# Dual dynamic scaling in deconfined quantum criticality

Yu-Rong Shu  School of Physics and Materials Science, Guangzhou University,
Guangzhou 510006, China  Shuai Yin<EMAIL_ADDRESS>School of Physics,
Sun Yat-Sen University, Guangzhou 510275, China

###### Abstract

Emergent symmetry is one of the characteristic phenomena in deconfined quantum
critical point (DQCP). As its nonequilibrium generalization, the dual dynamic
scaling was recently discovered in the nonequilibrium imaginary-time
relaxation dynamics in the DQCP of the $J$-$Q_{3}$ model. In this work, we
study the nonequilibrium imaginary-time relaxation dynamics in the $J$-$Q_{2}$
model, which also hosts a DQCP belonging to the same equilibrium universality
class. We not only verify the universality of the dual dynamic scaling at the
critical point, but also investigate the breakdown and the vestige of the dual
dynamic scaling when the tuning parameter is away from the critical point. We
also discuss its possible experimental realizations in devices of quantum
computers.

##  I Introduction

Symmetry plays pivotal roles in identifying, characterizing and classifying
phases and phase transitions in condensed matter physics Landau and Lifshitz ;
Sachdev (2011). Traditional Landau-Ginzburg-Wilson (LGW) theory of phase
transitions is based on the mechanism of spontaneous symmetry breaking, in
which the symmetry of the ordered phase is always lower than that of the
microscopic model Landau and Lifshitz . In contrast, many critical systems
show higher symmetry in the infrared limit than they do in the ultraviolet
limit. For instance, spacetime supersymmetry can emerge at the critical point
in some topological materials Lee (2007); Grover  _et al._ (2014); Jian  _et
al._ (2015); Huijse  _et al._ (2015); Rahmani  _et al._ (2015); Witczak-Krempa
and Maciejko (2016); Li  _et al._ (2017a); Jian  _et al._ (2017); O’Brien and
Fendley (2018); Yu  _et al._ (2019); Yin and Zuo (2020); Han  _et al._ (2021);
Lorentz symmetry can emerge in the superfluid-Mott insulator quantum phase
transition Pollet and Prokof’ev (2012); Chen  _et al._ (2013a), critical two-
subband quantum wires Sitte  _et al._ (2009), and phase transitions in Dirac
systems Li  _et al._ (2017b); Roy  _et al._ (2016); Torres  _et al._ (2018);
$\mathbb{SU}(3)$ symmetry can emerge in the critical spin-$2$ chain with
translational invariant interaction and in the critical spin-$1$ chain with
random bond interaction Chen  _et al._ (2015); Quito  _et al._ (2015);
extended $\mathbb{O}(N)$ symmetry can emerge at the multicritical point
Calabrese  _et al._ (2003); Janssen  _et al._ (2018); Roy  _et al._ (2018);
Torres  _et al._ (2020). In two-dimensional (2D) spin systems, a prominent
example in which the emergent symmetry arises as its characteristic phenomenon
is the deconfined quantum critical point (DQCP) Senthil  _et al._ (2004a, b);
Levin and Senthil (2004). The DQCP was proposed as a new mechanism of
continuous phase transition between two spontaneous symmetry breaking phases,
while the usual LGW paradigm asserts that this kind of phase transition should
be first ordered Senthil  _et al._ (2004a, b); Levin and Senthil (2004);
Tanaka and Hu (2005); Nogueira and Kleinert (2005); Senthil and Fisher (2006);
Sandvik (2007); Nogueira  _et al._ (2007); Kaul and Melko (2008); Melko and
Kaul (2008); Gellé  _et al._ (2008); Nogueira and Kleinert (2008); Lou  _et
al._ (2009); Lou and Sandvik (2009); Sandvik (2010a); Sandvik  _et al._
(2011); Nahum  _et al._ (2011); Bartosch (2013); Nogueira and Sudbø (2013);
Tang and Sandvik (2013); Pujari  _et al._ (2015); Nahum  _et al._ (2015a);
Shao  _et al._ (2016); D’Emidio and Kaul (2017); Qin  _et al._ (2017); Wang
_et al._ (2017); Shao  _et al._ (2017); Janssen and He (2017); You  _et al._
(2018); Thomson and Sachdev (2018); Zhang  _et al._ (2018); Jian  _et al._
(2018); Gazit  _et al._ (2018); Liu  _et al._ (2019); Li  _et al._ (2019);
Ihrig  _et al._ (2019); Janssen  _et al._ (2020); Sandvik and Zhao (2020);
Zayed  _et al._ (2017). It was shown that for the DQCP separating the
antiferromagnetic (AFM) phase and valence-bond-solid (VBS) phase in
$\mathbb{SU}(2)$-invariant quantum magnets, $\mathbb{SO}(5)$ symmetry emerges
to reconcile VBS and AFM order parameters Sandvik (2007); Lou and Sandvik
(2009); Nahum  _et al._ (2015b); Sreejith  _et al._ (2019); while for the DQCP
in easy-plane quantum magnets, $\mathbb{SO}(4)$ symmetry emerges Ma  _et al._
(2019).

The emergent symmetry and its breaking give rise to the intriguing critical
properties in DQCP. For the square lattice, on the VBS side, the emergent
continuous symmetry breaks down to discrete $\mathbb{Z}_{4}$ symmetry Senthil
_et al._ (2004a, b); Levin and Senthil (2004); Lou and Sandvik (2009).
Accordingly, the fugacity of the quadrupled coherent monopoles works as a
dangerously irrelevant scaling variable, which is irrelevant exactly at the
critical point, but relevant in the ordered VBS phase Oshikawa (2000); Lou
_et al._ (2007); Okubo  _et al._ (2015); Shao  _et al._ (2020); Patil  _et
al._ (2021); Léonard and Delamotte (2015). Pertinent to this variable, an
extra divergent length $\xi^{\prime}$, which measures the spinon confinement
length or the thickness of the VBS domain walls, develops, in addition to the
conventional correlation length $\xi$ Levin and Senthil (2004); Shao  _et al._
(2016). And they satisfy $\xi^{\prime}\propto\xi^{\nu^{\prime}/\nu}$ with
$\nu$ and $\nu^{\prime}$ being the corresponding critical exponents Oshikawa
(2000); Lou  _et al._ (2007); Okubo  _et al._ (2015); Shao  _et al._ (2020);
Patil  _et al._ (2021); Léonard and Delamotte (2015). It was plausibly shown
that the interplay between these two length scales may take responsibility for
some anomalous equilibrium scaling behaviors near the DQCP Nahum  _et al._
(2015a); Shao  _et al._ (2016), although a very weak first-order phase
transition with pseudo-critical phenomena cannot be ruled out Kuklov  _et al._
(2005, 2006, 2008); Jiang  _et al._ (2008); Chen  _et al._ (2013b); Gorbenko
_et al._ (2018a, b); Ma and Wang (2020); Nahum (2020); D’Emidio  _et al._
(2021).

On the other hand, from the inflating universe to the flowing rivers,
equilibrium phenomena are just the exception rather than the rule in nature.
Moreover, investigations on nonequilibrium critical properties are of
particular significance since universal time-dependent behaviors always appear
near a critical point Hohenberg and Halperin (1977); Dziarmaga (2010);
Polkovnikov  _et al._ (2011); D’Alessio  _et al._ (2016); Mitra (2018). In
classical systems, the theory of critical dynamics has been well established
by classifications of the dynamic universality classes Hohenberg and Halperin
(1977). Recently, spurred by the remarkable experimental progresses in
manipulating and detecting the nonequilibrium quantum process, the quantum
critical dynamics has attracted intensive attention from both theoretical and
experimental aspects Dziarmaga (2010); Polkovnikov  _et al._ (2011); D’Alessio
_et al._ (2016); Mitra (2018). Among these studies, it was shown that the
scaling properties of the imaginary-time relaxation dynamics near quantum
critical points resemble those in the classical short-time critical dynamics
Janssen  _et al._ (2014); Li  _et al._ (1995); Zheng (1996); Albano  _et al._
(2011); Yin  _et al._ (2014); Zhang  _et al._ (2014); Shu  _et al._ (2017);
Shu and Yin (2020); Zuo  _et al._ (2021).

Inspired by above intriguing issues, a nature question arises: how the
emergent symmetry and the associated scaling form with two length scales
affect the nonequilibrium dynamics in DQCP. In our previous work Shu  _et al._
(2022), we studied the imaginary-time relaxation dynamics at the DQCP of the
$J$-$Q_{3}$ model. We found that with an ordered initial VBS (Néel) state, the
relaxation dynamics of the VBS (Néel) order parameter is controlled by the
conventional correlation length $\xi$, while the dynamics of the Néel (VBS)
order parameter is controlled by the confinement length $\xi^{\prime}$. A dual
dynamic scaling, which states that the dynamic scaling forms change to their
dual partners as the initial states are changed to their dual counterpart, is
then proposed. This dual dynamics scaling can be regarded as the
nonequilibrium incarnation of the emergent symmetry in the equilibrium case.
Given the fact that emergent symmetry is a common phenomenon in DQCP Senthil
_et al._ (2004a, b); Levin and Senthil (2004); Lou and Sandvik (2009); Nahum
_et al._ (2015b); Sreejith  _et al._ (2019); Ma  _et al._ (2019), it is
imperative to explore the universality and robustness of the dual dynamic
scaling in other models that host a DQCP.

