Open-system Spin Transport and Operator Weight Dissipation in Spin Chains Yongchan Yoo1Christopher David White2 3and Brian Swingle4 1Department of Physics Condensed Matter Theory Center and Joint Quantum Institute

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Open-system Spin Transport and Operator Weight Dissipation in Spin Chains

Yongchan Yoo,1Christopher David White,2, 3 and Brian Swingle4

1Department of Physics, Condensed Matter Theory Center, and Joint Quantum Institute,

University of Maryland, College Park, Maryland 20742, USA

2Joint Center for Quantum Information and Computer Science,

University of Maryland, College Park, Md, 20742

3Condensed Matter Theory Center, University of Maryland, College Park, Md, 20742

4Department of Physics, Brandeis University, Waltham, Massachusetts, 02453

(Dated: March 14, 2023)

We use non-equilibrium steady states to study the eﬀect of dissipation-assisted operator evolution

(DAOE) on the scaling behavior of transport in one-dimensional spin chains. We consider three

models in the XXZ family: the XXZ model with staggered anisotropy, which is chaotic; XXZ

model with no external ﬁeld and tunable interaction, which is Bethe ansatz integrable and (in the

zero interaction limit) free fermion integrable; and the disordered XY model, which is free-fermion

integrable and Anderson localized. We ﬁnd evidence that DAOE’s eﬀect on transport is controlled by

its eﬀect on the system’s conserved quantities. To the extent that DAOE preserves those symmetries,

it preserves the scaling of the system’s transport properties; to the extent it breaks those conserved

quantities, it pushes the system towards diﬀusive scaling of transport.

I. INTRODUCTION

Quantum out-of-equilibrium dynamics is at the heart

of various areas of physics from condensed matter to high

energy physics and even quantum information science.

The dynamics of conserved quantities is particularly in-

teresting within this broad non-equilibrium setting. In

the solid-state context, measurements of transport of

conserved quantities like energy and charge provide a use-

ful window into the underlying dynamics of these com-

plex systems. In particular the scaling behavior of a sys-

tem’s transport properties—whether it is diﬀusive, sub-

diﬀusive, or superdiﬀusive, as well as details like the na-

ture of the scaling function—is intimately connected with

the strength of the system’s interactions [1], the presence

of kinetic constraints and higher-form symmetries [2–12],

and its integrable or chaotic nature. Recent experimental

developments in various platforms including cold atom

systems [13–18], quantum magnets [19], superconducting

quantum circuits [20], and heavy-ion collisions [21] are

also shedding light on the subject. Along with those ex-

perimental results, new theoretical approaches have been

developed to tackle the major challenge of calculating

and interpreting the observed transport phenomena. Due

to the breadth of the subject, theoretical developments

include a range of approaches from general frameworks

to techniques for speciﬁc situations (reviews include [22–

30]).

These new approaches are especially important for

strongly interacting systems where the physical interpre-

tation of transport phenomena is not well understood.

Numerical approaches are indispensable since there is of-

ten no simple analytical technique available. Tensor net-

work algorithms, especially matrix product state meth-

ods, can access transport physics close to the thermo-

dynamic limit [31–33]. For other commonly considered

problems (e.g. ground states of gapped local Hamilto-

nians and short-time evolution), matrix product state

methods are reliable because the states in question have

low entanglement. For short-time evolution in particu-

lar, TEBD [34,35] constitutes a controlled approxima-

tion. But matrix product state methods become expen-

sive for systems with slow dynamics (e.g., subdiﬀusive

transport [36,37]) or high amounts of entanglement.

Some alternate techniques have been suggested [38–49].

Many of those methods modify the dynamics to a non-

unitary time evolution not unlike a Lindblad dynamics.

By doing so, they cut oﬀ (notionally) less relevant parts

of the dynamics while preserving the essential transport

physics. From a tensor network perspective, one impor-

tant outcome of the modiﬁcation is to reduce the amount

of entanglement while preserving the physics of interest.

Developing a principled theory of when and why these

methods work is an active line of research [50–52].

