Superdiffusive Energy Transport in Kinetically Constrained Models

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Superdiﬀusive Energy Transport in Kinetically Constrained Models

Marko Ljubotina,1, ∗Jean-Yves Desaules,2, ∗Maksym Serbyn,1and Zlatko Papi´c2

1Institute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria

2School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK

(Dated: March 9, 2023)

Universal nonequilibrium properties of isolated quantum systems are typically probed by study-

ing transport of conserved quantities, such as charge or spin, while transport of energy has received

considerably less attention. Here, we study inﬁnite-temperature energy transport in the kinetically-

constrained PXP model describing Rydberg atom quantum simulators. Our state-of-the-art nu-

merical simulations, including exact diagonalization and time-evolving block decimation methods,

reveal the existence of two distinct transport regimes. At moderate times, the energy-energy cor-

relation function displays periodic oscillations due to families of eigenstates forming diﬀerent su(2)

representations hidden within the spectrum. These families of eigenstates generalize the quantum

many-body scarred states found in previous works and leave an imprint on the inﬁnite-temperature

energy transport. At later times, we observe a long-lived superdiﬀusive transport regime that we

attribute to the proximity of a nearby integrable point. While generic strong deformations of the

PXP model indeed restore diﬀusive transport, adding a strong chemical potential intriguingly gives

rise to a well-converged superdiﬀusive exponent z≈3/2. Our results suggest constrained models to

be potential hosts of novel transport regimes and call for developing an analytic understanding of

their energy transport.

I. INTRODUCTION

The understanding of out-of-equilibrium properties

of many-body systems is one of the central problems

in quantum statistical physics. The universal aspects

of nonequilibrium dynamics are commonly probed by

quantum transport at inﬁnite temperature. Generic

chaotic models typically exhibit diﬀusive transport of

conserved quantities such as spin [1–7], charge [1,4] or

energy [2,4,8–10]. On the other hand, disorder can give

rise to slower than diﬀusive (subdiﬀusive) dynamics or

even localization [3,11–16].

In contrast to (sub)diﬀusion, faster-than-diﬀusive

transport typically rests on the existence of special

structures. In one dimension, integrable models [17]

can support ballistic transport since their mascroscopic

number of conserved quantities may prevent currents

from decaying. Furthermore, intermediate behavior be-

tween diﬀusion and ballistic transport can arise in inte-

grable models with certain symmetries, where superdiﬀu-

sive Kardar-Parisi-Zhang (KPZ) dynamics has been ob-

served [5,7,18–28]. Importantly, all examples of faster-

than-diﬀusive dynamics in short range models rely on

integrability. The same, naturally, does not hold for

long-range models where superdiﬀusion has also been ob-

served and explained by classical arguments using L´evy

ﬂights [29,30].

In addition to disorder and integrability, it was re-

cently shown that kinetic constraints may also lead to

a diﬀerent class of dynamics known as “quantum many-

body scars” (QMBS) [31–33]. For instance, the so-called

PXP model [34,35] that describes constrained dynamics

∗M.L. and J.-Y.D. contributed equally to this work.

in Rydberg atom quantum simulators [36,37], displays

weak ergodicity breaking due to the presence of periodic

revivals in the dynamics and the existence of nonther-

malizing eigenstates in its spectrum [38–42]. These rare

nonergodic eigenstates were understood as forming a sin-

gle approximate su(2) algebra representation, embedded

into the spectrum of the Hamiltonian, even though the

latter has no SU(2) symmetry [43,44]. In addition to the

phenomenon of scarring, recent studies of inﬁnite tem-

perature charge transport in random constrained Floquet

models revealed several examples of slower-than-diﬀusive

dynamics [45,46].

In this paper we study energy transport in the PXP

model. To access dynamics at long times, we use time-

evolving block decimation (TEBD) [47,48] to calculate

the energy-energy correlation function. Intuitively, this

setup tracks the spreading of a small energy “hump” cre-

ated at the central site of the chain, atop of the inﬁnite

temperature density matrix. At short times the spread-

ing of the initial energy inhomogeneity is characterized

by oscillations that we attribute to the existence of multi-

ple approximate su(2) algebra representations hidden in

the spectrum of the PXP model. The eigenstates forming

these representations encompass the previously identiﬁed

QMBS eigenstates [38,44], but also include additional

eigenstates corresponding to lower-spin representations.

We relate the multiple su(2) representations to the oscil-

latory behavior observed in energy transport at inﬁnite

temperature, mirroring the QMBS revivals observed in

quenches from special initial states [36]. At later times,

the oscillations due to multiple su(2) representations are

damped and we observe faster-than-diﬀusive decay of the

energy density. The superdiﬀusive (possibly transient)

dynamics provides further evidence to the existence of an

integrable point, proximate to the PXP model [49]. Sur-

prisingly, while deforming the PXP model with a chem-

arXiv:2210.01146v2 [cond-mat.stat-mech] 8 Mar 2023

ical potential removes the remnants of integrability, in-

stead of restoring diﬀusion, it leads to a stable superdif-

fusive regime with a dynamical exponent z≈3/2.