In this paper, we study the nonequilibrium imaginary-time relaxation dynamics
in the $J$-$Q_{2}$ model Sandvik (2007).  The model shares the same
equilibrium universality class with the $J$-$Q_{3}$ model but has a weaker VBS
order Lou  _et al._ (2009); Tang and Sandvik (2011). After estimating the
critical point via the dynamics of the sign function of the order parameters,
we study the relaxation behaviors of the Néel and VBS order parameters from
different initial states at the critical point. By comparing the scaling forms
for different quantities, we verify the universality of the dual dynamic
scaling by showing that (i) the Néel and VBS order parameters are controlled
by different length scales for different initial states; (ii) the dynamic
scaling forms exchange under the  exchange of the initial states, similar to
the case of the $J$-$Q_{3}$ model. Moreover, we investigate dynamic scaling
properties when the tuning parameter is away from the critical point.
Strikingly, we find that in the short-time stage, the dual dynamic scaling,
with a proper generalization of the dual transformation, can exist even in the
presence of the off-critical-point effects, although in equilibrium the
emergent symmetry fades away  once the system is set away from its critical
point. A possible experimental realization based on programmable quantum
devices is also discussed.

The rest of the paper is organized as follows. In Sec. II, we introduce the
equilibrium properties of the $J$-$Q_{2}$ model, the protocol of the
imaginary-time relaxation dynamics and the numerical method used, Then, in
Sec. III, we give a brief review on the nonequilibrium dynamic scaling form.
After estimating the critical point of the $J$-$Q_{2}$ model via the
nonequilibrium scaling in Sec. IV, we explore the dual dynamic scaling at the
critical point for various initial states in Sec. V. In Sec. VI, we show the
dynamic scaling behavior with the off-critical-point effects. Then we discuss
the experimental realizations in Sec. VII. A summary is given in Sec. VIII.

##  II Model and Imaginary-time Relaxation Dynamics

The Hamiltonian of the Sandvik’s $J$-$Q_{2}$ model reads Sandvik (2007)

$H=-J\sum_{\langle ij\rangle}P_{ij}-Q\sum_{\langle
ijkl\rangle}{P_{ij}P_{kl}},$ (1)

in which $J>0$ and $Q>0$, $\langle ij\rangle$ and $\langle ijkl\rangle$ denote
nearest neighbors and two nearest-neighbor pairs in horizontal  rows or
vertical columns on the square lattice, respectively, and $P_{ij}$ denotes the
spin singlet operator defined as $P_{ij}\equiv\frac{1}{4}-{\bf S}_{i}\cdot{\bf
S}_{j}$ with $\bf S$ being the spin-$1/2$ operator. The system favors the Néel
phase with a finite order parameter ${\bf M}\equiv\sum_{r}{(-1)^{r}\bf
S}_{r}/N$ when  $q\equiv J/Q\gg q_{\rm c}$, while it favors the VBS phase with
a finite $\bf D$ when  $q\ll q_{\rm c}$ Sandvik (2007), in which ${\bf
D}=D_{x}\hat{x}+D_{y}\hat{y}$ with $D_{x(y)}\equiv\sum_{r}(-1)^{r_{x(y)}}{\bf
S}_{r}\cdot{\bf S}_{r+\hat{x}(\hat{y})}/N$. $N=L^{d}$ is the number of spins
and $\hat{x}$, $\hat{y}$ denote the unit lattice vector in the $x$ and $y$
direction, respectively. These two ordered phases break different symmetries:
the Néel order breaks the spin rotation symmetry, while the VBS order breaks
the translation symmetry. The phase transition between them happens at
$q=q_{\rm c}\simeq 0.045$ Sandvik (2007); Shao  _et al._ (2016); Sandvik and
Zhao (2020). According to the LGW theory, this phase transition should be
first ordered Landau and Lifshitz . However, plenty of numerical results with
scrutiny demonstrate that this phase transition is a continuous one satisfying
the DQCP theory Sandvik (2007, 2010a); Shao  _et al._ (2016); Sandvik and Zhao
(2020). Here we list the critical exponents of the $J$-$Q_{2}$ model relevant
to this study. The dynamic exponent $z$ is equal to $1$ Sandvik (2007). Recent
works show that the anomalous dimension $\eta\simeq 0.25$ Nahum  _et al._
(2015b), leading to $(2\beta/\nu)\simeq 1.25$ from the scaling laws and the
correlation length exponent is $\nu\simeq 0.455$ Sandvik and Zhao (2020).

Figure 1: Sketch of the quench dynamics in imaginary-time with different
initial states. The initial states are prepared as (i) the saturated VBS
state, (ii) the saturated AFM state, and (iii) the completely disordered
initial state. All these states have vanishing correlation length and
correspond to the fixed points of the initial states under the renormalization
group transformation.

Remarkably, it was shown that an emergent $\mathbb{SO}(5)$ symmetry appears at
the critical point of this Néel-VBS transition Senthil and Fisher (2006);
Nahum  _et al._ (2015b); Sreejith  _et al._ (2019). This symmetry is induced
by the conservation of the monopole defects Motrunich and Vishwanath (2004),
indicating that the critical point is described by the non-compact $(2+1)$D
quantum electrodynamics with the deconfined spinons as its matter field
Senthil  _et al._ (2004a, b); Levin and Senthil (2004). Moreover, in the VBS
phase, confined spinon-pairs are formed and the discrete $\mathbb{Z}_{4}$
symmetry is broken Haldane (1988); Read and Sachdev (1990, 1991). To reconcile
the emergent continuous symmetry at $q=q_{\rm c}$ and the discrete symmetry
for $q<q_{\rm c}$, the fugacity of the monopole defects should take
responsibility as a dangerously irrelevant scaling variable. Accordingly,
besides the usual correlation length $\xi\propto|\delta|^{-\nu}$ with
$\delta=q-q_{\rm c}$ the distance to the critical point, the confinement
length $\xi^{\prime}\propto|\delta|^{-\nu^{\prime}}$ with $\nu^{\prime}\simeq
0.585$ Shao  _et al._ (2016), measuring the averaged distance between two
spinons, also plays significant roles Oshikawa (2000); Lou  _et al._ (2007);
Okubo  _et al._ (2015); Shao  _et al._ (2020); Patil  _et al._ (2021); Léonard
and Delamotte (2015). It was shown that the interplay between these two
characteristic scales can explain the anomalous scaling behaviors for the
domain wall density and the susceptibility Shao  _et al._ (2016).

For the imaginary-time relaxation dynamics, the evolution of the wave function
$|\psi(\tau)\rangle$ obeys the imaginary-time Schrödinger equation

$-\frac{\partial}{\partial\tau}|\psi(\tau)\rangle=H|\psi(\tau)\rangle,$ (2)

with the normalization condition $Z=\langle\psi(\tau)|\psi(\tau)\rangle=1$ Yin
_et al._ (2014); Zhang  _et al._ (2014).  The formal solution of the
Schrödinger equation is given by

$|\psi(\tau)\rangle=\frac{1}{Z}U(\tau)|\psi(\tau_{0})\rangle,$ (3)

in which $U(\tau)\equiv\textrm{exp}(-\tau H)$ is the imaginary-time evolution
operator and $\tau_{0}$ is the starting point of the evolution. In studies of
relaxation dynamics, one focuses on the dynamical scaling behaviors of
different quantities when relaxed from a given state. Therefore, the initial
state is of key significance.  In the present work, as illustrated in Fig. 1,
we will consider three kinds of uncorrelated initial states: (i) the saturated
VBS state, (ii) the saturated AFM state, and (iii) the completely disordered
state. All these three states have vanishing correlation length and correspond
to the fixed points of the initial states under the renormalization group
transformation.

A wide range of quantum Monte Carlo (QMC) methods, including the stochastic
series expansion, projector QMC, and world line methods, have natural
connections to simulations of the imaginary-time evolution of quantum spin
systems Sandvik (2010b). In particular, the projector QMC has proven a
powerful tool in pursuing the imaginary-time dynamics Sandvik (2010b); Liu
_et al._ (2013); Grandi  _et al._ (2013); Farhi  _et al._ (2012); Shu  _et
al._ (2017); Shu and Yin (2020); Shao  _et al._ (2015); Weinberg and Sandvik
(2017); Shu  _et al._ (2022). In the projector QMC method, the imaginary-time
evolution operator is Taylor-expanded and the normalization can then be
written as the sum of the operator sequence acting on some suitable basis
states, such as the $S^{z}$ basis, the valence bond basis etc Tang and Sandvik
(2011); Beach and Sandvik (2006). The sum over the operator sequence, along
with the basis states, and the expansion power are then importance sampled.
Local and global operator-loop updating schemes are developed to improve the
efficiency of the Monte Carlo sampling. The expansion order is truncated to
some maximum length, which is not strictly necessary in principle but brings
significant convenience to implementations of the method. Note that such
truncation of the expansion order causes no detectable errors. Expectation
values of physical quantities are then estimated in the finial state
$|\psi(\tau)\rangle$ that is propagated from $|\psi(\tau_{0})\rangle$ with the
sampled operator sequence and the corresponding basis states. In QMC
simulations, to achieve a given initial state for the system, one need to fix
the boundaries of the imaginary-time propagation direction. Therefore,
different basis are applied for convenience. For the saturated VBS state, the
valence bond basis is used Tang and Sandvik (2011); Beach and Sandvik (2006),
while for the AFM and disordered state, the $S^{z}$ basis is used Farhi  _et
al._ (2012); Shu  _et al._ (2017); Shu and Yin (2020); Shu  _et al._ (2022);
Weinberg and Sandvik (2017); Liu  _et al._ (2013); Grandi  _et al._ (2013);
Tang and Sandvik (2011); Sandvik (2010b). For more detailed introduction of
the method, we refer to the literature Sandvik (2010b); Liu  _et al._ (2013);
Grandi  _et al._ (2013); Farhi  _et al._ (2012); Shu  _et al._ (2017); Tang
and Sandvik (2011); Sandvik (2007); Sandvik and Evertz (2010).