One of these new tensor network methods, dissipation

assisted operator evolution (DAOE) [44] employs an ar-

tiﬁcial dissipation based on operator weight to overcome

the entanglement barrier in unitary simulations. Here op-

erator weight refers to the number of non-identity single-

qubit operators contained in a many-body operator;

suppressing high-weight operators—that is, suppressing

many-point correlations—suppresses many-body entan-

glement. Because the conserved quantities and their cur-

rents are local operators, the artiﬁcial dissipation does

not directly modify those quantities or currents. In a

chaotic system, the expectation values of conserved quan-

tities and currents determine the state of the system, so

one expects the artiﬁcial dissipation not to substantially

modify the system’s state or dynamics. Moreover, be-

cause DAOE directly manipulates the operator weight

distribution, it is possible to study the inﬂuence of oper-

ator growth [53–58] on transport physics. Fig. 1gives a

schematic of DAOE as implemented with matrix product

operators.

We investigate the eﬀect of operator weight dissipa-

tion on the scaling behavior of spin transport in one-

arXiv:2210.06494v2 [cond-mat.str-el] 13 Mar 2023

dimensional lattice models by combining two sources of

non-unitarity: DAOE and boundary-driven open system

dynamics. The physical quantity of interest is the scaling

exponent relating the average spin current to the system

size. Diﬀusive transport gives one characteristic value of

the exponent, and the exponent allows us to characterize

the transport away from the diﬀusive case.

First, as a benchmark, we apply the method to the

anisotropic XXZ model with a staggered ﬁeld, which is

chaotic and possesses normal diﬀusive relaxation of the

spin current. We ﬁnd that for any operator dissipation

parameters, the normal diﬀusive transport is maintained.

Next, we study the clean XXZ model in three diﬀerent

regimes. In the weak interaction regime, the modiﬁed

transport shows superdiﬀusive transport up to the sys-

tem size we calculated (N∼256), whereas the unitary

limit is believed to exhibit ballistic transport [59–62]. At

the isotropic point (∆ = 1) where the non-dissipative

transport exhibits a superdiﬀusive relaxation, the trans-

port under DAOE is still superdiﬀusive but the scaling

exponents vary depending on the operator cut-oﬀ length.

In the strong interaction regime (∆ >1), the unitary sys-

tem’s diﬀusive transport is retained for all operator cut-

oﬀ lengths up to the largest system size. Lastly, we treat

the disordered XY model. There we observe behavior

consistent with coherent transport on length scales given

by the DAOE cut-oﬀ length and diﬀusive transport on

longer length scales; we explain this in terms of DAOE’s

eﬀect on Anderson orbitals.

Taken together, these results point to the following

conclusions. First, as a technical point, DAOE can be

usefully combined with open system dynamics. This in-

troduces a need to extrapolate to the physical limit, but

the NESS is generally easier to obtain and more stable

in the presence of artiﬁcial dissipation. Second, DAOE

tends to push the dynamics towards diﬀusive transport,

all other things being equal. It maintains diﬀusivity for

generic chaotic models and typically breaks integrability

in non-chaotic models. Third, how well DAOE captures

the underlying unitary dynamics depends sensitively on

the number and nature of the symmetries it preserves.

We elaborate on these points in the discussion.

The rest of the paper is structured as follows. In Sec-

tion II we introduce the model and the framework for

analyzing spin currents. Next, in Section III we describe

the methods combining DAOE with open system dynam-

ics. In Section IV we present our main results which in-

clude various one-dimensional spin models and crossovers

between diﬀerent transport types. Lastly, we discuss the

results and possible future directions in Section V.

II. MODEL AND QUANTITIES OF INTEREST

A. Model

We study spin transport in three variations of an XXZ

spin chain. The general form of the Hamiltonian is

N−1

i=1

Hi,i+1,(1a)

Hi,i+1 =σx

iσx

i+1 +σy

iσy

i+1 + ∆σz

iσz

i+1

2(hiσz

i+hi+1σz

i+1),(1b)

where σα

i’s are Pauli matrices, ∆ controls the anisotropy,

and hiis the magnitude of z-directed ﬁeld at site i.

