Reviving product states in the disordered Heisenberg chain Henrik Wilming1Tobias J. Osborne1Kevin S.C. Decker2and Christoph Karrasch2 1Leibniz Universit at Hannover Appelstraße 2 30167 Hannover Germany

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Reviving product states in the disordered Heisenberg chain

Henrik Wilming,1, ∗Tobias J. Osborne,1Kevin S.C. Decker,2and Christoph Karrasch2

1Leibniz Universit¨

at Hannover, Appelstraße 2, 30167 Hannover, Germany

2Technische Universit¨

at Braunschweig, Institut f¨

ur Mathematische Physik, Mendelssohnstraße 3, 38106 Braunschweig, Germany

When a generic quantum system is prepared in a simple initial condition, it typically equilibrates toward a

state that can be described by a thermal ensemble. A known exception are localized systems that are non-ergodic

and do not thermalize, however, local observables are still believed to become stationary. Here we demonstrate

that this general picture is incomplete by constructing product states which feature periodic high-ﬁdelity revivals

of the full wavefunction and local observables that oscillate indeﬁnitely. The system neither equilibrates nor

thermalizes. This is analogous to the phenomenon of weak ergodicity breaking due to many-body scars and

challenges aspects of the current phenomenology of many-body localization, such as the logarithmic growth of

the entanglement entropy. To support our claim, we combine analytic arguments with large-scale tensor network

numerics for the disordered Heisenberg chain. Our results hold for arbitrarily long times in chains of 160 sites

up to machine precision.

I. INTRODUCTION

When a large, closed, interacting quantum many-body sys-

tem is initialized in a simple initial condition, it typically ap-

proaches a state that is stationary when only observed with

coarse-grained (e.g, local) observables – the system equili-

brates [1, 2]. In addition, the stationary state of the coarse-

grained observables is often well-described by statistical (e.g.,

canonical) ensembles – the system thermalizes [2–5]. While

thermalization is a generic phenomenon and aids the theoreti-

cal description, it is not inevitable. One of the most intensely

debated exceptions is that of many-body localization (MBL),

which is realized in interacting quantum models with a suf-

ﬁciently strong disorder potential [6–9]. Systems exhibiting

MBL provide generic examples of non-ergodic systems that

fail to thermalize due to a memory of the local initial con-

ditions, yet they are still equilibrating [10, 11]. Other key

features of MBL phases include an unbounded growth of the

entanglement during quantum quenches [12–14] and peculiar

transport properties [15–17]. There are now a variety of exper-

imental realizations exhibiting signatures of MBL, including,

cold atoms [18, 19] and photonic systems [20].

The existence of MBL as a stable phase of matter has re-

cently been questioned and it has been suggested that ther-

malization actually eventually occurs [21–25]. However, it

is fair to say that a conclusive picture has not yet emerged

[26–31]. A key obstacle is that many studies are based on

an exact diagonalization of small systems and might thus not

be representative for the behavior in the thermodynamic limit

[32, 33]. Approaching the problem from the perspective of

quantum avalanches has been a major recent direction [34–

42].

Another exception to the rule of equilibration and ther-

malization was recently discovered: In so-called many-body

scarred systems, there exists a relatively small set of initial

product states which may show indeﬁnite revivals of the full

many-body wavefunction. When the system is initialized in

such an initial state, all physical observables (including local

∗henrik.wilming@itp.uni-hannover.de

FIG. 1. Indeﬁnitely oscillating spins. Top: The disordered Heisen-

berg chain (L= 20) is initialized in a deformed domain wall product

state that has an overlap >0.994 with a superposition of two en-

ergy eigenstates. Middle: Under the unitary time evolution, the local

spins remain almost uncorrelated and start to oscillate in the region

around the domain wall interface. The solid line shows the expecta-

tion value of the Pauli-X observable ⟨2ˆ

j⟩at the center spin in the

superposed energy eigenstates. The dynamics of the actual product

state is within the associated shaded regions due to its large overlap

with the superposition of eigenstates. The blue dotted line indicates

the certiﬁed amplitude, which provides a lower bound for the mag-

nitude of the oscillations in the inﬁnite-time limit. Bottom: Overlay

of different snapshots in time of the expectation values of the local

spin operators around the domain wall interface, visualized as arrows

within their respective Bloch spheres.

ones) show periodic oscillations and the system neither ther-

malizes nor equilibrates [43–50]. The revivals of the wave-

function are connected to the existence of a small set of high-

energy eigenstates that exhibit atypically low entanglement,

dubbed “quantum (many-body) scars”. Conversely, if all en-

ergy eigenstates are sufﬁciently entangled, then initial product

states generically equilibrate [51–53].

