘restricted’ Fokker-Planck equation for the out-of-time ordered correlator (OTOC) in

the large-Nlimit, which permits a classical probability description strictly in the incoherent sector

of the global operator-space. We then exploit this feature to describe the OTOC in terms of a

Fisher-Kolmogorov-Petrovsky-Piskun (FKPP)-equation which couples the operator with the charge

and is valid at all time-scales and for arbitrary charge-density proﬁles. The coupled equations

obtained belong to a class of models also used to describe the population dynamics of bacteria

embedded in a diﬀusive media. We simulate them to explore operator-dynamics in a background

of non-uniform charge conﬁguration, which reveals that the charge transport can strongly aﬀect

dynamics of operators, including those that have no overlap with the charge.

CONTENTS

I. Introduction 1

II. Summary of the main results 2

III. Charged operator basis 4

IV. General procedure and emergent SU(4)

algebra 5

V. Map from operator strings to SU(4) basis 6

VI. Quantum operator coherence due to U(1)

symmetry 7

VII. The large-Nformalism and the ‘restricted’

Fokker-Planck equation 7

VIII. The OTOC at large N9

A. Free-fermionic chain 9

B. Charge-dependent FKPP equation 9

1. Constant charge density 10

2. Charge-transport and late-time

behavior 10

IX. A case study: operator dynamics in a domain

wall density background 11

A. Asymmetric butterﬂy velocity 11

B. Conserved vs. non-conserved operator 11

C. Charge dynamics inﬂuences the dynamics of

non-conserved operators 12

X. Conclusion and Discussions 12

XI. Acknowledgement 12

References 13

Appendix A. Anti-commuting fermions on

multiple time-contours 15

Appendix B. Charge-resolution of

null-eigenstates 15

Appendix C. Solving the restricted Fokker-Planck

equation 16

Appendix D. Analytic solution of the

free-fermionic chain 16

1. Charge transport at ﬁnite-N16

2. OTOC at ﬁnite-N18

I. INTRODUCTION

In many-body dynamics, a ﬁne-grained approach to

study thermalization [1–4] comes from the study of

operator-dynamics [5–7]. This is also related to the con-

cept of information scrambling, where local information

ﬂows to non-local degrees of freedom and is not retriev-

able via local measurements [8–13]. The notions of op-

erator growth and information scrambling can be pre-

cisely measured using the out-of-time-ordered correlator

(OTOC):

F(W(t), V ) = 1

tr(I)tr(W†(t)V†W(t)V)(1)

This quantity is often used to measure the size of the

evolving operator W(t) and shows interesting features

arXiv:2210.14828v3 [cond-mat.str-el] 30 Nov 2022

such as Lyapunov growth and butterﬂy velocity [11,14–

27]. The OTOC is experimentally accessible on various

platforms [28–38], where it is often used as a measure of

information scrambling.

Previous work on random circuits and noisy-driven

models has established several related eﬀective classical

models of operator dynamics, including biased random

walk [5], reaction diﬀusion process [39] and population

dynamics [40]. These classical models are used to ﬁt and

understand experimental data of the OTOC on quantum

platforms [34,41]. The models are obtained by mapping

the unitary operator dynamics to a classical stochastic

process by disorder average. While this approach is fea-

sible for usual noisy/random models without symmetry,

adding charge conservation makes the problem more dif-

ﬁcult as it causes quantum coherence to persist at the

operator level, even after disorder-averaging (Fig. 1(e)).

However, such a classical description is important to ob-

tain for charged models, as thermalization in systems

with charge conservation has recently become accessible

on quantum simulators [42].

There is extensive literature on operator-dynamics [43–

55] and spectral form factors [56–61] in the presence

of symmetries. The primary mechanism through which

charge, or other conserved quantities, can inﬂuence oper-

ator growth is by conﬁning the access of growing opera-

tors to a speciﬁc (symmetry) sector of the Hilbert space.

In addition, within extended systems, the conservation

law leads to transport of local conserved quantities, which

couples to the operator dynamics. Therefore the correct

semi-classical picture of operator dynamics in the pres-

ence of a symmetry should at least contain two dynamical

variables, the operator size and the local density of the

conserved quantity.