The model (1) displays a rich variety of spin-transport

behaviors. At ∆ = 0 it is free-fermion integrable, so

it displays ballistic transport if the hjare uniform and

Anderson localization if the hjare random. It can also

exhibit a transition between the two behaviors if the hj

are appropriately quasiperiodic [63].

For ∆ 6= 0 and hj= 0 uniform, the model is Bethe-

ansatz integrable. At half ﬁlling it is ballistic for ∆ ≤1

and diﬀusive for ∆ >1 [64–89]. ([65] Sec. 6 has a use-

ful, concise summary of this literature.) At the isotropic

point ∆ = 1, h = 0 the model is SU(2) symmetric; this

symmetry appears to protect the superdiﬀusive behavior,

which remains even for large SU(2)-symmetric perturba-

tions [90].

For ∆ 6= 0 and hirandom, the spin transport is not well

understood. For small to moderate disorder, the model

appears to display anomalous diﬀusion [36,91–97]. This

anomalous diﬀusion may be due to Griﬃths rare region

eﬀects [98–102], but other possible scenarios include ir-

regular scaling of the matrix elements [103], multifractal-

ity of eigenstates [104], and a non-Griﬃths phenomeno-

logical theory of the resistance distribution [37]. The sys-

tem may undergo a many-body localization (MBL) [105–

107] transition at h≈7.6, but recent work has cast

doubt on the location and indeed existence of the tran-

sition [108–115]. Detailed reviews are available for the

MBL phases [29,116,117].

In this paper, we consider three parameter regimes:

one chaotic, one Bethe ansatz integrable, and one An-

derson localized. DAOE [44] was designed to compute

transport coeﬃcients in the ﬁrst regime, for chaotic one-

dimensional quantum systems. To test the method in

this case we consider the anisotropic XXZ model with

anisotropy ∆ = 0.5 and staggered ﬁeld h2i=−0.5 and

h2i+1 = 0. With these parameters, the model is non-

integrable and shows diﬀusive transport [118].

We then pick two cases to study anomalous transport:

the zero-ﬁeld XXZ model and the disordered XX model.

The XXZ model at zero ﬁeld disorder h= 0 exhibits var-

ious transport types as the anisotropy ∆ increases from

zero. For weak anisotropy ∆ <1, one ﬁnds ballistic

transport; in the opposite regime ∆ >1, the model ex-

hibits normal diﬀusive transport. The critical point is

FIG. 1. Schematics of the combined method of the boundary

driven open quantum system and DAOE. It describes one

period of the artiﬁcial dissipation superoperator application.

The gradation of |ρii from red to blue is for the spin imbalance

by the Markovian spin bath setting.

at the isotropic point, ∆ = 1, where superdiﬀusive but

subballistic transport occurs. The disordered XY model

(∆ = 0, h6= 0) also displays an interacting crossover

in its transport. The clean limit has ballistic trans-

port thanks to a dual free fermion description, whereas

any non-zero disorder brings Anderson localization in the

thermodynamic limit [119]. But the model always pos-

sesses an extensive number of conserved quantities, and

the physical size of these conserved quantities in the spin

language varies with the disorder strength.

B. Spin Current Analysis

Suppose a system has a conserved quantity Q=PiQi,

Qilocal. The corresponding local current Jiis derived

from the continuity equation and the Heisenberg equa-

tions of motion:

∂Qi

∂t =−i[Qi, H] = −(Ji−Ji+1).(2)

The model (1) has a conserved quantity Qz=Piσz

i, the

total z-spin; the current is Ji=σx

iσy

i+1 −σy

iσx

i+1.

For systems exhibiting diﬀusive transport, the discrete

Fourier’s law hJii=−D(hQi+1i−hQii) relates the cur-

rent and the corresponding charge density. Here Dis

the diﬀusion constant in lattice units. If such a diﬀusive

system is subject to a bias hQLi−hQRi, where hQL,Ri

denote ﬁxed values of the spin density at the left and

right ends of the sample, the current through the sample

scales as

hJi=−DhQLi−hQRi

N,(3)

where Nis the length of the system.