##  III Brief review of the dynamic scaling in $J$-$Q_{3}$ model

To study the nonequilibrium imaginary-time critical dynamics, one should at
first clarify the scaling relation between the imaginary-time $\tau$ and the
correlation length $\xi$. For usual critical point with one single divergent
length scale, the scaling relation between $\tau$ and $\xi$ satisfies
$\xi\propto\tau^{1/z}$. In contrast, for the criticality with two length
scales, there are two possibilities: (i) $\xi\propto\tau^{1/z}$ and
$\xi^{\prime}\propto\tau^{\nu^{\prime}/\nu z}$; and (ii)
$\xi^{\prime}\propto\tau^{1/z}$ and $\xi\propto\tau^{1/z_{u}}$ with $z_{u}$
being $z_{u}\equiv z\nu^{\prime}/\nu$ (The subscript $u$ means the usual
correlation length). It has been shown that scenario (ii) is selected by the
DQCP in the $J$-$Q_{3}$ model Shu  _et al._ (2022).

With scenario (ii), the imaginary-time dynamics for saturated ordered and
completely disordered initial states should obey the scaling form

$Y(\tau,\delta,L)=\tau^{\frac{s}{\tilde{z}}}f(\delta\tau^{\frac{1}{\tilde{\nu}\tilde{z}}},\tau
L^{-z},\tau L^{-z_{u}}),$ (4)

in which $Y$ is an arbitrary operator, $s$ is the exponent related to $Y$,
$\delta\equiv q-q_{\rm c}$ is the distance to the critical point, $L$ is the
lattice size, and $\tilde{z}$ is the dynamic exponent, which can be $z$ or
$z_{u}$, or their combination, depending on the operator $Y$ and the dynamic
process, similarly, $\tilde{\nu}$ can be $\nu$ or $\nu^{\prime}$ or
combination of both, $f$ is the scaling function. For the three kinds of
initial states introduced above, the initial state information does not appear
explicitly in Eq. (4), since all of them are the fixed points of the initial
states. In contrast, for other initial states with finite initial order
parameter and initial correlations, these initial conditions should be
included in the scaling form Janssen  _et al._ (2014); Li  _et al._ (1995);
Zheng (1996); Albano  _et al._ (2011); Yin  _et al._ (2014); Zhang  _et al._
(2014); Shu  _et al._ (2017); Shu and Yin (2020); Zuo  _et al._ (2021).

If $z_{u}=z$, Eq. (4) recovers the usual single-length-scale relaxation
scaling theory, in which, for instance, at the critical point, i.e.,
$\delta=0$, for a saturated initial state the order parameter scales as
$M^{2}=\tau^{-2\beta/\nu z_{u}}f({\tau L^{-z_{u}}})$ Li  _et al._ (1995);
Albano  _et al._ (2011); Shu  _et al._ (2022), while for a disordered initial
state $M^{2}=L^{-d}\tau^{d/z_{u}-2\beta/\nu z_{u}}f({\tau L^{-z_{u}}})$ in
which the factor $L^{-d}$ stems from the random distribution of the initial
state Albano  _et al._ (2011); Shu  _et al._ (2022). For both cases, in the
long-time limit,  the scaling form recovers the equilibrium case, namely,
$M^{2}\propto L^{-2\beta/\nu}$ Albano  _et al._ (2011); Shu  _et al._ (2022).

In contrast, at the critical point of $J$-$Q_{3}$ model, a _dual dynamic
scaling_ appears Shu  _et al._ (2022). Specifically, from the saturated VBS
initial state, $D^{2}$ is controlled by $\xi$ and obeys the scaling form

$D^{2}(\tau,L)=\tau^{-\frac{\beta}{\nu z_{u}}}f(\tau L^{-z_{u}}),$ (5)

while $M^{2}$ is controlled by $\xi^{\prime}$ and obeys the scaling form

$M^{2}=L^{-d}\tau^{\frac{d}{z}-\frac{2\beta}{\nu z}}f({\tau L^{-z}}).$ (6)

As a dual case, from the saturated AFM initial state, $M^{2}$ is controlled by
$\xi$ and obeys the scaling form

$M^{2}(\tau,L)=\tau^{-\frac{\beta}{\nu z_{u}}}f(\tau L^{-z_{u}}),$ (7)

while $D^{2}$ is controlled by $\xi^{\prime}$ and obeys the scaling form

$D^{2}=L^{-d}\tau^{\frac{d}{z}-\frac{2\beta}{\nu z}}f({\tau L^{-z}}).$ (8)

In addition, for the disordered initial state, which is  equivalent to the
Néel and the VBS phase, both $D^{2}$ and $M^{2}$ are controlled by
$\xi^{\prime}$ and obey similar scaling forms

$P^{2}=L^{-d}\tau^{\frac{d}{z}-\frac{2\beta}{\nu z}}f({\tau L^{-z}}),$ (9)

with $P$ represents $D$ and $M$. In the long-time limit, all these equations
tend to the same equilibrium form, $D^{2}\sim M^{2}\sim L^{-2\beta/\nu}$
Sandvik (2007). These scaling forms demonstrate a remarkable _dual dynamic
scaling_ that the dynamic scaling behaviors change to their dual partners
correspondingly when the initial states turn to their dual counterpart. This
dual dynamic scaling reflects the equivalence between the Néel order and the
VBS order at the critical point, and can thus be regarded as the noneqilibrium
incarnation of the equilibrium emergent $\mathbb{SO}(5)$ symmetry.

Figure 2: Estimation of the critical point via the dynamic scaling of the sign
function of the order parameter $I_{D(M)}$. The crossing points of curves of
$I_{D(M)}$ versus $q$ for $L$ and $2L$ and fixed $\tau L^{-1}=1/4$ converge to
the critical point as $L\rightarrow\infty$, giving $q_{\rm c}=0.0449(7)$ from
$I_{D}$ and $q_{\rm c}=0.0453(5)$ from $I_{M}$. The solid lines indicate fits
of the form $q_{\rm c}(L)=q_{\rm c}+aL^{-\omega}$.  Figure 3: Evolution of the
distance between two spinons $\Lambda$ at the critical point with the initial
state set as the one with a triplet embedded in the VBS background. The curves
for different $L$ before and after rescaled are shown in (a) and (b),
respectively. The solid line is a power-law fit that gives an exponent of
$0.953$, close to $1/z$.

##  IV Estimation of the critical point

In this section, we employ the nonequilibrium scaling to estimate the critical
point of the $J$-$Q_{2}$ model (1). In equilibrium, for usual criticality with
single length scale, any arbitrary dimensionless quantity $A$ scales as
$A=f(\delta L^{1/\nu})$. Accordingly, for different system sizes, curves of
$A$ versus $\delta$ will cross at the critical point  if the scaling
corrections are neglected. One can use this scaling feature of dimensionless
quantities to estimate the critical point Sandvik (2010b). For the DQCP with
two length scales, in general $A=f(\delta L^{1/\tilde{\nu}})$,  in which the
choice of $\tilde{\nu}$ depends on the quantity . For example,  in
equilibrium, when $A$ is the Binder ratio of the order parameter,
$\tilde{\nu}=\nu$; when $A$ is the ratio of the distance between two spinons
to the lattice, $\tilde{\nu}=\nu^{\prime}$ Shao  _et al._ (2016). Both can be
employed to determine the critical point Shao  _et al._ (2016).

In generalizing the scaling form of $A$ to the nonequilibrium case, according
to Eq. (4), one find that in general $A=f(\tau L^{-z},\tau L^{-z_{u}},\delta
L^{1/\tilde{\nu}})$, in which there are two additional time-dependent
variables: $\tau L^{-z}$ and $\tau L^{-z_{u}}$. Without a priori wisdom, one
does not know which one dominates. In the $J$-$Q_{3}$ model, it was found that
the sign function of the VBS order parameter from the VBS initial state,
$I_{D}\equiv\langle{\rm sgn}(D)\rangle$ is dominated by $\tau L^{-z}$ and so
does the sign function of the Néel order parameter from the AFM initial state,
$I_{M}\equiv\langle{\rm sgn}(M)\rangle$ Shu  _et al._ (2022).

To verify the universality of the dynamic properties of $I_{D(M)}$ and
determine the critical point of the $J$-$Q_{2}$ model, here we calculate
$I_{D}$ and $I_{M}$ versus $\delta$ for various lattice sizes with fixed $\tau
L^{-z}=1/4$ from the VBS and the AFM initial states, respectively.  Note that
in principle, different values of the ratio $\tau L^{-z}$ should not deliver
distinguishable results of $q_{\rm c}$ as long as $L$ is large enough.

From Fig. 2, one finds that the  size dependence of the crossing points of
$I_{D}$ versus $q$ for $L$ and $2L$ decrease monotonously as $L$ increases and
converge to a point $q=q_{\rm c}=0.0449(7)$ in the thermodynamic limit, which
is close to the known results of the critical point Sandvik (2010a); Shao  _et
al._ (2016); Sandvik and Zhao (2020). Moreover, Fig. 2 also shows that the
crossing points of the curves of $I_{M}$ versus $q$ for $L$ and $2L$ increases
as $L$ increases and converge to $q=q_{\rm c}=0.0453(5)$, which is almost
identical as that in $I_{D}$ case within the error bar. These results not only
determine the critical point accurately, but also demonstrate that both
$I_{D}$ and $I_{M}$ are dominated by $\tau L^{-z}$ in the relaxation process
from  their respective ordered initial states, similar to the case of the
$J$-$Q_{3}$ model. In the following calculations, we use $q_{\rm c}=0.045$,
which is closed to the average value of $q_{\rm c}$ obtained by $I_{D}$ and
$I_{M}$.

To further examine the value of the critical point, we calculate the dynamic
scaling of the spinon confinement length at the critical point $q_{\rm
c}=0.045$. By setting the initial state as that with a triplet in the VBS
background, we calculate the averaged distance between two unpaired spinons,
$\Lambda$, which is proportional to the confinement length $\xi^{\prime}$ Shao
_et al._ (2016). Figure 3 (a) shows that in the short-time stage,
$\Lambda\propto\tau^{0.953}$ with the exponent close to $1$. Moreover, Fig. 3
(b) shows that the rescaled curves of $\Lambda L^{-1}$ versus $\tau L^{-z}$
collapse onto each other well. These results not only confirm the value of the
critical point, but also demonstrate that the scaling relation
$\xi^{\prime}\propto\tau^{1/z}$ is universal in both the $J$-$Q_{2}$ and the
$J$-$Q_{3}$ model.

