Anatomy of Dynamical Quantum Phase Transitions Maarten Van Damme 1Jean-Yves Desaules 2Zlatko Papi c 2and Jad C. Halimeh3 4 1Department of Physics and Astronomy University of Ghent Krijgslaan 281 9000 Gent Belgium

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Anatomy of Dynamical Quantum Phase Transitions

Maarten Van Damme ,1, ∗Jean-Yves Desaules ,2, ∗Zlatko Papi´c ,2and Jad C. Halimeh 3, 4, †

1Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium

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

3Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC),

Ludwig-Maximilians-Universit¨at M¨unchen, Theresienstraße 37, D-80333 M¨unchen, Germany

4Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, D-80799 M¨unchen, Germany

(Dated: July 12, 2023)

Global quenches of quantum many-body models can give rise to periodic dynamical quantum

phase transitions (DQPTs) directly connected to the zeros of a Landau order parameter (OP). The

associated dynamics has been argued to bear close resemblance to Rabi oscillations characteristic

of two-level systems. Here, we address the question of whether this DQPT behavior is merely a

manifestation of the limit of an eﬀective two-level system or if it can arise as part of a more complex

dynamics. We focus on quantum many-body scarring as a useful toy model allowing us to naturally

study state transfer in an otherwise chaotic system. We ﬁnd that a DQPT signals a change in

the dominant contribution to the wave function in the degenerate initial-state manifold, with a

direct relation to an OP zero only in the special case of occurring at the midpoint of an evenly

degenerate manifold. Our work generalizes previous results and reveals that, in general, periodic

DQPTs comprise complex many-body dynamics fundamentally beyond that of two-level systems.

I. INTRODUCTION

One of the central goals of far-from-equilibrium quan-

tum many-body physics is the understanding of dynam-

ical quantum universality classes, the pursuit of which

has led to the introduction of several concepts of dynam-

ical phase transitions [1–4]. Extending the concept of

spontaneous symmetry breaking in equilibrium, one con-

cept of dynamical phase transitions is characterized by

the order parameter (OP) of the long-time steady state

following a quench in a control parameter after start-

ing in an ordered (symmetry-broken) initial state [5–7].

The critical value of the quench parameter separates a

symmetry-broken from a symmetric steady state. In ad-

dition to this Landau type of dynamical phase transi-

tions, another concept has been introduced, known as

dynamical quantum phase transitions (DQPTs), that re-

lies on a connection to thermal phase transitions [8]. It is

based on recognizing that the overlap ⟨ψ0|ψ(t)⟩between

the initial state |ψ0⟩and the time-evolved wave function

|ψ(t)⟩=e−iˆ

Ht |ψ0⟩, with ˆ

Hthe quench Hamiltonian, is

a boundary partition function with complexiﬁed time it

representing inverse temperature. Equivalently, the re-

turn rate,−limL→∞ L−1ln|⟨ψ0|ψ(t)⟩|2, with Lthe sys-

tem size, becomes a dynamical analog of the thermal free

energy, with a DQPT formally deﬁned as a nonanalytic-

ity in it at a critical time tc.

In a wide variety of quantum many-body models host-

ing a global symmetry, DQPTs in the wake of suﬃciently

large quenches starting in the ordered phase have been

shown to be directly connected to zeros of the OP dynam-

ics [9–16]. In the seminal work introducing DQPTs, the

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

†jad.halimeh@physik.lmu.de

FIG. 1. (Color online). Schematic of the main conclusions

of our work. During “state-transfer” quench dynamics in the

spin-SU(1) QLM initialized in an extreme vacuum, DQPTs

(red diamonds) arise in the return rate (RR) when there is

a shift in wave function-overlap dominance between compo-

nents of the degenerate vacuum manifold, where each compo-

nent vacuum is denoted by mz∈{−S,...,S}(orange dots) in

the total Hilbert space H. (a) For S=3/2 we ﬁnd a DQPT oc-

curring at the same time as a zero in the OP E(t) (blue dot).

This DQPT signals state transfer between the intermediate

vacua mz=±1/2. Other DQPTs show no such connection.

This can be generalized for any half-integer S. (b) For S=1, a

DQPT and an OP zero do not occur simultaneously. Instead,

the OP zero is halfway between two consecutive DQPTs that

signal state transfer to and away from the middle vacuum

mz=0. This holds for all integer S. This picture generalizes

previous results and highlights the complex many-body dy-

namics comprising DQPTs.

model showing this behavior is the integrable transverse-

ﬁeld Ising chain (TFIC) [8]. This model can be solved

exactly by mapping it to a two-band free fermionic model

using a Jordan–Wigner transformation, where each dis-

arXiv:2210.02453v2 [quant-ph] 11 Jul 2023

connected momentum sector is a two-level system. For

nonintegrable models, e.g., the long-range interacting

TFIC or two-dimensional quantum Ising models, the di-

rect connection between DQPTs and OP zeros occurs

for large quenches, where the transverse-ﬁeld strength is

much larger than the coupling constant, and which can be

considered perturbatively close to classical precession in

two-level systems during the short timescales where this

behavior is prominent [10,12,15]. Indeed, for interme-

diate quenches where this perturbative treatment is no

longer valid, this connection breaks down [17–19], and

for small quenches within the ordered phase, anomalous

DQPTs can appear even without any OP zeros occurring

over all investigated timescales [10,11,20]. As such, it

has been argued that the periodic DQPT behavior seen

for large quenches may be a mere manifestation of eﬀec-

tive two-level system dynamics [21].