In fact, MBL systems also exhibit quantum many-body

scarring, and they do so in a most dramatic way: Not just a

arXiv:2210.03153v2 [quant-ph] 16 Nov 2023

few, but all high-energy eigenstates have atypically low en-

tanglement, since the entanglement entropy features an area

law [54–59] instead of a volume law. (This is the generic sit-

uation in interacting systems [60–68].)

To summarize: Many-body scarred systems host a few

slightly entangled eigenstates, and these can be sufﬁcient for

a complete breakdown of equilibration in certain initial prod-

uct states. Conversely, all energy eigenstates in MBL systems

are low-entangled. This leads to a natural question: Can MBL

systems also host initial product states that show high-ﬁdelity

revivals of the wavefunction with corresponding local observ-

ables that oscillate indeﬁnitely?

If the answer to this question is “yes”, then – contrary to

current belief – MBL systems do not generally equilibrate

from product states and hence also do not thermalize. More-

over, a further hallmark feature of MBL, namely the slow (log-

arithmic) but unbounded growth of the entanglement entropy,

would be violated for these particular initial conditions.

It is, however, unclear how to approach this problem and

how to ﬁnd such initial conditions for a given MBL Hamilto-

nian. In particular, there are two key difﬁculties to be over-

come: i) The product state might have to be ﬁne-tuned to the

details of the Hamiltonian such as the disorder conﬁguration.

However, the set of product states is a continuum, so we can-

not simply search through all of them. Furthermore, we can-

not exploit algebraic structures (such as symmetries) to guide

us; ii) Even given a candidate initial state, how could we make

sure that it does not equilibrate? In principle, the revival could

happen at arbitrary long times, which cannot be accessed an-

alytically or numerically (even for MBL systems).

In this work, we overcome these difﬁculties and demon-

strate that one can ﬁnd initial product states featuring high-

ﬁdelity revivals and local observables that oscillate indeﬁ-

nitely. We combine analytical arguments with state-of-the-art

tensor network calculations. Importantly, our approach works

for arbitrarily long times, and we can treat systems of up to

160 sites with machine precision.

II. RESULTS

We focus on the paradigmatic disordered spin-1/2 Heisen-

berg model on Llattice sites,

H=−

L−1

j=1

Sj·S

Sj+1 +

j=1

hjˆ

S(z)

j,(1)

where S

Sj= ( ˆ

S(x)

j,ˆ

S(y)

j,ˆ

S(z)

j)⊤is the vector of spin-1/2 an-

gular momentum operators at site j. The local magnetic ﬁelds

hj∈[−W, W ]are sampled independently from a uniform

distribution; Wis the disorder strength. Exact diagonaliza-

tion of small systems predicts a crossover from an ergodic to

an MBL phase around W∼3.5[33]. In the main part of this

work, we set W= 8.

First, we show that if we can ﬁnd two eigenstates whose

superposition is well approximated by a product state, then

one can construct a local observable which oscillates indeﬁ-

nitely with an amplitude that is lower-bounded by a certiﬁed

amplitude Acert.(Sec. II A).

Secondly, we use large-scale tensor network numerics to

construct such eigenstates for the disordered Heisenberg chain

(Sec. II B). We present data for systems of up to L= 160 sites

and, up to machine precision, provide a rigorous certiﬁcate for

the indeﬁnite oscillations of a local observable (Sec. II C).

Lastly, we present theoretical arguments suggesting that

large systems may in fact host a ﬁnite density of locally os-

cillating excitations (Sec. II D).

Our results are illustrated in Fig. 1. To keep the discussion

concise, we delegate most technical details to the Methods

(Sec. IV) and the Supplementary Information.