However, different from the usual criticality with  single length scale, in
the DQCP with two length scales, not every dimensionless variable exhibits
scaling properties controlled by the same scaling variable. In Sec. V.1, we
will show that the dynamics of the Binder ratios of the order parameters can
be dominated by both $\tau L^{-z}$ and $\tau L^{-z_{u}}$, depending on the
initial states.

Figure 4: Relaxation dynamics of the order parameters with the saturated VBS
initial state. Evolution of $D_{x}^{2}$, $D_{y}^{2}$, and $M^{2}$ for various
lattice sizes $L$ indicated are shown in top (a)-(c), middle (d)-(f), and
bottom (g)-(i) rows, respectively. In the left column (a),(d),(g), curves of
the order parameter versus $\tau$ are fitted in the short-time stage by the
power function. In addition, curves after rescaled are shown in the middle
(b),(e),(h) and right (c),(f),(i) columns. For both columns, $D_{x}^{2}$,
$D_{y}^{2}$, and $M^{2}$ are rescaled as
$D_{x}^{2}L^{2\beta/\nu}$,$D_{y}^{2}L^{2\beta/\nu}$ and $M^{2}L^{2\beta/\nu}$,
respectively. To compare, $\tau$ is rescaled according to $\tau L^{-z_{u}}$
and $\tau L^{-z}$ in the middle (b),(e),(h) and right (c),(f),(i) columns,
respectively.

##  V Dual dynamic scaling at the critical point

In this section, we will explore the imaginary-time relaxation dynamics at the
critical point from different initial states. We will show that the dual
dynamic scaling is a universal behavior in the relaxation dynamics of the DQCP
since it also appears at the critical point of the $J$-$Q_{2}$ model.

###  V.1 Dynamics with the VBS initial state

Figure 5: Relaxation dynamics of the Binder ratios with the VBS initial state.
Evolutions of $U_{D}$ and $U_{M}$ for various lattice sizes $L$ indicated are
shown in top (a)-(c) and bottom (d)-(f) rows, respectively. For comparison,
$\tau$ is rescaled according to $\tau L^{-z_{u}}$ in the middle column
(b),(e); while $\tau$ is rescaled according to $\tau L^{-z}$ in the right
column (c),(f).

The VBS order breaks the $\mathbb{Z}_{4}$ discrete symmetry. The initial VBS
state is chosen as that with horizontal dimers occupying every other bond as
shown in Fig. 1. Accordingly, at $\tau=0$, $D_{x}$ has a saturated value
$D_{x,0}=3/8$, while $D_{y}=0$.

For the VBS order parameter $D_{x}$, we find from Fig. 4 (a) that it relaxes
according to $D_{x}^{2}\propto\tau^{-0.847}$ in the short-time stage. This
exponent is close to $2\beta/\nu z_{u}\simeq 0.881$ rather than $2\beta/\nu
z\simeq 1.228$, in which $z_{u}=z\nu^{\prime}/\nu\simeq 1.398$.  Here, we use
$\nu/\nu^{\prime}\simeq 0.715$ obtained from direct fitting of the exponent
ratio in Ref. Shao  _et al._ (2016). Therefore, in general, the full scaling
form of $D_{x}^{2}$ in the whole relaxation process should be
$D_{x}^{2}=\tau^{-2\beta/\nu z_{u}}f(\tau L^{-z_{u}},\tau L^{-z})$. Moreover,
previous studies showed that in equilibrium $D^{2}\propto L^{-2\beta/\nu}$
Sandvik (2007, 2010a). Thus one can infer that the full scaling form of
$D_{x}^{2}$ should also be $D_{x}^{2}=L^{-2\beta/\nu}g(\tau L^{-z_{u}},\tau
L^{-z})$,  in which $g$ is another scaling function. These two full scaling
forms should be consistent with each other. This constraint indicates that the
dominant scaling variable in the scaling function is $\tau L^{-z_{u}}$ rather
than $\tau L^{-z}$, so that the scaling form can be converted from
$D_{x}^{2}=\tau^{-2\beta/\nu z_{u}}f(\tau L^{-z_{u}})$ to
$D_{x}^{2}=L^{-2\beta/\nu}g(\tau L^{-z_{u}})$ by recognizing the relation
$f(\tau L^{-z_{u}})=(\tau L^{-z_{u}})^{2\beta/\nu z_{u}}g(\tau L^{-z_{u}})$.
To verify the scaling theory, in Figs. 4 (b) and (c), we plot the rescaled
curves of $D_{x}^{2}L^{2\beta/\nu}$ versus $\tau L^{-z_{u}}$ and compare them
with those of $D_{x}^{2}L^{2\beta/\nu}$ versus $\tau L^{-z}$.  It is clear
that with $\tau$ scaled with $L^{-z_{u}}$, the curves collapse better than
with $L^{-z}$, indicating that it is $\tau L^{-z_{u}}$, or $\xi$, that governs
the scaling form.  These results show that the scaling form of $D_{x}^{2}$
from the VBS initial state also satisfied Eq. (5) found in the $J$-$Q_{3}$
model Shu  _et al._ (2022), confirming the universality of the scaling of
$D_{x}^{2}$ with the horizontal VBS initial state in the Néel-VBS transition.

Note that here one may argue that $D_{x}^{2}$ can also be expressed as
$D_{x}^{2}(\tau,L)\propto\tau^{-2\beta/\nu^{\prime}z}$. In this case,
$2\beta/\nu^{\prime}\simeq 0.881$ is smaller than $1$. Accordingly, the
scaling law $2\beta/\nu^{\prime}-1=\eta^{\prime}$ gives a negative anomalous
dimension $\eta^{\prime}$, which would imply a nonunitary theory Nahum  _et
al._ (2015a); Ferrara  _et al._ (1974); Mack (1977). Alternatively, to satisfy
the unitarity bound of the critical point Ferrara  _et al._ (1974); Mack
(1977), we choose to adopt $D_{x}^{2}(\tau,L)\propto\tau^{-2\beta/\nu z_{u}}$
in which ${2\beta/\nu}$ keeps intact while an additional dynamic exponent
$z_{u}$ is introduced.

From the perspective of the Néel order, the saturated VBS state  plays a
similar role to a disordered state, since both the VBS state and the
disordered state keep the spin rotation symmetry and have vanishing
correlation length. For the Néel order parameter $M$, we find from Fig. 4 (g)
that it relaxes according to $M^{2}\propto L^{-d}\tau^{0.737}$ in the short-
time stage. The exponent $0.737$ is close to $(d/z-2\beta/\nu z)=0.772$. Thus
the general full scaling form characterizing the whole relaxation process
should be $M^{2}=L^{-d}\tau^{d/z-2\beta/\nu z}f(\tau L^{-z_{u}},\tau L^{-z})$.
Similar to $D_{x}^{2}$, the full scaling form can also be expressed as
$M^{2}=L^{-2\beta/\nu}g(\tau L^{-z_{u}},\tau L^{-z})$, since in equilibrium
$M^{2}\propto L^{-2\beta/\nu}$. The consistence between these two full scaling
forms dictates that the dominant scaling variable in the scaling function is
$\tau L^{-z}$ rather than $\tau L^{-z_{u}}$,  such that the scaling form can
be converted from $M^{2}=L^{-d}\tau^{d/z-2\beta/\nu z}f(\tau L^{-z})$ to
$M^{2}=L^{-2\beta/\nu}g(\tau L^{-z})$ by recognizing the relation  $f(\tau
L^{-z})=(\tau L^{-z})^{-d/z+2\beta/\nu z}g(\tau L^{-z})$. Moreover, by
comparing the rescaled curves of $M^{2}L^{2\beta/\nu}$ versus $\tau
L^{-z_{u}}$ and $\tau L^{-z}$, one finds from Figs. 4 (h)-(i) that the
rescaled curves collapse onto each other with $\tau L^{-z}$, while deviate
from each other with $\tau L^{-z_{u}}$. These results show that in the
$J$-$Q_{2}$ model the dynamics of $M^{2}$ from the VBS initial state is
controlled by the confinement length scale $\xi^{\prime}$ and satisfied Eq.
(6), same as the case of the $J$-$Q_{3}$ model Shu  _et al._ (2022),
confirming the universality of the dynamic critical behavior of $M^{2}$ in
DQCP.

In addition, we calculate the VBS order parameter in the vertical direction
$D_{y}$. From Fig. 4 (d) we find that it relaxes according to
$D_{y}^{2}\propto L^{-d}\tau^{0.73}$ in the short-time stage. The exponent is
close to $(d/z-2\beta/\nu z)$, in analogy to the case of $M^{2}$. Similar
analyses give that the full scaling forms should be
$D_{y}^{2}=L^{-d}\tau^{d/z-2\beta/\nu z}f(\tau L^{-z})$ and
$D_{y}^{2}=L^{-2\beta/\nu}g(\tau L^{-z})$. Moreover, by comparing the rescaled
curves of $D_{y}^{2}L^{2\beta/\nu}$ versus $\tau L^{-z_{u}}$ and $\tau
L^{-z}$, one finds that the rescaled curves collapse better with $\tau
L^{-z}$, as shown in Figs. 4 (e)-(f). From the above results, one can see that
in the $J$-$Q_{2}$ model, with the horizontal VBS initial state, the dynamics
of $D_{y}^{2}$ is also controlled by the confinement length scale, same as the
case of $M^{2}$. Such results illustrate the equivalence under the rotation
between $M$ and $D_{y}$ when the initial state is saturated in $D_{x}$.