More generally, if the system exhibits anomalous trans-

port, the above relation is modiﬁed by introducing a scal-

ing exponent χ,

hJi=−DχhQLi−hQRi

Nχ(4)

We assume that χin Eq. (4) is the only scaling exponent

that characterizes the transport. Other than the normal

diﬀusive transport (χ= 1), possible types of anomalous

transport are (i) ballistic transport (χ= 0), (ii) superdif-

fusive transport (0 <χ<1), and (iii) subdiﬀusive trans-

port (χ > 1). In a localized state, this power-law ansatz

does not provide a good description of the spin trans-

port. We can heuristically understand localized systems

as having χ→ ∞.

III. METHODS

We extract quantum transport properties with dissi-

pation assisted operator evolution (DAOE) simulations

of non-equilibrium steady states (NESS). Each method

takes advantage of non-unitary evolution to make simu-

lating a system’s dynamics tractable. We ﬁnd that com-

bining them gives new insights into both the systems’

physics and the eﬀect of DAOE on that physics. In this

section we describe the two methods.

A. Master Equation and NESS

In a NESS experiment on a spin chain we attach leads

with slightly diﬀerent chemical potentials to the two ends

of the system. The leads thermalize the system, so in the

long-time limit its state should have an eﬃcient MPO

representation[120,121]. But because the leads’ chem-

ical potentials diﬀer, they induce a small spin current;

how this current scales with system size characterizes the

model’s transport properties (cf Sec. II B).

Formally the NESS is the ﬁxed point solution

dρ∞/dt = 0 of the Gorini-Kossakowski-Lindblad-

Sudarshan (GKLS) master equation [122,123]

dρ

dt =L(ρ)≡i[ρ, H] + X

νLνρL†

ν−1

2L†

νLν, ρ.(5)

The NESS is generated by full Hamiltonian Hand Lind-

blad operators Lν, which model the leads. Explicitly, the

Lindblad operators are

L1,±=p1±µ σ±

LN,±=p1∓µ σ±

(6)

where σ±=1

2(σx±iσy).

Under the right conditions, the GKLS equation has

exactly one steady-state solution [124], but even when

this is the case, there may still be many slowly-decaying

almost steady states, especially when the jump operators

only aﬀect the edges of the sample. We expect to have

a unique NESS ρ∞which is accessible in the long-time

limit t→ ∞, but the presence of slow modes means we

must be careful about convergence in time.

B. Artiﬁcial Dissipation Superoperator

Dissipation assisted operator evolution (DAOE) [44]

is a tensor network-based algorithm that reduces the

weight of operators longer than a given cut-oﬀ length

[125]. These long operators are responsible for the en-

tanglement growth that makes MPS simulations infeasi-

ble. By gently reducing them, DAOE makes long-time

simulations possible.

DAOE is implemented by an artiﬁcial dissipation su-

peroperator acting on the operator Hilbert space. The

operator Hilbert space of our N-site system is spanned by

a basis of 4NPauli strings. Each element (Pauli string)

Sin the basis is represented by the tensor product of

single-site Pauli matrices σ0, σx, σy, σz. The length `Sof

a string Sis the number of non-trivial Pauli matrices in

S. In this notation the artiﬁcial dissipation superopera-

tor is

D`∗,γ [S] = (Sif `S≤`∗

e−γ(`S−`∗)Sif `S> `∗.(7)

Periodically applying this superoperator generates a non-

unitary quantum evolution that can be heuristically un-

derstood as a global ‘bath’. Just as a bath—consider in

particular the depolarizing channel—reduces the expec-

tation value of a string of `nontrivial Pauli operators by

an amount ∝`, the DAOE superoperator (7) reduces the

expectation value of a string of `nontrivial Pauli opera-

tors by an amount ∝max(`−`∗,0).

DAOE as presented in [44] uses the above artiﬁcial op-

erator dissipation to modify the Heisenberg picture dy-

namics of observables. For example, starting from an ini-

tial state ρ0with some spatially varying proﬁle for the av-

erage spin density tr(Sz

rρ0), the spin diﬀusivity can be ex-

tracted from the time-dependent spin proﬁle tr(Sz

r(t)ρ0)

where Sz

r(t) is the Heisenberg evolution of Szat site r.

