Imaginary-time Quantum Relaxation Critical Dynamics with Semi-ordered Initial States Zhi-Xuan Li1Shuai Yin2and Yu-Rong Shu1

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Imaginary-time Quantum Relaxation Critical Dynamics with Semi-ordered Initial

States

Zhi-Xuan Li,1Shuai Yin,2and Yu-Rong Shu1, ∗

1School of Physics and Materials Science, Guangzhou University, Guangzhou 510006, China

2School of Physics, Sun Yat-Sen University, Guangzhou 510275, China

(Dated: May 9, 2023)

We explore the imaginary-time relaxation dynamics near quantum critical points with semi-

ordered initial states. Diﬀerent from the case with homogeneous ordered initial states, in which

the order parameter Mdecays homogeneously as M∝τ−β/νz , here Mdepends on the location

x, showing rich scaling behaviors. Similar to the classical relaxation dynamics with an initial do-

main wall in Model A, which describes the purely dissipative dynamics, here as the imaginary time

evolves, the domain wall expands into an interfacial region with growing size. In the interfacial re-

gion, the local order parameter decays as M∝τ−β1/νz, with β1being an additional dynamic critical

exponent. Far away from the interfacial region the local order parameter decays as M∝τ−β/νz in

the short-time stage, then crosses over to the scaling behavior of M∝τ−β1/νz when the location

xis absorbed in the interfacial region. A full scaling form characterizing these scaling properties

is developed. The quantum Ising model in both one and two dimensions are taken as examples to

verify the scaling theory. In addition, we ﬁnd that for the quantum Ising model the scaling function

is an analytical function and β1is not an independent exponent.

The understanding of nonequilibrium dynamics in

quantum many-body systems is attracting increasing at-

tentions in recent years inspired by the fast develop-

ments of quantum computers [1–3] and ultra-cold atom

experimental techniques [4]. In particular, the study

of imaginary-time quantum critical dynamics is heat-

ing up thanks to the recent progresses made by quan-

tum computers in realizing imaginary-time evolution in

quantum systems [5,6]. Grasping critical properties on

the imaginary-time relaxation avenue down to the crit-

ical ground state not only saves computational eﬀorts

but also yields dynamical properties that is beyond the

reach of equilibrium studies. The critical relaxation dy-

namics is one of the most simple but important member

of the nonequilibrium quantum critical dynamics family.

In the past few years, a scaling theory for the imaginary-

time quantum critical relaxation dynamics has been de-

veloped [7,8] in analogy to the critical relaxation dy-

namics in classical systems [9–14]. It has been shown

that the initial information can aﬀect the critical re-

laxation dynamics in the macroscopic time scale owing

to the divergence of the correlation time at the critical

point. Universal behaviors have been found during the

imaginary-time relaxation process with a homogeneous

initial state. For saturated ordered initial state, the or-

der parameter Mchanges with the imaginary-time τas

M∝τ−β/νz [7], in which βis the order parameter ex-

ponent deﬁned as M∝ |g|−βwith gbeing the distance

to the critical point, νis the correlation length expo-

nent deﬁned as ξ∝ |g|−νwith ξbeing the correlation

length, and zis the dynamic exponent deﬁned as ζ∝ξz

with ζbeing the correlation time. For initial state with

a small initial order parameter M0, the order parame-

ter increases according to M∝M0τθwith θthe critical

∗yrshu@gzhu.edu.cn

initial slip exponent in the short-time stage, then decays

as M∝τ−β/νz in the long-time stage. For initial state

with arbitrary order parameter, a universal characteristic

function is introduced to describe the universal eﬀects in-

duced by the initial state [8,14]. These phenomena have

been investigated in various phase transitions within and

beyond the Landau paradigm [7,8,15–18]. It has been

shown that the imaginary-time relaxation dynamics of

quantum systems can be diﬀerent from its classical coun-

terpart. For instance, the critical initial slip exponent θ

is 0.373 [7] and 0.209 [15] for the one-dimensional (1D)

and two-dimensional (2D) quantum Ising model, while

θis 0.191 [19] and 0.108 [20] for their classical counter-

parts, respectively. Therefore, the imaginary-time relax-

ation dynamics deserves special attentions. The ﬂourish-

ing developments in this issue also inspire us to explore

the eﬀects induced by other kinds of initial states, like

the inhomogeneous initial state, in the imaginary-time

evolution.

Critical properties in the presence of the inhomoge-

neous interfacial regions have raised long-term attentions

in various systems, since phase coexistence is a common

phenomenon in nature [21]. In particular, critical relax-

ation dynamics with a domain interface in a semi-ordered

initial state was studied [22,23]. These works showed

that diﬀerent from the relaxation dynamics with a homo-

geneous initial state, the initial domain wall can expand

into a growing interfacial region, and in this region the

order parameter decays obeying a distinct scaling rela-

tion M∝t−β1/νz with β1being an additional dynamic

exponent. In quantum systems, exotic prethermal dy-

namics induced by the interface in the 2D quantum Ising

model was discovered [24–26].