To further reveal the dynamic scaling behaviors from the VBS initial state, we
also study the Binder ratios for the VBS and Néel order parameters, defined as
$U_{D}\equiv\frac{3}{2}-\frac{\langle D_{x}^{4}\rangle}{2\langle
D_{x}^{2}\rangle^{2}}$ and $U_{M}\equiv\frac{3}{2}-\frac{\langle
M^{4}\rangle}{2\langle M^{2}\rangle^{2}}$, respectively. As a dimensionless
quantity, the general scaling forms are $U_{D(M)}=f_{D(M)}(\tau
L^{-z_{u}},\tau L^{-z})$. By comparing $U_{D(M)}$ versus $\tau L^{-z_{u}}$ and
$\tau L^{-z}$ in Fig. 5, one finds that $U_{D}$ is controlled by the usual
length scale $\xi$ and its dynamics obeys $U_{D}=f_{D}(\tau L^{-z_{u}})$,
while $U_{M}$ is controlled by the confinement length scale $\xi^{\prime}$ and
its dynamics obeys $U_{M}=f_{M}(\tau L^{-z})$. These scaling properties are
also consistent with the results that $D_{x}^{2}$ and $M^{2}$ are dominated by
$\tau L^{-z_{u}}$ and $\tau L^{-z}$, respectively.  Therefore, special
attention should be paid when one employs the short-time dynamics of the
Binder ratios $U_{D}$ and $U_{M}$ to determine the critical properties in DQCP
with two length scales, although this method is often used in usual LGW phase
transitions Li  _et al._ (1995). Combining with the scaling properties of the
sign function $I_{D(M)}$, we find that in the nonequilibrium dynamics of DQCP
with two length scales, dimensionless quantities can be controlled by
different length scales, depending on the physical quantities themselves and
the initial states applied.

###  V.2 Dynamics with the AFM initial state

Figure 6: Relaxation dynamics of order parameters with the AFM initial state.
Evolutions of $D^{2}$ and $M_{z}^{2}$ for various lattice sizes $L$ indicated
are plotted in top (a)-(c) and bottom (d)-(f) rows, respectively. In the left
column (a),(d), curves of the order parameter versus $\tau$ are fitted in the
universal short-time stage by the power function. In addition, curves after
rescaled are shown in the middle (b),(e) and right (c),(f) columns. For both
columns, $D^{2}$ and $M^{2}$ is rescaled as $D^{2}L^{2\beta/\nu}$ and
$M_{z}^{2}L^{2\beta/\nu}$, respectively. In contrast, $\tau$ is rescaled
according to $\tau L^{-z_{u}}$ and $\tau L^{-z}$in the middle (b),(e) and
right (c),(f) columns, respectively.

The AFM state breaks the spin $\mathbb{SO}(3)$ rotation symmetry. We choose
the initial state as the one with saturated component in the $z$-direction, as
shown in Fig. 1.

In analogy to the situation of the Néel order with the saturated VBS initial
state, from the perspective of the VBS order, the saturated AFM state  plays a
similar role to a disordered state, since both of them contribute zero VBS
order parameter and vanishing correlation length. We find in Fig. 6 (a) that
the VBS order parameter evolves according to $D^{2}\propto L^{-d}\tau^{0.73}$
in the universal short-time stage.  Here, $D^{2}=D_{x}^{2}+D_{y}^{2}$. This
exponent is close to $(d/z-2\beta/\nu z)$. By considering the equilibrium
scaling $D^{2}\propto L^{-2\beta/\nu}$ simultaneously Sandvik (2007, 2010a),
one finds that the confinement length dominates the dynamics and the scaling
form of $D^{2}$ is $D^{2}=L^{-d}\tau^{d/z-2\beta/\nu z}f(\tau L^{-z})$. This
scaling form can be converted to $D^{2}=L^{-2\beta/\nu}g(\tau L^{-z})$ by
substituting  $f(\tau L^{-z})=(\tau L^{-z})^{-d/z+2\beta/\nu z}g(\tau L^{-z})$
into the former equation. Moreover, by comparing the rescaled curves of
$D^{2}L^{2\beta/\nu}$ versus $\tau L^{-z_{u}}$ and $\tau L^{-z}$, as shown in
Figs. 6 (b)-(c), one finds that the rescaled curves collapse better with $\tau
L^{-z}$. These results show that in the $J$-$Q_{2}$ model the dynamics of
$D^{2}$ from the AFM initial state is controlled by the confinement length
scale and satisfied Eq. (8). These results are the same as the case of the
$J$-$Q_{3}$ model Shu  _et al._ (2022), confirming the universality of the
dynamic critical behavior of $D^{2}$ in DQCP.

Figure 7: Relaxation dynamics of order parameters with the disordered initial
state. Evolutions of $D^{2}$ and $M^{2}$ for various lattice sizes $L$
indicated are plotted in top (a)-(c) and bottom (d)-(f) rows, respectively. In
the left column (a),(d), curves of the order parameter versus $\tau$ are
fitted in the short-time stage by the power function. In addition, curves
after rescaled are shown in the middle (b),(e) and right (c),(f) columns. For
both columns, $D^{2}$ and $M^{2}$ is rescaled as $D^{2}L^{2\beta/\nu}$ and
$M_{z}^{2}L^{2\beta/\nu}$, respectively. In contrast, $\tau$ is rescaled
according to $\tau L^{-z_{u}}$ and in the middle (b),(e) and right (c),(f)
columns, respectively.

For the Néel order parameter, similar to the VBS order parameter from the
saturated VBS initial state, it decays with $\tau$ as
$M_{z}^{2}\propto\tau^{-0.866}$ with the exponent close to $2\beta/\nu z_{u}$
as shown in Fig. 6 (d). Combining with the equilibrium finite-size scaling
$M_{z}^{2}\propto L^{-2\beta/\nu}$ Sandvik (2007, 2010a), one deduces that the
full scaling form should be $M_{z}^{2}=\tau^{-2\beta/\nu z_{u}}f(\tau
L^{-z_{u}})$, or equivalently, $M_{z}^{2}=L^{-2\beta/\nu}g(\tau L^{-z_{u}})$.
As seen in Figs. 6 (e)-(f), by rescaling $M_{z}^{2}$ and $\tau$ with
$L^{2\beta/\nu}$ and $L^{-z_{u}}$, we find the rescaled curves match with each
other. However, when $\tau$ is rescaled with $L^{-z}$, the rescaled curves
deviates. These results confirm that the dynamics of $M_{z}^{2}$ with the
$z$-component AFM initial state is controlled by the usual correlation length
scale and satisfied Eq. (7). These results are also the same as the case of
the $J$-$Q_{3}$ model Shu  _et al._ (2022).

###  V.3 Dynamics with the disordered initial state

The disordered initial state can be prepared in very high temperature region.
From the perspective of the disordered state, either the Néel or the VBS order
looks equivalent. Accordingly, it is expected that both $D^{2}$ and $M^{2}$
should satisfy the same scaling form: $P^{2}=L^{-d}\tau^{d/z-2\beta/\nu
z}f(\tau L^{-z})$, in which $P$ represents $D$ and $M$. This dynamic scaling
is similar to the cases of $D^{2}$ with the AFM initial state and $M^{2}$ with
the VBS initial state. To verify these scaling properties, in Figs. 7 (a) and
(d), we plot the evolution of $D^{2}$ and $M^{2}$. We find that in the
universal short-time stage $P^{2}\propto L^{-d}\tau^{d/z-2\beta/\nu z}$. In
addition, by noting that $P^{2}=L^{-d}\tau^{d/z-2\beta/\nu z}f(\tau L^{-z})$
can be transformed to $P^{2}=L^{-2\beta/\nu z}f(\tau L^{-z})$, we compare the
curves of $P^{2}L^{2\beta/\nu}$ versus $\tau L^{-z}$ and $\tau L^{-z_{u}}$,
respectively. From Figs. 7 (b)-(c) and (e)-(f), one finds that with $\tau
L^{-z}$, the curves collapse better, verifying the full scaling forms
mentioned above. Moreover, these results demonstrate that both $M^{2}$ and
$D^{2}$ are controlled by the confinement length $\xi^{\prime}$. The same
scaling properties are also found in the $J$-$Q_{3}$ model as shown in Eq.
(9). Therefore one concludes that these scaling properties are universal as
well.

###  V.4 Dual dynamic scaling

The appearance of the emergent symmetry is a characteristic critical property
of the DQCP Senthil  _et al._ (2004a, b); Sandvik (2007); Lou and Sandvik
(2009); Nahum  _et al._ (2015b); Sreejith  _et al._ (2019); Ma  _et al._
(2019). For  the class of the $\mathbb{SU}(2)$ $J$-$Q$ model, the emergent
symmetry at the critical point is the $\mathbb{SO}(5)$ symmetry, which
describes the rotation symmetry between the components of the superspin
$\mathcal{S}=(M_{x},M_{y},M_{z},D_{x},D_{y})$, including the components of the
Néel and the VBS order parameters Senthil  _et al._ (2004a, b); Sandvik
(2007); Lou and Sandvik (2009); Nahum  _et al._ (2015b); Sreejith  _et al._
(2019). The emergent symmetry is broken in both ordered phases. In particular,
along with the appearance of the discrete $\mathbb{Z}_{4}$ symmetry, scaling
properties with two length scales arise on the VBS side.

In the imaginary-time relaxation dynamics, the above numerical results for the
$J$-$Q_{2}$ model (1) show a remarkable dual dynamic scaling behavior: the
scaling forms of the order parameters exchange as the initial state is rotated
in the superspin space, similar to the results in the $J$-$Q_{3}$ model Shu
_et al._ (2022). Specifically, when the initial state is rotated from the VBS
state to the AFM state, the dynamic scaling form of the VBS (Néel) order
parameter changes to that of the Néel (VBS) order parameter, and vice versa.
Additionally, for the disordered initial state which keeps invariant under the
rotation between the superspin components, both the VBS and the Néel order
parameters show similar scaling behaviors. Comparing with the equilibrium
emergent symmetry, we find that the dual dynamic scaling also reflects the
rotation symmetry between the superspin components. Moreover, these results
also show that effects induced by the interplay between two length scales are
naturally included in the dual dynamic scaling.