DAOE is then used to modify the dynamics of Sz

r(t) to

render it more tractable to an entanglement-constrained

tensor network simulation, with the true physics obtained

from an extrapolation in γ.

However, more than just modifying the particular dy-

namics with the introduction of γ, DAOE signiﬁcantly

alters the basic rules of quantum evolution. This is most

easily seen in the Schrodinger picture formulation, where

the fact that DAOE reduces expectation values of long

operators without reducing expectation values of short

operators means it can break positivity of density matri-

ces. Consider applying D`∗=1,γ to the density matrix of

the two-site state |↑↑i: it becomes

4D`∗=1,γ hI+σz

1+σz

2+σ1

zσ2

=I+σz

1+σz

2+e−γσ1

zσ2

(8)

which has one negative eigenvalue 1

4(e−γ−1).

Crucially, the Heisenberg and Schrodinger pictures re-

main equivalent even in the presence of DAOE’s artiﬁcial

operator dissipation [126]. In the Heisenberg picture, the

time evolution of some operator Aby a Lindbladian L

becomes

A(t) = D`∗,γ e−iLτt/τ A(0) .(9)

But the DAOE superoperator, like the Lindblad time

evolution superoperator eiLτ, is linear; indeed D`∗,γ

is Hermitian under the trace inner product hA, Bi=

tr A†B. So the operator expectation value hA(t)iobeys

tr A(t)ρ(0) = tr Aρ(t) (10)

where Ais the Schr¨odinger picture operator and

ρ(t) = D`∗,γ ρ(0)eiLτt/τ .(11)

Hence, the Schr¨odinger and Heisenberg pictures give

identical dynamics, so we can speak of DAOE “failing

to preserve positivity”. Moreover we can use NESS sim-

ulations to examine how DAOE changes a system’s dy-

namics, with full conﬁdence that the results apply to

Heisenberg-picture experiments like those of [44].

Given the signiﬁcant modiﬁcations that DAOE makes

to the quantum dynamics, it is important to under-

stand when and why DAOE gives a good approxima-

tion of the transport coeﬃcients. Ref. [50] analyses the

“operator backﬂow” process, in which information con-

tained in large-diameter, non-local operators ﬂows into

the subspace of short operators as the system evolves.

For chaotic models, combinatoric scattering amplitude

arguments and numerical experiments conﬁrm the expo-

nential suppression of the backﬂow process contribution

to correlation functions between local operators. Con-

sequently, the estimated error of DAOE-produced trans-

port coeﬃcients is also exponentially small in such sys-

tems.

But this backﬂow analysis will not go through for inte-

grable systems. In the language of [51], the backﬂow

analysis assumes that the dynamics of long operators

is chaotic. But in the integrable system the tower of

local conserved quantities will strongly constrain that

dynamics—it cannot be treated as chaotic. Additionally,

the interplay of DAOE with more complex non-Abelian

symmetries and with integrability has not yet been stud-

ied. Some of our results below address these open prob-

lems.

C. Tensor Network Implementation

Both NESS and DAOE can be eﬃciently realized in

the language of tensor networks. In this formalism, a

vector in the operator Hilbert space directly expresses

the corresponding density matrix; we call such a vector

asuperket state |ρii. Physical operators can act on ρin

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Open-systemSpinTransportandOperatorWeightDissipationinSpinChainsYongchanYoo,1ChristopherDavidWhite,2,3andBrianSwingle41DepartmentofPhysics,CondensedMatterTheoryCenter,andJointQuantumInstitute,UniversityofMaryland,CollegePark,Maryland20742,USA2JointCenterforQuantumInformationandComputerScience,Univer...

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Open-system Spin Transport and Operator Weight Dissipation in Spin Chains Yongchan Yoo1Christopher David White2 3and Brian Swingle4 1Department of Physics Condensed Matter Theory Center and Joint Quantum Institute.pdf

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Open-system Spin Transport and Operator Weight Dissipation in Spin Chains Yongchan Yoo1Christopher David White2 3and Brian Swingle4 1Department of Physics Condensed Matter Theory Center and Joint Quantum Institute

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