A. Locally oscillating product states

Let us consider two eigenstates |E1⟩and |E2⟩. Their time-

evolved equal superposition

|Ψ(t)±⟩=1

√2e−iE1t|E1⟩ ± e−iE2t|E2⟩(2)

shows perfect revivals at even multiples of the period τ=

π/(E1−E2). Now suppose there is a product state

|Φ(0)±⟩=|ϕ(1)

±⟩⊗···⊗|ϕ(L)

±⟩(3)

that approximates |Ψ±(0)⟩in the sense that its overlap fulﬁlls

±=|⟨Ψ(0)±|Φ(0)±⟩|2≥1−ϵwith ϵsmall. This implic-

itly deﬁnes the local quantum states |ϕ(k)

±⟩. The simple but

key observation of our approach is that then the time-evolved

state |Φ(t)±⟩= exp(−iˆ

Ht)|Φ(0)±⟩will necessarily also

show high-ﬁdelity revivals:

|⟨Φ(0)±|Φ(2kτ)±⟩|2≥1−4ϵ(4)

for any integer k. Moreover, let j= argmink|⟨ϕ(k)

+|ϕ(k)

−⟩|.

Then the observable

A=⊗|ϕ(j)

+⟩⟨ϕ(j)

+|−|ϕ(j)

−⟩⟨ϕ(j)

−|⊗(5)

is supported on a single site and its time-dependent expecta-

tion value in the state |Φ+(t)⟩oscillates with period τ:

|⟨Φ+|ˆ

A(2kτ)|Φ+⟩−⟨Φ+|ˆ

A((2k+ 1)τ)|Φ+⟩| ≥ Acert.

(6)

for any integer k.ˆ

A(t)refers to the Heisenberg picture. The

certiﬁed amplitude Acert.is given by

Acert.= max{1−f2−2p(1 −f2)ϵ, 0},(7)

where f2= minj|⟨ϕ(j)

+|ϕ(j)

−⟩|2measures the minimal lo-

cal overlap between |Φ(0)+⟩and |Φ(0)−⟩(assuming that

each |ϕ(j)

±⟩is normalized). A detailed proof can be found in

Sec. IV A.

As a next step, we demonstrate how to ﬁnd pairs of en-

ergy eigenstates whose equal superpositions are well approx-

imated by product states. It is reasonable to hypothesize that

such states must have a low entanglement with respect to

any bipartition. Therefore we performed a structured search

on small systems using exact diagonalization and targeting

energy eigenstates whose sub-lattice entanglement entropy

(ABABAB...-bipartition) is small, see Supplementary Ma-

terial for more details. Targeting small sub-lattice entangle-

ment is a heuristic choice motivated by the following con-

siderations: i) Product states have vanishing sub-lattice en-

tanglement entropy and therefore any state sufﬁciently close

to a product state should have small sublattice entanglement

and ii) even generic translationally invariant matrix-product

states (MPS) [69, 70], which are commonly considered to

be low-entangled, have extensive sub-lattice entanglement en-

tropies [71]. Therefore small sub-lattice entanglement heuris-

tically indicates an amount of entanglement that is small

even compared to MPS. Our preliminary analysis showed that

pairs of energy eigenstates whose equal superpositions are

well approximated by product states exist and that one class

of them comes in the form of deformed domain walls (see

Fig. 1). This knowledge then allows us to devise an efﬁ-

cient tensor-network based algorithm to study large systems,

which we now brieﬂy explain (further details may be found in

Sec. IV B).

B. Numerical construction

At sufﬁciently strong disorder, the eigenstates of ˆ

Hfeature

an area-law entanglement and may be represented faithfully

as MPS [57], whose explicit representation can be determined

using the DMRG-X algorithm [72]. The algorithm starts with

a “seed” state |m1⟩⊗···⊗|mL⟩, where |mj⟩ ∈ {|↑⟩,|↓⟩}

denote the eigenstates of ˆ

S(z)

j. These seeds are the eigenstates

of ˆ

Hin the limit of W→ ∞. DMRG-X then iteratively de-

termines an (approximate) eigenstate at ﬁnite Wthat is, in a

sense, closest to the initial seed. The main numerical con-

trol parameter is the so-called bond dimension χ, which we

choose so that high-energy eigenstates are obtained up to ma-

chine precision.