The remainder of this paper is organized as follows.

In Sec. II we introduce the PXP model and present the

results for its energy transport at inﬁnite temperature.

In Sec. III we discuss the short-time regime of the energy

transport, which is characterized by oscillations that we

attribute to multiple su(2) representations. In Sec. IV we

analyze the late-time energy transport by applying inte-

grable deformations to the PXP model. The eﬀect of the

chemical potential is studied in Sec. V, where it is shown

to lead to a robust regime of superdiﬀusive transport at

the accessible time scales. Our conclusions are presented

in Sec. VI, while the appendices contain further details

on the numerics simulations, construction of the multiple

su(2) representations, and larger-range Rydberg blockade

that leads to diﬀusive energy transport.

II. INFINITE-TEMPERATURE ENERGY

TRANSPORT IN THE PXP MODEL

The Hamiltonian of the PXP model [35]

HPXP = Ω X

Pi−1σx

iPi+1,(1)

operates on a chain of Nspins-1/2, where Pi=|↓iih↓|iis

a local projector on the |↓i state and σx

iis the correspond-

ing Pauli matrix. The projectors Piencode the Rydberg

blockade mechanism [50] and lead to a block-diagonal

structure of HPXP, also known as Hilbert space frag-

mentation [51–53]: any two consecutive up spins, |↑↑i,

remain frozen under the dynamics generated by HPXP,

eﬀectively disconnecting the chain at that point. In what

follows, we will set the Rabi frequency Ω = 1 and work

in the reduced Hilbert space, i.e., the largest sector that

excludes any consecutive pairs of up spins. The PXP

model in the reduced Hilbert space exhibits the repul-

sion of energy levels typical of chaotic systems [38].

We probe energy transport via the connected energy

correlation function

hh0(0)h`(t)ic=hh0(0)h`(t)i−hh0(0)ihh`(t)i,(2)

where h`(0) = P`−1σx

`P`+1 is the energy density operator

at site `, and h`(t) = eiHPXP th`(0)e−iHPXP t. Crucially,

the expectation values in Eq. (2) are evaluated by taking

the trace with respect to the inﬁnite temperature density

matrix within the reduced Hilbert space. Speciﬁcally, we

deﬁne expectation value of a given operator O

hOi ≡ tr(PO . . .),(3)

where the global projector P=Qi(1i,i+1 −nini+1), with

ni=|↑iih↑|i, annihilates any states with two neighboring

up spins.

The projection on the reduced Hilbert space is a cru-

cial diﬀerence with respect to the earlier studies of par-

ticle transport in constrained models, e.g., in Ref. [45].

100101102

10−2

10−1

100

hh0(0)h0(t)ic

I II

(a)

0 100 200 300

1/z

(b)

50 100 200

0.1

0.2

0.4

1/z −1/2

(c)

Figure 1. (a) Connected energy autocorrelation function in

the PXP model has a short-time oscillatory regime (I) fol-

lowed by power-law decay (II). (b) The inverse instantaneous

dynamical exponent extracted from the correlation function

approaches the ballistic value of one at t∼100, followed by

slow decay at later times that appears to saturate to a su-

perdiﬀusive value 1/z ≈2/3. (c) Double-logarithmic plot of

the data in (b) shows that power-law convergence to diﬀusion

1/z = 1/2 is also consistent with the data. Dashed line cor-

responds to ∝t−0.8dependence. The data is for a chain with

N= 1024 sites and bond dimension of χ= 512.

The existence of many disconnected sectors in the Hilbert

space is expected to slow down the transport. Indeed,

below we observe diﬀusive or superdiﬀusive transport, in

contrast to slower dynamics observed in Ref. [45].

Accessing energy transport in the thermodynamic

limit requires the evaluation of the connected correlation

function (2) in large systems at late times. To access the

required system sizes and times, we use a state-of-the-

art massively parallel implementation of the TEBD algo-

rithm [47,48] based on the ITensor library [54]. This al-

lows us to simulate operator dynamics in the PXP model

up to times exceeding t&300, requiring N= 1024 lat-

tice sites to avoid the operator spreading reaching the

boundary of the system (see Appendix Afor further de-

tails on the implementation and convergence of the data

with bond dimension).

Fig. 1(a) highlights two distinct regimes in the decay

of the connected energy autocorrelation function for the

PXP model. At short times, marked by the shaded area,

we observe oscillatory behavior which we explain in the

following section. At long times, these oscillations disap-

pear and the correlation function settles to a power-law

like decay. This decay is conveniently probed via the

instantaneous dynamical exponent

z−1(t) = −d lnhh0(0)h0(t)ic

d ln t,(4)

that gives the running exponent of the power-law decay.

Fig. 1(b) shows that 1/z ﬁrst approaches the ballistic

value z= 1 before relaxing slowly to a smaller value. De-

spite the decrease of 1/z, its value remains superdiﬀusive

(z < 2) even at extremely long times t≈300, at which

the correlation function has spread approximately 300

sites from the center. Nevertheless, plotting the data on

a log-log scale in Fig. 1(c), one cannot rule out power-law

relaxation of zto diﬀusion at times that are inaccessible

to our numerics.