In this work, we study operator dynamics in the pres-

ence of charge transport. We derive the required semi-

classical equations which couple the charge and the oper-

ator, that are valid even in inhomogeneous and dynamical

charge-density backgrounds:

∂tρ=∂2

rρ

∂tξ=∂2

rξ+ 2g2ξ(ξ2−ρ(1 −ρ)) (2)

Here ρ(r, t) is the charge density, which obeys the diﬀu-

sion equation, while ξ(r, t) is the analog of operator-size

in models with symmetries that measures local scram-

bling of the operator. The dynamics of ξis described by a

diﬀusion-reaction equation that depends on the dynami-

cal charge density. In particular, the local density bounds

the range of ξfrom 0 to pρ(1 −ρ). We note that this

class of equations has been independently studied in the

context of bacterial population growth in diﬀusive media,

where it is denoted as the ‘Diﬀusive Fisher–Kolmogorov

equation’ [62].

To derive these equations, we use the complex Brow-

nian SYK model on a lattice with U(1) symmetry. SYK

models, which are all-to-all interacting models with ran-

dom couplings [63], are well-known for their connection

to black-hole physics [64,65] and for their exactly solv-

able nature in the large-Nlimit [17–20]. Their Brownian

limit, which considers time-dependent couplings, has also

been studied in various contexts [48,50,66–70]. The dy-

namics of Brownian models can be mapped to a stochas-

tic process [39,69,71], or the imaginary time dynamics of

bosonic-models [67]. It has also been shown that Brown-

ian SYK models, in particular, give rise to emergent sym-

metry structures after the disorder averaging procedure

which can be used to compute the OTOC both at large

ﬁnite Nand in the inﬁnite-Nlimit [50]. In this work,

we show that the extended complex Brownian model is

mapped to a quantum SU(4) spin chain with inter-site

Heisenberg coupling and intra-site interaction. In the

large Nlimit, the microscopic quantum model is reduced

to the semi-classical equations in Eq. (2). We provide a

complete picture of the coupled dynamics between oper-

ators and charge. Our approach can also be extended to

systems with other symmetries.

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

provides a brief summary of the main results. In Sec. III,

we discuss the operator-basis which respects U(1) sym-

metry and can therefore be used to precisely characterize

the four dynamical charges involved in the computation

of the OTOC. In Sec. IV, we describe the details of the

complex SYK model, and the mapping of the eﬀective

imaginary-time evolution in the Brownian model to an

SU(4) spin-chain. In Sec. Vwe describe the mapping of

operator-states of complex fermions to GT-patterns in

the SU(4) algebra. We also map the physical charges to

the weights of the algebra. Sec. VI covers the primary dis-

tinguishing feature of operator dynamics restricted by a

U(1) symmetry, namely the presence of transitions which

introduce quantum operator coherence. We describe the

procedure to compute the inﬁnite-Nlimit in Sec. VII,

within which we also describe the general structure of the

‘restricted’ Fokker-Planck equation obtained for quan-

tum many-body models with charge conservation. In

Sec. VIII, we derive the equation governing the OTOC,

namely the charge-dependent FKPP equation. This re-

veals some interesting features of operator-dynamics in

the presence of symmetries, as we show in section Sec. IX,

where we examine operator-dynamics in diﬀerent charge

domain-wall backgrounds and observe the strong inﬂu-

ence of charge dynamics on even non-conserved opera-

tors. Sec. Xconcludes this work with a brief summary

and some discussions on a contrasting view of charge vs.

energy conservation.

II. SUMMARY OF THE MAIN RESULTS

The primary result of our work is captured by Eq (2).