A prominent question is why the relaxation processes of different superspin
components can be  governed by different length scales, albeit their
equilibrium finite-size scaling forms are the same. By inspecting  the
relaxation behaviors under different initial conditions, one finds that for
the dominant component with a saturated initial value, i.e., the component
accordant with the saturated initial component, its relaxation dynamics is
controlled by the usual correlation length; while for the complementary
component with zero initial value, i.e., the component orthogonal to the
saturated initial component, its relaxation dynamics is controlled by the the
confinement length. For the  dominant component, the relaxation dynamics is
associated with the local fluctuations whose characteristic length scale is
measured by the usual correlation length $\xi$. For  the complementary
component, the average value of the component keeps zero in the relaxation
process due to the symmetry of the Hamiltonian, while its average squared
value in the short-time stage is proportional to $L^{-d}$ at a given evolution
time $\tau$. Since this average squared value is directly related to the
lattice size, one can infer that its dynamics is associated with the global
fluctuations, which is related to the topological properties of the system.
Different from the situation in the usual critical point where both the local
and the global fluctuations have the same characterized length scale $\xi$
Albano  _et al._ (2011); Shu  _et al._ (2022), in the DQCP  of the $J$-$Q$
model  class, the global fluctuations can have a different typical length
scale. To be specific, the global fluctuations correspond to the excitations
with the spinons living in the vortices of the VBS domain walls and the
distance between these spinons is characterized by the confinement length
$\xi^{\prime}$ Levin and Senthil (2004). Accordingly, it is expected that the
complementary order parameter component with a vanishing initial value should
be controlled by the confinement length $\xi^{\prime}$. This argument is also
supported by the findings that with a disordered initial state,  all superspin
components are governed by global fluctuations, so that both $M^{2}$ and
$D^{2}$ are controlled by $\xi^{\prime}$.

Besides the order parameters, other quantities can also obey the dual dynamic
scaling. For instance, it is shown that the sign function $I_{D(M)}$ from the
VBS (Néel) saturated initial state is controlled by the confinement length
$\xi^{\prime}$. By rotating $D$ to $M$, the initial states and dynamic scaling
forms change correspondingly. Note that here the scaling form for $I_{D}$ and
$I_{M}$ are the same. Moreover, although $D^{2}$ and $M^{2}$ are controlled by
the usual length scale $\xi$ for their respective ordered initial states,
their sign functions are dictated by the global flip of the order parameter
over entire lattice range. Accordingly, $I_{D(M)}$ is controlled by the
confinement length.

Figure 8: Relaxation dynamics away from the critical point with the VBS
initial state. Top row: for $\delta>0$, curves of $\Delta D_{x}^{2}$, $\Delta
D_{y}^{2}$ and, $\Delta M^{2}$ versus $\tau$ are plotted in (a), (b), and (c),
respectively. Bottom row: for $\delta<0$, curves of $\Delta D_{x}^{2}$,
$\Delta D_{y}^{2}$ and, $\Delta M^{2}$ versus $\tau$ are plotted in (d), (e),
and (f), respectively. Power law fittings are implemented for all curves in
the universal short-time stage. Double-logarithmic scales are used.

Here we discuss another intriguing puzzle on the dual dynamic scaling.
According to the usual critical theory in the presence of a dangerously
irrelevant scaling variable, the additional length scale $\xi^{\prime}$ only
play roles on one side of the critical point. Accordingly, asymmetric scaling
properties arise between two sides of the critical point in equilibrium
Léonard and Delamotte (2015).  In the DQCP, for the relaxation dynamics from
an ordered initial state to the critical point, this selected ordered state
breaks the rotation symmetry in the superspin space. Therefore, the relaxation
process should encode the information of the initial ordered phase during its
way down to the ground state. As a result, it is expected that with different
ordered initial states, asymmetric dynamic scaling should appear rather than
the dual dynamic scaling. Namely, following this argument, for the dominant
order parameter, different scaling forms should be observed for the dominant
order parameter on the exchange of the saturated AFM and VBS initial state. So
does the complementary ones. However, such prediction is inconsistent with the
numerical results found here and in Ref. Shu  _et al._ (2022).

To solve this puzzle, we note that in equilibrium, only the ground state
contributes to the expectation value and brings about the asymmetric scaling
properties. In contrast, in the universal stage of nonequilibrium process, the
relaxation state is the superposition of the initial state, the ground state
and the low-energy excited states. Specifically, for the AFM initial state,
besides the usual spin wave excitation,  excited states with quadrupled
monopoles , which are directly connect to the VBS order, also play important
roles at the critical point Haldane (1988); Read and Sachdev (1990, 1991).
Although the spin wave excitations near the AFM initial state exhibit the
usual correlation length $\xi$, the excited states with quadrupled monopoles
should be characterized by the confinement length scale $\xi^{\prime}$.
Similarly, for the VBS initial state, besides the usual spin triplet
excitation, excited states with spinons staying at the vortices of the VBS
domain walls  that have direct connection to the Néel order, are also
important at the critical point Levin and Senthil (2004). The correlation of
spin triplet excitations exhibits the usual correlation length $\xi$, while
the excited states combining deconfined spinons and vortices of the VBS domain
walls are characterized by the confinement length scale $\xi^{\prime}$.
Therefore, the relaxation state  is an intertwining state of both length
scales, despite of the choice of ordered initial state. This argument may help
to explain the appearance of the dual dynamic scaling under the superspin
rotation.

##  VI Dynamic scaling away from the critical point

In equilibrium, when the tuning parameter $\delta$ deviates from the critical
point, the emergent symmetry fades away. As the nonequilibrium incarnation of
the emergent symmetry, the fate of the dual dynamic scaling in the presence of
a finite $\delta$ is studied in this section.

###  VI.1 Off-critical-point dynamic scaling from the VBS initial state

According to Eq. (4), with the saturated horizontal VBS initial state and a
small $\delta$, $D_{x}^{2}$ obeys the scaling form

$D_{x}^{2}(\tau,\delta,L)=\tau^{-\frac{\beta}{\nu
z_{u}}}f(\delta\tau^{\frac{1}{\tilde{\nu}\tilde{z}}},\tau L^{-z_{u}}),$ (10)

in which $\tilde{\nu}$ is $\nu$ or $\nu^{\prime}$ and $\tilde{z}$ is $z$ or
$z_{u}$. However, the specific choices are unknown. To determine them, we
expand the scaling function $f$ in term of
$\delta\tau^{1/\tilde{\nu}\tilde{z}}$ and obtain $\Delta D_{x}^{2}\equiv
D_{x}^{2}(\tau,\delta,L)-D_{x}^{2}(\tau,L)$ up to the leading order as

$\Delta D_{x}^{2}(\tau,\delta)\simeq\delta\tau^{-\frac{\beta}{\nu
z_{u}}+\frac{1}{\tilde{\nu}\tilde{z}}}.$ (11)

Note that in Eq. (11) the dependence on $L$ is ignored since in the short-time
stage $\xi$ and $\xi^{\prime}$ are smaller than $L$. Similarly, $\Delta
D_{y}^{2}\equiv D_{y}^{2}(\tau,\delta,L)-D_{y}^{2}(\tau,L)$ and $\Delta
M^{2}\equiv M^{2}(\tau,\delta,L)-M^{2}(\tau,L)$ obey

$\Delta D_{y}^{2}(\tau,\delta)\simeq
L^{-d}\delta\tau^{\frac{d}{z}-\frac{2\beta}{\nu
z}+\frac{1}{\tilde{\nu}\tilde{z}}},$ (12)

and

$\Delta M^{2}(\tau,\delta)\simeq
L^{-d}\delta\tau^{\frac{d}{z}-\frac{2\beta}{\nu
z}+\frac{1}{\tilde{\nu}\tilde{z}}},$ (13)

respectively. By analyzing the dependence of $\Delta D_{x}^{2}$, $\Delta
D_{y}^{2}$ and $\Delta M^{2}$ on $\tau$, we can determine the specific choices
of $\tilde{\nu}$ and $\tilde{z}$ for different quantities. For clarification,
we enumerate the value of all possible combinations of
$1/\tilde{\nu}\tilde{z}$: $1/\nu z\simeq 2.198$, $1/\nu^{\prime}z=1/\nu
z_{u}\simeq 1.572$ and $1/\nu^{\prime}z_{u}\simeq 1.125$. In the following
calculations, we choose $\delta=\pm 0.005$. We have checked that for
$\delta=\pm 0.01$, there is no qualitative difference in the universal
properties.

At first, we study the case for $\delta<0$ with the ground state sitting in
the VBS side. From Fig. 8 (a), one finds that for $\delta=-0.005$, $\Delta
D^{2}_{x}$ evolves according to $\Delta D^{2}_{x}\propto\tau^{0.34}$. This
exponent is close to $-2\beta/\nu z_{u}+1/\nu^{\prime}z_{u}\simeq 0.246$.  The
deviation between the result in Fig. 8(a) and the value $0.246$ can result
from finite size effect. In addition, Figs. 8 (b) and (c) shows that $\Delta
D_{y}^{2}\propto\tau^{1.93}$ and $\Delta M^{2}\propto\tau^{1.844}$. Both their
exponents are close to $d/z-2\beta/\nu z+1/\nu^{\prime}z_{u}\simeq 1.897$.
These results show that for $D_{x}^{2}$, $D_{y}^{2}$ and $M^{2}$, their off-
critical-point effects in the short-time stage are all controlled by
$\delta\tau^{1/\nu^{\prime}z_{u}}$. Since
$\xi^{\prime}\propto|\delta|^{-\nu^{\prime}}$ and $\xi\propto\tau^{1/z_{u}}$,
the term $\delta\tau^{1/\nu^{\prime}z_{u}}$ combines the two arguments and
reflects an effect induced by the interplay of two length scales.