III. SHORT-TIME OSCILLATIONS AND

MULTIPLE SU(2) REPRESENTATIONS

Unlike generic thermalizing models, the PXP model

displays unusual sensitivity of its dynamics on the ini-

tial state, which has attracted considerable attention [31].

While most initial states undergo fast thermalizing dy-

namics in the PXP model, special states such as the

N´eel state, |Z2i=|↑↓↑↓. . .i, feature long-lived quantum

revivals [36], accompanied by a slow growth of entan-

glement [38]. The non-thermalizing dynamics were ex-

plained by the existence of N+ 1 special QMBS eigen-

states embedded throughout the spectrum [38]. How-

ever, these QMBS eigenstates constitute a vanishing frac-

tion of the total number of states in the exponentially

large Hilbert space, hence they are not expected to aﬀect

transport properties in the thermodynamic limit. Re-

markably, the short-time oscillations in the regime I of

Fig. 1(a) bear striking parallels with the scarred quantum

revivals. In what follows we show that the PXP model

hosts a much larger set of non-thermalizing eigenstates

that cluster around approximately equidistant energies.

These “towers of states” account for the oscillations in

Fig. 1.

To reveal the bulk spectral properties of the PXP

model, we consider a smoothened density of states

(sDOS) and spectral form factor (SFF) [55]. The sDOS

is deﬁned as

ρσ2(E) = (1/D)X

exp−(E−En)2/2σ2/√2πσ2,(5)

where Enare eigenenergies, Dis the reduced Hilbert

space dimension, and σsets the smoothing interval.

Fig. 2(a) shows that sDOS has tiny oscillations in the

middle of the spectrum. These oscillations can be made

more prominent by subtracting the sDOS at low and high

variances, ∆ρ=ρσ2=0.06 −ρσ2=0.5, plotted in the inset.

The energy diﬀerence between the peaks in sDOS roughly

coincides with the oscillation period of the energy corre-

lation function in Fig. 1(a). A similar timescale is also

observed in the SFF, deﬁned as

K(t) = X

n,m

e−i(En−Em)t,(6)

plotted in Fig. 2(b) for a range of system sizes. The SFF

shows clear peaks at times t≈5.1 and t≈10.2, indicated

Figure 2. Signatures of multiple su(2) representations in

the PXP model. (a) Oscillations in the smoothened density

of states for N= 28 sites, with the inset showing the dif-

ference between diﬀerent sDOS (see text). (b) Spectral form

factor for various system sizes. The dashed lines indicate the

approximate times of the peaks in Fig. 1and Fig. 4below.

erator and the PXP eigenstates. The red squares indicate

the primary QMBS eigenstates with high overlap on the N´eel

state. All data is obtained by exact diagonalization of the

PXP model with periodic boundary conditions.

by dashed lines. Although the SFF, in general, is not a

self-averaging quantity, the peaks appear converged in

system size, and thus they are expected to persist in the

thermodynamic limit.

Oscillations in sDOS and SFF noted above can be

explained by the clustering of eigenstates into towers

that are approximately equally spaced in energy. We

explain these towers of states as stemming from addi-

tional approximate su(2) representations that general-

ize the family of N+ 1 QMBS eigenstates identiﬁed in

Ref. [43]. To explicitly construct multiple su(2) repre-

sentations, we use the dimer picture from Ref. [41] and

project the free spin-1 paramagnet with N/2 particles

onto the constrained Hilbert space [56]. The equiva-

lent spin-1 model is obtained from PXP by mapping the

Hilbert space of adjacent pairs of spin-1/2 onto that of

spin-1 as |↑↓i=|−i,|↓↓i=|0i, and |↓↑i=|+i. The global

spin operators in the spin-1 representation are deﬁned

as Sα=Pb∈ΛBSα

b, where Λbis the set of non over-

lapping spin-1/2 pairs and Sα

bare spin-1 operators (see

Appendix Cfor details). For the PXP Hamiltonian in

Eq. (1), the projection of the maximal total spin eigen-

states of Sxgives an excellent approximation to the orig-

inal N+ 1 QMBS states of the PXP model. Crucially,

we ﬁnd that other eigenstates of Sxwith a large, but

smaller than maximum total spin, |S|=|Sx|=N/2−d,

with dN, also provide a good approximation to the

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SuperdiusiveEnergyTransportinKineticallyConstrainedModelsMarkoLjubotina,1,Jean-YvesDesaules,2,MaksymSerbyn,1andZlatkoPapic21InstituteofScienceandTechnologyAustria(ISTA),AmCampus1,3400Klosterneuburg,Austria2SchoolofPhysicsandAstronomy,UniversityofLeeds,LeedsLS29JT,UK(Dated:March9,2023)Universalno...

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Superdiffusive Energy Transport in Kinetically Constrained Models

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