Motivated by the above intriguing issues, we inves-

tigate the imaginary-time relaxation dynamics with a

semi-ordered initial state, in which two completely or-

dered domains with opposite spin direction sandwich a

arXiv:2210.03979v2 [cond-mat.stat-mech] 8 May 2023

sharp domain wall, as shown in Fig. 1. We ﬁnd that sim-

ilar to the classical case [22,23], as the time evolves, the

sharp domain wall will blur and expand into an inter-

facial region. Let us focus on the behavior of the local

parameter at the position x, with xdenotes the distance

to the initial domain wall. When xis far away from the

interfacial region, the local order parameter decays as

M(τ, x)∝τ−β/νz . As time elapses, the interfacial region

spreads to the position x. Accordingly, the order param-

eter changes to decay as M(τ, x)∝τ−β1/νz . Here, β1is

a purely dynamic exponent, since it has no equilibrium

counterpart, similar to the classical case [22,23]. A full

scaling form is then developed to explain this behavior.

We take the 1D and 2D quantum Ising models as ex-

amples to verify this scaling theory. From the numerical

results, we ﬁnd that the scaling function is an analytical

function and β1seems not an independent critical expo-

nent. Instead, it satisﬁes β1/νz =β/νz + 1, in analogy

to the classical case [22,23].

Spin down Spin up

Interfacial

…

FIG. 1. Sketch of the imaginary-time relaxation dynamics

from a semi-ordered initial state. A sharp domain wall sepa-

rates the spin-up and -down domains in the initial state. As

time evolves, the domain wall extends to an interfacial region.

For the imaginary-time relaxation dynamics, the evo-

lution of the wave function |ψ(τ)iobeys the imaginary-

time Schr¨odinger equation −∂τ|ψ(τ)i=H|ψ(τ)iwith

the normalization condition hψ(τ)|ψ(τ)i= 1 [27,28].

The formal solution of the Schr¨odinger equation is given

by |ψ(τ)i= exp(−τH)|ψ(τ0)i/Z, in which τ0is the ini-

tial time of the evolution and Z=kexp(−τ H)|ψ(τ0)ik,

with k · k being the modulo operation. Since both the

imaginary-time dynamics and the classical dynamics de-

scribed by Model A, which refers to a set of models with

no conservation laws [29], are purely dissipative dynam-

ics, one expects that their dynamic scaling behaviors near

the critical point are similar [7]. For instance, from the

saturated ordered initial state, the order parameter M

follows the same power-law decay with the evolution time

with the exponent β/νz for both classical model A dy-

namics and quantum imaginary-time relaxation dynam-

ics.

Here we study the inﬂuence of a diﬀerent type of initial

state to the quantum imaginary-time relaxation dynam-

ics. The initial state is set as a semi-ordered state with

two fully-ordered domains with opposite spin directions,

separated by a sharp domain wall, as shown in Fig. 1.

Since the translational symmetry is broken by the ini-

tial state, it is expected that the evolution behavior of

Mdepends on the distance to the initial domain wall x.

Near the domain wall, the spin at small xfeels a stronger

spin-ﬂip intention from the other domain where the spins

orientated in the opposite direction than its homogeneous

ordered environment. Accordingly, one anticipates that

at the critical point, for small x, the local order parame-

ter should follows

M(x, τ)∝τ−β1/νz,(1)

with β1being an additional critical exponent which is

larger than β, since the other domain lures the spin at

xto ﬂip. Note that similar to the classical case [22,23],

here β1is a purely nonequilibrium critical exponent, since

it has no static counterpart.

In contrast, for large x, the dynamic scaling behavior of

M(x, τ) is much richer as a result of the spread of the ef-

fects induced by the domain wall, as illustrated in Fig. 1.

In the short-time stage, the domain wall region is too far

away to control the spin at xand thus the local order pa-

rameter Mdecays according to M(x, τ)∝τ−β/νz , sim-

ilar to the case with homogeneous ordered initial state.

Therefore, this stage is referred to as the ‘homogeneous

region’. As time passes by, the domain wall extends into

an ‘interfacial region’ with growing size. When the lo-

cation xis absorbed into this interfacial region, Mwill

evolve following Eq. (1).

Scaling theory. For the relaxation critical dynamics

with homogeneous initial state, a characteristic scaling

behavior is the appearance of the critical initial slip char-

acterized by an independent critical exponent θ. A na-

ture question is whether or not β1is another independent

exponent. To answer this question, we develop a full scal-

ing form to describe the whole imaginary-time relaxation

process with the semi-ordered initial state. In analogy

with the classical situation [22,23], the scaling form of

the local order parameter Mat a quantum critical point

is given by

M(τ, x) = τ−β/νz f(xτ−1/z ),(2)

in which f(xτ−1/z) is the scaling function. By compar-

ing Eq. (2) with the imaginary-time relaxation scaling

theory with the homogeneous initial state [7], one ﬁnds

that the initial order parameter is absent in Eq. (2). The

reason is that the initial state keeps invariant under the

renormalization transformation, as illustrated in Fig. 2.

Similarly, the initial correlation is also not included since

both the initial correlation length and correlation time

are zero.

It is expected that the scaling behaviors discussed

above should be covered by the full scaling form Eq. (2).

This gives some constraints on the scaling function

f(xτ−1/z ): (i) f(xτ −1/z) should be an odd function of

xsince Mshould change sign on switching the spin

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Imaginary-timeQuantumRelaxationCriticalDynamicswithSemi-orderedInitialStatesZhi-XuanLi,1ShuaiYin,2andYu-RongShu1,1SchoolofPhysicsandMaterialsScience,GuangzhouUniversity,Guangzhou510006,China2SchoolofPhysics,SunYat-SenUniversity,Guangzhou510275,China(Dated:May9,2023)Weexploretheimaginary-timerelaxat...

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Imaginary-time Quantum Relaxation Critical Dynamics with Semi-ordered Initial States Zhi-Xuan Li1Shuai Yin2and Yu-Rong Shu1

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