Then, let us turn to the case for $\delta>0$, in which the system is relaxed
from the VBS initial state to a AFM ground state. From Fig. 8 (d), one finds
that for $\delta=0.005$, $\Delta D^{2}_{x}$ evolves according to $\Delta
D^{2}_{x}\propto\tau^{0.32}$. This exponent is close to $-2\beta/\nu
z_{u}+1/\nu^{\prime}z_{u}\simeq 0.246$, similar to the case for $\delta<0$. In
contrast, for $\Delta D_{y}^{2}$ and $\Delta M^{2}$, we find that $\Delta
D_{y}^{2}\propto\tau^{2.23}$ and $\Delta M^{2}\propto\tau^{2.26}$, as shown in
Figs. 8 (e) and (f). Both exponents are close to $d/z-2\beta/\nu z+1/\nu
z_{u}$ or $d/z-2\beta/\nu z+1/\nu^{\prime}z$,  which are close to 2.344. These
results show that for $\delta>0$, the off-critical-point effects for $\Delta
D_{x}^{2}$, $\Delta D_{y}^{2}$ and $\Delta M^{2}$ are controlled by different
scaling variables: $\delta\tau^{1/\nu^{\prime}z_{u}}$ dominates in $\Delta
D_{x}^{2}$, while $\delta\tau^{1/\nu z_{u}}$ or $\delta\tau^{1/\nu^{\prime}z}$
dominates in $\Delta D_{y}^{2}$ and $\Delta M^{2}$.

In addition, for $\Delta D_{y}^{2}$ and $\Delta M^{2}$, from the present
numerical results, one cannot clarify which one of $\delta\tau^{1/\nu z_{u}}$
and $\delta\tau^{1/\nu^{\prime}z}$ dominates the scaling form for $\delta>0$.
Here we argue that both should be taken into account in the short-time stage.
For the former, both $\delta^{-\nu}$ and $\tau^{1/z_{u}}$ are related to the
usual length scale $\xi$, and $\delta\tau^{1/\nu z_{u}}$ represents the ratio
between the contributions from $\delta$ and $\tau$ to $\xi$; while for the
latter, both $\delta^{-\nu^{\prime}}$ and $\tau^{1/z}$ are related to the
confinement length $\xi^{\prime}$, and $\delta\tau^{1/\nu^{\prime}z}$
represents the ratio between the contributions from $\delta$ and $\tau$ to
$\xi^{\prime}$. These ratios can have similar order of magnitude in the
universal short-time stage for finite $\delta$.  However, in the long-time
stage, $\delta\tau^{1/\nu z_{u}}$ should dominate due to the asymmetric
appearance of the dangerously irrelevant scaling variable. In the equilibrium
situation, the dangerously irrelevant scaling variable only appear on the VBS
side, causing scaling behaviors as a result of the interplay of the usual
length scale $\xi$ and the additional confinement length scale $\xi^{\prime}$.
When relaxed to the AFM side, there is no dangerously irrelevant scaling
variable in the equilibrium AFM ground state. Therefore, only the usual length
scale, namely $\delta\tau^{1/\nu z_{u}}$, should governs the scaling form.

Thus, we conclude the choice of $1/\tilde{\nu}\tilde{z}$ for different cases:

(i) when $\delta<0$, for $\Delta D_{x}^{2}$, $\Delta D_{y}^{2}$ and $\Delta
M^{2}$, $1/\tilde{\nu}\tilde{z}$ should be $1/\nu^{\prime}z_{u}$, which
reflects the interplay of two length scales.

(ii) when $\delta>0$, for $\Delta D_{x}^{2}$, $1/\tilde{\nu}\tilde{z}$ is
$1/\nu^{\prime}z_{u}$ as well while for $\Delta D_{y}^{2}$ and $\Delta M^{2}$,
$1/\tilde{\nu}\tilde{z}$ can be $1/\nu z_{u}$ or $1/\nu^{\prime}z$.

###  VI.2 Off-critical-point dynamic scaling from the AFM initial state

Figure 9: Relaxation dynamics away from the critical point with the AFM
initial state. Top row: for $\delta>0$, curves of $\Delta D^{2}$ and $\Delta
M_{z}^{2}$ versus $\tau$ are plotted in (a) and (b), respectively. Bottom row:
for $\delta<0$, curves of $\Delta D^{2}$ and $\Delta M_{z}^{2}$ versus $\tau$
are plotted in (c) and (d), respectively. Power law fittings are implemented
for all curves in the universal short-time stage. Double-logarithmic scales
are used.

From the saturated AFM initial state, similar analyses give that for a small
$\delta$, in the universal short-time stage, $\Delta D^{2}$ obeys the
following scaling relation

$\Delta D^{2}(\tau,\delta)\simeq
L^{-d}\delta\tau^{\frac{d}{z}-\frac{2\beta}{\nu
z}+\frac{1}{\tilde{\nu}\tilde{z}}},$ (14)

and $\Delta M_{z}^{2}$ obeys

$\Delta M_{z}^{2}(\tau,\delta)\simeq\delta\tau^{-\frac{2\beta}{\nu
z_{u}}+\frac{1}{\tilde{\nu}\tilde{z}}}.$ (15)

to the leading order of $\delta\tau^{1/\tilde{\nu}\tilde{z}}$.

Firstly, we study the case for $\delta<0$ with the ground state sitting in the
VBS side. From Fig. 9 (a), one finds that for $\delta=-0.005$, $\Delta
M_{z}^{2}$ evolves according to $\Delta M_{z}^{2}\propto\tau^{0.36}$. This
exponent is close to $-2\beta/\nu z_{u}+1/\nu^{\prime}z_{u}\simeq 0.246$,
similar to the scaling relation of $\Delta D^{2}_{x}$ from the VBS initial
state. In contrast, from Fig. 9 (b), we find that $\Delta
D^{2}\propto\tau^{2.47}$ with its exponent close to $d/z-2\beta/\nu z+1/\nu
z_{u}$ and $d/z-2\beta/\nu z+1/\nu^{\prime}z$,  both close to 2.344. Similar
to the case of $\Delta M^{2}$ with $\delta>0$ from the VBS initial state, we
argue that both $\delta\tau^{1/\nu z_{u}}$ and $\delta\tau^{1/\nu^{\prime}z}$
should be taken into account in the universal short-time stage.  However, in
the long-time stage, different from the case when the system is relaxed from
the saturated VBS initial state to the AFM side, from the saturated AFM
initial state to the VBS side, besides $\delta\tau^{1/\nu z_{u}}$,
$\delta\tau^{1/\nu^{\prime}z}$ should also play impotant roles.

Then, we turn to the case for $\delta>0$ and the ground state is in the Néel
phase. From Fig. 9 (c), one finds that for $\delta=0.005$, $\Delta M_{z}^{2}$
evolves according to $\Delta M_{z}^{2}\propto\tau^{0.37}$. This exponent is
close to $-2\beta/\nu z_{u}+1/\nu^{\prime}z_{u}\simeq 0.246$, similar to
previous scaling relation of $\Delta M_{z}^{2}$ for $\delta<0$. Moreover, Fig.
9 (d) shows that $\Delta D^{2}\propto\tau^{1.97}$ with the exponent close to
$d/z-2\beta/\nu z+1/\nu^{\prime}z_{u}\simeq 1.897$. These results show that
for $\Delta D^{2}$ and $\Delta M_{z}^{2}$, the off-critical-point effects in
the universal short-time stage is controlled by
$\delta\tau^{1/\nu^{\prime}z_{u}}$ when $\delta<0$.

Thus, we conclude the choice of $1/\tilde{\nu}\tilde{z}$ for different cases:

(i) when $\delta<0$, for $\Delta M_{z}^{2}$, $1/\tilde{\nu}\tilde{z}$ is
$1/\nu^{\prime}z_{u}$ while for $\Delta D^{2}$, the choice can be $1/\nu
z_{u}$ or $1/\nu^{\prime}z$.

(ii) when $\delta>0$, for $\Delta M_{z}^{2}$ and $\Delta D^{2}$,
$1/\tilde{\nu}\tilde{z}$ should be $1/\nu^{\prime}z_{u}$.

###  VI.3 Off-critical-point dynamic scaling from the disordered initial state

Figure 10: Relaxation dynamics away from the critical point with the
disordered initial state. Top row: for $\delta>0$, curves of $\Delta D^{2}$
and $\Delta M^{2}$ versus $\tau$ are plotted in (a) and (b), respectively.
Bottom row: for $\delta<0$, curves of $\Delta D^{2}$ and $\Delta M^{2}$ versus
$\tau$ are plotted in (c) and (d), respectively. Power law fittings are
implemented for all curves in the universal short-time stage. Double-
logarithmic scales are used.

For the disordered initial state, similar analyses give that for a small
$\delta$, in the universal short-time stage, $\Delta D^{2}$ obeys

$\Delta D^{2}(\tau,\delta)\simeq
L^{-d}\delta\tau^{\frac{d}{z}-\frac{2\beta}{\nu
z}+\frac{1}{\tilde{\nu}\tilde{z}}},$ (16)

and $\Delta M^{2}$ obeys

$\Delta M^{2}(\tau,\delta)\simeq
L^{-d}\delta\tau^{\frac{d}{z}-\frac{2\beta}{\nu
z}+\frac{1}{\tilde{\nu}\tilde{z}}},$ (17)

to the leading order of $\delta\tau^{1/\tilde{\nu}\tilde{z}}$.

For $\delta<0$ with the VBS ground state, Figs. 10 (a)-(b) shows that for
$\delta=-0.005$, $\Delta D^{2}$ and $\Delta M^{2}$ evolve according to $\Delta
D^{2}\propto\tau^{2.4}$ and $\Delta M^{2}\propto\tau^{2.24}$, respectively.
The two exponents are close to $d/z-2\beta/\nu z+1/\nu z_{u}$ and
$d/z-2\beta/\nu z+1/\nu^{\prime}z$, both close to $2.344$. As discussed above,
both $\delta\tau^{1/\nu z_{u}}$ and $\delta\tau^{1/\nu^{\prime}z}$ should make
contributions in the short-time regime. For $\delta>0$ with the AFM ground
state, from Fig. 10 (c)-(d), one finds that $\Delta D^{2}\propto\tau^{1.95}$
and $\Delta M^{2}\propto\tau^{1.87}$ when $\delta=0.005$. Both these exponents
are close to $d/z-2\beta/\nu z+1/\nu^{\prime}z_{u}=1.897$, demonstrating the
off-critical-point effects are mainly contributed by
$\delta\tau^{1/\nu^{\prime}z_{u}}$.