The equation eﬀectively models the evolution of the

OTOC depicted in Eq. (1), for a charge-conserving

fermionic model deﬁned on a chain (Fig. 1(a)). The steps

via which this connection is established are:

•To begin, one speciﬁes the charge density on each

1 2

…L

…

Coherent Incoherent

FIG. 1. (a) The Brownian SYK model with Lclusters of

Ncomplex fermions, with on-site interaction Jand near-

est neighbor hopping K. The total fermion number operator

Pnr,i is the conserved operator, leading to a U(1) symmetric

circuit. (b-d) The time evolution of charge ρ(b),the OTOC

F(W(t), χr) (c), and OTOC F(W(t), nr) (d), when the ini-

tial operator Whas an initial charge distribution as shown

in (b). In each of the graphs the darkness of the plots in-

creases with increasing times. The charge (b) and conserved

part of the operator (d) have diﬀusive behavior, while the

OTOC F(W(t), χr) (c) and the uncharged part of OTOC

F(W(t), nr) propagates ballistically. (e) Schematic of the

time evolution of two copies of the operator, which is involved

in the OTOC computation. Due to the U(1) symmetry, the

super-operator develops coherence, which makes the hydro-

dynamic description hard. In this work, we argue that for

certain probing operators, the correction from this induced

coherence is 1/N suppressed. In the N→ ∞ limit, we di-

rectly obtain the hydrodynamical equations 2, which lead to

the dynamical evolution in (b-d).

site r. This ﬁxes the proﬁle of ρ(r, t = 0).

•Because the initial charge on every site has been

ﬁxed, the simplest initial operator is the projec-

tion operator onto a symmetry sector with a given

charge density on every site.

•In models where the charge is conserved on each

site, this projector is static. However, once the

charge is allowed to ﬂow from site to site, this op-

erator becomes dynamical as well.

•Just as there are multiple states within each charge

sector, there are also multiple operators [50]. The

variable ξcontrols the choice of the operator once

the charge is ﬁxed. For example, the maximal value

of ξ=pρ(1 −ρ) represents a simple operator such

as a projector, while ξ= 0 represents a local scram-

bled operator within the charge sector. This is con-

sistent with ξ= 0 being a stable solution of Eq. (2),

as all simple operators will eventually evolve into

the most complex one.

•Fixing both ρ(t= 0) and ξ(t= 0) also completely

ﬁxes the initial global operator W(t= 0) in Eq. (1).

Hence the operator Wcarries information about

two separate modes, the charge, and the ‘complex-

ity’ of the operator within the charge subspace. Fol-

lowing this, the operator is evolved using Eq. (2),

where it is observed that the mode ξspreads bal-

listically while ρis governed by diﬀusion.

•The evolved operator is then measured via the use

of a probing operator V. The choice of whether Vis

chosen to be a conserved (has overlap with charge)

or non-conserved operator determines which modes

the OTOC detects:

F(t)|V=χr0=ξ(r0, t)

F(t)|V=nr0=ξ2(r0, t) + ρ2(r0, t)(3)

The operator χr0refers to the local creation operator

on site r0of the chain, while nr0is the local number oper-

ator, which has overlap with the charge. This formalism

distinguishes the time-ordered Green’s function from the

OTOC, as the former can only detect the charged mode

and not the ﬁxed-charge operator transitions (Fig. 1).

It is important to mention that Wis a non-local oper-

ator as it ﬁxes the initial charge proﬁle (Sec. III). This

is necessary if one wishes to obtain a precise descrip-

tion of how information dynamics are related to charge

dynamics, as local operators do not have a well-deﬁned

charge. This is perhaps also related to other works which

have observed that locality has non-trivial implications

in the presence of symmetries [72]. The equations in

Eq. (2) correctly reproduce all known features of charged

chaotic models, such as the charge-dependent Lyapunov

exponent/butterﬂy velocity [47,48], and the late-time

diﬀusive tail of the OTOC [44,45,73]. Moreover, since

they are valid for arbitrary initial charge/operator pro-

ﬁle, we simulate the coupled equations in inhomogeneous

backgrounds to obtain new phenomena.

The formalism developed in this work encodes the mi-

croscopic operator dynamics in terms of transitions be-

tween states in a particular irrep of the SU(4) algebra.