In our case, we ﬁnd the energy eigenstates |E:k⟩associ-

ated with seeds in domain-wall form

|dw : k⟩=|↓⟩ ⊗ ··· ⊗ |↓⟩

| {z }

ktimes

⊗|↑⟩ ⊗ ··· ⊗ |↑⟩.(8)

We then form the superposition of the energy eigenstates re-

sulting from neighboring domain-walls,

Ψ(k)

±E=1

√2(|E:k⟩ ± |E:k+ 1 ⟩),(9)

and ﬁnally construct their product-state approximation |Φ(k)

±⟩.

This allows us to calculate the certiﬁed amplitude Acert.of

Eq. (7). All of these operations can be implemented efﬁciently

and accurately in the MPS representation (see Sec. IV B for

further details). We stress that at this point it is not clear why

the states Ψ(k)

±Eshould be close to product states apart from

the fact that we found revolving product states with a sim-

ilar structure in our small scale exact-diagonalization numer-

ics (see Supplementary Material). Our main results in the next

section show that for domain-wall seeds, closeness to a prod-

uct state is indeed a generic case for sufﬁciently strong dis-

order. This in turn immediately implies the non-equilibrating

behavior for the associated product states.

C. Main results

In Fig. 2 our aggregated numerical data for the certiﬁed am-

plitude at varying system sizes up to L= 160 and at a disor-

der strength W= 8 with 100 disorder realizations per system

size is depicted (the corresponding ﬁdelities are discussed in

Supplementary Note 1). We ﬁnd median certiﬁed amplitudes

of the order of 0.7, essentially independent of the system size

with decreasing ﬂuctuations as Lincreases. Moreover, the

maximum certiﬁed amplitudes for domain-wall states with in-

terface in the middle half of the system (sites k=L/4to

k= 3L/4) slowly increase with system size, with all sam-

pled realizations reaching Acert.>0.91 for L= 160. The

restriction to states with the interface in the middle half of the

system excludes states that can be interpreted as being close

to single-particle excitations (see below and Supplementary

Note 2). We emphasize that the certiﬁed amplitude provides

a lower bound to the magnitude of the oscillations of ˆ

Aand

that there may exist local operators which oscillate with even

higher amplitude.

In a nutshell, Fig. 2 conclusively demonstrates the (generic)

existence of initial product states that host high-ﬁdelity re-

vivals and the existence of local, indeﬁnitely-oscillating, ob-

servables in a system of up to 160 sites. The overall shape of

these product states is of the form of two domain walls sep-

arated by a spin pointing roughly in ±x-direction at their in-

terface. Moving away from the interface, the spins still point

away from their original ±z-directions, but with decreasing

components in the x−y-plane. This is visualized in Fig. 1. As

a side remark, we mention that the Hamiltonian ˆ

Hmay also

be interpreted as a Hamiltonian of interacting fermions by a

Jordan-Wigner transformation. However, in this picture the

parity super-selection rule forbids our reviving product states,

since they correspond to super-positions of states with differ-

ent fermion-number parity.

The fact that we ﬁnd oscillating deformed domain walls

is particularly interesting since previous results indicate that

the bare domain-wall states |dw : k⟩approach a steady-state

with a smeared-out interface, a process known as domain-wall

melting [73]. An interface spin pointing away from the z-axis

therefore protects against this mechanism.

Since the DMRG-X algorithm outputs the energy eigen-

states |E:k⟩as MPS, we can compute the expectation values

⟨Ψ(k)

±|ˆ

B(t)|Ψ(k)

±⟩of any local operator ˆ

Bexactly for arbitrary

times t(see Sec. IV B). This in turn allows us to quantitatively

estimate the ﬁnite-time expectation value ⟨Φ(k)

±|ˆ

B(t)|Φ(k)

±⟩

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RevivingproductstatesinthedisorderedHeisenbergchainHenrikWilming,1,∗TobiasJ.Osborne,1KevinS.C.Decker,2andChristophKarrasch21LeibnizUniversit¨atHannover,Appelstraße2,30167Hannover,Germany2TechnischeUniversit¨atBraunschweig,Institutf¨urMathematischePhysik,Mendelssohnstraße3,38106Braunschweig,GermanyWh...

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Reviving product states in the disordered Heisenberg chain Henrik Wilming1Tobias J. Osborne1Kevin S.C. Decker2and Christoph Karrasch2 1Leibniz Universit at Hannover Appelstraße 2 30167 Hannover Germany

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