Thus, we conclude that with the disordered initial state,

(i) when $\delta<0$, for both $\Delta M^{2}$ and $\Delta D^{2}$,
$1/\tilde{\nu}\tilde{z}$ can be $1/\nu z_{u}$ or $1/\nu^{\prime}z$.

(ii) when $\delta>0$, for both $\Delta M^{2}$ and $\Delta D^{2}$,
$1/\tilde{\nu}\tilde{z}$ is $1/\nu^{\prime}z_{u}$.

###  VI.4 Breakdown and vestige of the dual dynamic scaling

From the above results, we find that the dual dynamic scaling breaks down
under the rotation between the superspin components when the tuning parameter
is away from its critical point. In the long-time limit, the system reaches
its equilibrium ground state, in which the asymmetric scaling properties
between the Néel order and the VBS order appear Léonard and Delamotte (2015);
Lou  _et al._ (2007) and meanwhile the emergent symmetry fades out Lou and
Sandvik (2009). In the short-time stage, from the ordered initial state, for a
fixed $\delta\neq 0$, under the rotation between $D$ and $M$, one finds that
the dual dynamic scaling also breaks down. Although  the dominant order
parameters of the ordered initial states obey similar scaling behavior,  the
complementary order parameters behave quite differently. When relaxed from the
saturated VBS initial state to the VBS side, i.e. $\delta<0$,  for the
complementary order parameter $M$, its off-critical-point effects $\Delta
M^{2}$ is dominated by $\delta\tau^{d/z-2\beta/\nu z+1/\nu^{\prime}z_{u}}$; as
the dual case, from the AFM initial state, in contrast,  for the complementary
order parameter $D$, its off-critical-point effects $\Delta D^{2}$ is
dominated by $\delta\tau^{d/z-2\beta/\nu z+1/\nu^{\prime}z}$ and  /or
$\delta\tau^{d/z-2\beta/\nu z+1/\nu z_{u}}$. Similarly, when relaxed from the
saturated VBS initial state to the AFM side, i.e. $\delta>0$, $\Delta M^{2}$
is dominated by $\delta\tau^{d/z-2\beta/\nu z+1/\nu^{\prime}z}$ and  /or
$\delta\tau^{d/z-2\beta/\nu z+1/\nu z_{u}}$; as its dual case, from the AFM
initial state, in contrast, $\Delta D^{2}$ is dominated by
$\delta\tau^{d/z-2\beta/\nu z+1/\nu^{\prime}z}$. This demonstrates that the
dual dynamic scaling breaks when the parameter is tuned away from the critical
point.

However, strikingly, for the ordered initial states, we find that the dual
dynamic scaling in the universal short-time stage can restore once the
transformation $M\Leftrightarrow D$ and $\delta\Leftrightarrow-\delta$ are
simultaneously implemented. Under this joint transformation, the relaxation
dynamics of the VBS (Néel) order parameter for $\delta<0$ with the VBS intial
state can be dualized to the relaxation dynamics of the Néel (VBS) order
parameter for $\delta>0$ with the Néel initial state, and vice versa.
Accordingly, the scaling behaviors of both the  dominant and the complementary
order parameters keep invariant under this joint dual transformation.  In this
sense, the dual dynamic scaling is maintained.

The vestige of the dual dynamic scaling in the short-time stage demonstrates
that the nonequilibrium dynamics can even exhibit much higher symmetry than
the equilibrium state.  Therefore, unlike the fade of the emergent symmetry on
running away from the critical point in equilibrium, the dual dynamic scaling
can survive up to some finite $\delta$ under the joint dual transformation. A
possible reason is that the nonequilibrium dynamics mixes various elements
together to form a state which is covariant under the dual transformation,
although each single element is asymmetric under the same condition. For
instance, relaxing from the saturated VBS initial state to the VBS side with a
finite $\delta<0$, the relaxation dynamics mixes the the VBS background, the
VBS ground state, and the topological defects with spinons living at the
vortices of the VBS domain walls; while relaxing from the AFM initial state to
the AFM side with a finite $\delta>0$, the relaxation dynamics mixes the local
fluctuations with the background in the AFM pattern, the AFM ground state, and
the topological defects with quadrupled monopole excitations. The interplay
between these elements may give rise to the dual dynamic scaling. However,
from the disordered initial states, in the short-time stage, only global
fluctuations contribute to $M^{2}$ and $D^{2}$. Thus the dual dynamic scaling
in the short-time stage still exists for the transformation of
$M\Leftrightarrow D$. However, it breaks down when
$\delta\Leftrightarrow-\delta$ is added simultaneously, since in this case,
the ground state information, which breaks the emergent symmetry of the
superspin rotation, should be simultaneously included.

##  VII Possible experimental realization

Figure 11: Realization of imaginary-time relaxation dynamics via unitary
quantum gates.

Recently, it was shown that quantum computers has become a new vivifying
platform to realizing various ‘experiments’ ranging from high-energy physics
Martinez  _et al._ (2016); Bauer  _et al._ (2021) to condensed matter physics
Satzinger  _et al._ (2021); Semeghini  _et al._ (2021). In particular,
nonequilibrium quantum critical dynamics has been observed in the D-wave
devices Weinberg  _et al._ (2020) and the noisy intermediate-scale quantum
device based on the Rigetti superconducting quantum chip Dupont and Moore
(2021). Moreover, imaginary-time relaxation was also implemented in the
various devices as a promising approach to find the ground state Motta  _et
al._ (2020). As illustrated in Fig. 11, imaginary-time evolution in a small
time interval $\Delta\tau$ with the Hamiltonian $H$ can be approximated by the
real-time unitary evolution with the auxiliary Hamiltonian $\mathcal{H}$. By
minimizing the following residual form,

${\rm Res}\equiv\left\|\frac{e^{-\Delta\tau
H}|\psi\rangle}{\langle\psi|e^{-2\Delta\tau H}|\psi\rangle}-e^{-i\Delta
t\mathcal{H}}|\psi\rangle\right\|^{2},$ (18)

one can determine $\mathcal{H}$ Motta  _et al._ (2020); Nishi  _et al._
(2021).  It was shown that the interaction length in $\mathcal{H}$ is only
required to have the same order of magnitude as the correlation length $\xi$.
For our present case, the initial state is set as the uncorrelated state with
vanishing correlation length Motta  _et al._ (2020). Thus it is promising that
our results can be detected in these systems in the future.

##  VIII Summary

In summary, we have studied the nonequilibrium imaginary-time dynamics of the
DQCP in the $2$D $J$-$Q_{2}$ model. At the critical point, we have generalized
the emergent symmetry in equilibrium to the dual dynamic scaling in the
relaxation process. In particular, we have found that in the relaxation
dynamics from the ordered initial states, although the dominant and the
complementary order parameters are controlled by different length scales, they
can be  dualized to each other under the rotation transformation in the
superspin space. In addition, we have shown that for the disordered initial
state the dual dynamic scaling dictates that both the Néel and the VBS order
parameter obey the same scaling form. By comparing the results of the
$J$-$Q_{3}$ model Shu  _et al._ (2022), we have confirmed that the dual
dynamic scaling is a universal nonequilibrium phenomenon in the DQCP that
separates the Néel order and the VBS order. Furthermore, we have studied the
relaxation dynamics away from the critical point and found that the dual
dynamic scaling can even exist when the emergent symmetry in the ground state
has been broken. We have attributed the appearance of the dual dynamic scaling
at and near the DQCP to the superposition of information included in the
initial state, ground state and the low-energy excited states. In addition, we
have discussed the possible experimental realizations in the fast-developing
programmable quantum devices.

In this paper, we have only focused on the situation where the initial state
is at its fixed points. In usual critical point with one single length scale,
it was shown that the initial state can induce interesting scaling behaviors
Janssen  _et al._ (2014); Zheng (1996); Yin  _et al._ (2014); Zhang  _et al._
(2014); Shu  _et al._ (2017); Shu and Yin (2020). In our previous work Shu
_et al._ (2022), we have shown that in DQCP, the critical initial slip
exponent is negative, while in usual phase transitions, this exponent is
positive. Thus it is instructive to explore more general effects induced by
different initial states.

Moreover, at present, directly attacks on the nonequilibrium dynamics in two-
dimensional quantum systems is still confronted with sever difficulties.
Although the direct analytic continuations seem inadequate to obtain the
results of the real-time dynamics, it has been shown that some universal
behaviors are shared in both the imaginary-time and real-time dynamics De
Grandi  _et al._ (2011); Grandi  _et al._ (2013); Liu  _et al._ (2013);
Kolodrubetz  _et al._ (2017). In addition, it was shown that the real-time
relaxation has the similar scaling relations with the imaginary-time
relaxation, but with different critical exponent Dalla Torre  _et al._ (2013);
Gagel  _et al._ (2014); Maraga  _et al._ (2015); Chiocchetta  _et al._ (2015,
2016, 2017); Jian  _et al._ (2019); Yin and Jian (2021); Halimeh and Maghrebi
(2021). Therefore, it is expected that our present work should provide
insights to the real-time dynamics of the DQCP.

## Acknowledgements

Y.R.S. acknowledges support from the National Natural Science Foundation of
China (Grants No. 11947035 and No. 12104109), the Science and Technology
Projects in Guangzhou (202102020933). S.Y. is supported by the National
Natural Science Foundation of China (Grant No. 12075324).

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