Since the OTOC is computed on four time-contours, this

allows us to track the dynamics of the four corresponding

conserved charges in terms of the weights of the SU(4)

algebra, and explains the emergence of the coupled equa-

tions describing the OTOC in terms of a single charged

mode (ρ) and a non-conserved ballistic mode (ξ). This

derivation reveals new features of operator dynamics in

the presence of conservation laws. Namely, in contrast

with models that do not have continuous symmetries, the

operator dynamics in the model with a U(1) symmetry

allow for transitions that introduce quantum coherence at

the operator level. Therefore, while usual noisy/random

models are well described by a classical stochastic pro-

cess, for U(1) symmetric models the time-evolution of the

operator can only be modeled by a probability in a ﬁxed

subspace, and in the inﬁnite-Nlimit, this gives rise to a

Fokker-Planck equation which only describes a conserved

quantity in the ‘incoherent’ sector of the operator-states.

III. CHARGED OPERATOR BASIS

In this work we will be concerned with computing the

OTOC, which can be viewed through the lens of operator

spreading. Due to the charge conservation, we work with

a speciﬁc choice of operator basis. We assume the degrees

of freedom are represented by complex fermions, which

satisfy the anti-commutation relations:

{χ†

i, χj}=δij ;{χi, χj}= 0

The relevant operators we consider are left and right

eigen-operators with respect to the U(1) symmetry:

X

niO=qaO;OX

ni=qbO(4)

where (n=χ†χ) is the number operator. It can be

easily checked that for a model with U(1) conservation,

the charges (qa, qb) are a constant of motion during the

dynamics of the operator string. For a single fermionic

operator, we choose the relevant four dimensional eigen-

operator basis and the charges take the following respec-

tive values:

χ†: (1,0) χ: (0,1) n: (1,1) ¯n: (0,0) (5)

where ¯n=I−n. Following this insight, we will work with

operator strings Swhen the system contains Nfermions,

where each element of the string is picked from the U(1)

operator basis deﬁned above:

S= 2N/2s1s2···sN, si∈ {χ†

i, χi, ni,¯ni}.(6)

This ensures that the entire operator string is also an

eigen-operator of the global U(1) symmetry. The factor

of 2N/2in the string is picked to ensure the following

orthogonal and completeness relations

trItr(S†S0) = δ(S0,S),1

trIX

SS†

mnSpq =δmq δnp.(7)

The procedure to compute the OTOC, on the other

hand, begins by rearranging the correlator to write it in

FIG. 2. Figure adapted from [50]. In the ﬁrst part of (a),

the process of resolving the identity in the output state to

ensure a proper overlap with the input state is depicted. For

the Brownian model, disorder-averaging after this procedure

turns the OTOC into an amplitude, computed after the states

are evolved in imaginary time using an eﬀective Hamiltonian

H. (b) and (c) depict the null-eigenstates on four-time con-

tours, which are a consequence of the dynamics being Unitary.

While (b) is simply the double-copy of the identity, (c) rep-

resents the fully-scrambled steady state PS|S†⊗ Si if the

dynamics are also chaotic.

the operator state language:

F(W(t), V ) = 1

trItr(W†(t)V†W(t)V) = trIhout|U|ini

|ini=1

trIXW†

mnWpq |m⊗n⊗p⊗qi

|outi=1

trIXV†

mqVpn |m⊗n⊗p⊗qi

U=U⊗U∗⊗U⊗U∗

(8)

Hence the computation of the OTOC involves four copies

of the Unitary and in a charge-conserving model, four

corresponding conserved charges as well. These con-

served charges, which we label as (qa, qb, qc, qd), are the

left and right charges of the double-copy of the operator-

state involved in the OTOC. Therefore, an exact dynam-

ical description of the OTOC would depend on the evo-

lution of these four independent charges as well.

To complete the formalism and ensure a proper over-

lap between the input and output state, we insert the

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Chargetransport,informationscramblingandquantumoperator-coherenceinamany-bodysystemwithU(1)symmetryLakshyaAgarwal,1SubhayanSahu,2,3andShenglongXu11DepartmentofPhysics&Astronomy,TexasA&MUniversity,CollegeStation,Texas77843,USA2CondensedMatterTheoryCenterandDepartmentofPhysics,UniversityofMaryland,Col...

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Charge transport information scrambling and quantum operator-coherence in a many-body system with U1 symmetry Lakshya Agarwal1Subhayan Sahu2 3and Shenglong Xu1

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