Here, we address this argument by investigating

DQPTs in models exhibiting “state transfer”—a dynam-

ical process where the wave function evolves cyclically

between the states of a given manifold—due to quantum

many-body scarring [22–24]. Such models possess a small

number of anomalous eigenstates throughout their spec-

trum, leading to oscillatory dynamics from a few speciﬁc

states in an otherwise thermalizing system. We will fo-

cus on a formulation of the lattice Schwinger model [25]

known as the spin-SU(1) quantum link model (QLM)

[26,27]. This model has been shown to exhibit quan-

tum many-body scarring for massless quenches starting

in the maximal-ﬂux (extreme) vacua for a wide range of

spin values [28,29]. We show in these models that peri-

odic DQPTs arise within complex many-body dynamics

that is beyond two-level systems, and where a DQPT sig-

nals state transfer from one vacuum to another within a

(2S+1)-fold degenerate vacuum manifold; see Fig. 1. We

ﬁnd no direct connection between DQPTs and OP ze-

ros for integer S. For half-integer S, a DQPT is directly

connected to an OP zero only when the DQPT signals

a transfer between intermediate minimal-ﬂux vacua of

opposite ﬂux sign. We further show that models where

DQPT behavior resembles two-level system dynamics are

a special case of our general picture.

II. MODEL

We consider the spin-SU(1) QLM, given by the Hamil-

tonian [26,27,30]

j=1 J

2pS(S+ 1)ˆσ−

jˆs+

j,j+1 ˆσ−

j+1 + H.c.

+µˆσz

j+κ2

2ˆsz

j,j+12,(1)

where we have adopted particle-hole and Jordan–Wigner

transformations [31,32]. The Pauli operator ˆσz

jdescribes

the matter occupation on site jwith mass µ, and the

spin-Soperators ˆs+

j,j+1/pS(S+ 1) and ˆsz

j,j+1 represent

the gauge and electric ﬁelds, respectively, residing on the

link between sites jand j+ 1. The tunneling constant is

J, which we shall set to unity as the overall energy scale,

κis the gauge-coupling strength, and Lis the number of

sites.

The generator of the U(1) gauge symmetry of Hamil-

tonian (1) is

Gj= (−1)jˆsz

j−1,j + ˆsz

j,j+1 +ˆσz

j+ˆ

2,(2)

and gauge-invariant states |ϕ⟩satisfy ˆ

Gj|ϕ⟩=gj|ϕ⟩,∀j,

where gj/(−1)j∈{−2S, . . . , 2S+1}. We will work in the

physical sector gj=0,∀j.

A building block of the U(1) QLM has been exper-

imentally realized for S→∞ in a cold-atom setup [33].

Large-scale implementations of the spin-1/2 U(1) QLM

on a Bose–Hubbard superlattice have been employed to

observe gauge invariance [34] and thermalization dynam-

ics [35].

III. QUENCH DYNAMICS

We now present time-evolution results obtained

through the inﬁnite matrix product state (iMPS) tech-

nique based on the time-dependent variational principle

[36–39]. This technique works directly in the thermody-

namic limit, and also allows us to directly detect DQPTs

as level crossings between the logarithms of the eigen-

values of the MPS transfer matrix [10,40,41], without

any need for ﬁnite-size scaling with L. DQPTs for gauge

theories have already been studied in the context of the

spin-1/2 U(1) QLM [13,42,43], and also for S≥1/2 [44]

as well as in the Schwinger model [14,45], but not in the

context of quantum many-body scarring. For the most

stringent calculations that we have performed in iMPS

for this work, we ﬁnd convergence for a maximal bond

dimension of 550 and a time-step of 0.0005/J.

We are interested in the dynamics of the experimen-

tally relevant return rate (RR)

λ(t) = min

mzλmz(t),(3a)

λmz(t) = −lim

L→∞

Lln 

⟨ψmz

0|ψ(t)⟩



2,(3b)

which has recently been used to identify DQPTs in a

trapped-ion experiment [46]. Here, |ψmz

0⟩is the set

of vacua with mz∈−S, . . . , S, which are the (2S+1)-

fold degenerate ground states of Hamiltonian (1) for

κ/J=0 and µ/J→∞. We call a vacuum extreme when

|mz|=S, and intermediate when |mz|<S. We can rep-

resent a vacuum state on the two-site two-link unit cell

as |ψmz

0⟩=|−1, mz,−1,−mz⟩, indicating the eigenvalue

−1 of ˆσz

jat each (empty) matter site and the eigenvalue

mzof ˆsz

j,j+1 on each link, but we will often refer to this

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AnatomyofDynamicalQuantumPhaseTransitionsMaartenVanDamme,1,∗Jean-YvesDesaules,2,∗ZlatkoPapi´c,2andJadC.Halimeh3,4,†1DepartmentofPhysicsandAstronomy,UniversityofGhent,Krijgslaan281,9000Gent,Belgium2SchoolofPhysicsandAstronomy,UniversityofLeeds,LeedsLS29JT,UK3DepartmentofPhysicsandArnoldSommerfeldCent...

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Anatomy of Dynamical Quantum Phase Transitions Maarten Van Damme 1Jean-Yves Desaules 2Zlatko Papi c 2and Jad C. Halimeh3 4 1Department of Physics and Astronomy University of Ghent Krijgslaan 281 9000 Gent Belgium

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