# Finite-time Scaling beyond the Kibble-Zurek Prerequisite: Driven Critical
Dynamics in Strongly Interacting Dirac Systems

Zhi Zeng1,2111These authors contribute equally to this work. These authors
contribute equally to this work.  Yin-Kai Yu1,2111These authors contribute
equally to this work. These authors contribute equally to this work.  Zhi-Xuan
Li1,2111These authors contribute equally to this work. These authors
contribute equally to this work.  Zi-Xiang Li3,4111These authors contribute
equally to this work<EMAIL_ADDRESS>Shuai Yin1,2
<EMAIL_ADDRESS>1Guangdong Provincial Key Laboratory of
Magnetoelectric Physics and Devices, School of Physics, Sun Yat-Sen
University, Guangzhou 510275, China  2School of Physics, Sun Yat-Sen
University, Guangzhou 510275, China  3Beijing National Laboratory for
Condensed Matter Physics & Institute of Physics, Chinese Academy of Sciences,
Beijing 100190, China  4University of Chinese Academy of Sciences, Beijing
100049, China

###### Abstract

In conventional quantum critical point (QCP) characterized by order parameter
fluctuations, the celebrated Kibble-Zurek mechanism (KZM) and finite-time
scaling (FTS) theory provide universal descriptions of the driven critical
dynamics. However, in strongly correlated fermionic systems where gapless
fermions are usually present in vicinity of QCP, the driven dynamics has
rarely been explored. In this Letter, we investigate the driven critical
dynamics in two-dimensional Dirac systems, which harbor semimetal and Mott
insulator phases separated by the QCP triggered by the interplay between
fluctuations of gapless Dirac fermions and order-parameter bosons. By studying
the evolution of physical quantities for different driving rates through
large-scale quantum Monte Carlo simulation, we confirm that the driven
dynamics is described by the FTS form. Accordingly, our results significantly
generalize the KZM theory by relaxing its requirement for a gapped initial
state to the system accommodating gapless Dirac fermionic excitation. Through
successfully extending the KZM and FTS theory to Dirac QCP, our work not only
brings new fundamental perspective into the nonequilibrium critical dynamics,
but also provides a novel theoretical approach to fathom quantum critical
properties in fermionic systems.

Introduction— Fathoming nonequilibrium universal properties near a quantum
critical point (QCP) is one of the central issues in modern physics Dziarmaga
(2010); Polkovnikov  _et al._ (2011); D’Alessio  _et al._ (2016); Mitra
(2018). Although the general organizing principle for the nonequilibrum
critical dynamics is still elusive, a unified framework for understanding the
generation of topological defects after the linear quench was proposed by
Kibble in cosmological physics and then generalized by Zurek in condensed
matter systems Kibble (1976); Zurek (1985). This celebrated Kibble-Zurek
mechanism (KZM) has aroused intensive investigations from both theoretical and
experimental aspects, exerting far-reaching significance in both classical and
quantum phase transitions Kibble (1976); Zurek (1985); Laguna and Zurek
(1997); Hindmarsh and Rajantie (2000); Chuang  _et al._ (1991); Dziarmaga
(1998); Zurek  _et al._ (2005); Dziarmaga (2005); Damski and Zurek (2007);
Lamporesi  _et al._ (2013a); Navon  _et al._ (2015); Du  _et al._ (2023); Ko
_et al._ (2019); Ulčakar  _et al._ (2020); Crowley  _et al._ (2020); Gómez-
Ruiz  _et al._ (2020); Goo  _et al._ (2021); Maegochi  _et al._ (2022);
Dziarmaga  _et al._ (2022); Sherman  _et al._ (2023); Balducci  _et al._
(2023); Zeng  _et al._ (2023); Lamporesi  _et al._ (2013b); Barends  _et al._
(2016); King  _et al._ (2022); Keesling  _et al._ (2019); Ebadi  _et al._
(2021); Yao  _et al._ (2022); Qiu  _et al._ (2020); Ebadi  _et al._ (2022);
Sunami  _et al._ (2023); Li  _et al._ (2023). More interestingly, it was found
that scaling behaviors can also manifest themselves in the whole driven
process Zhong and Xu (2005); Deng  _et al._ (2009); De Grandi  _et al._
(2010); Chandran  _et al._ (2012); Clark  _et al._ (2016); Kolodrubetz  _et
al._ (2012). As a generalization of the KZM, a finite-time scaling (FTS)
theory was proposed to systematically understand the full scaling properties
Gong  _et al._ (2010); Feng  _et al._ (2016). These full scaling forms have
been verified in various systems from numerical to experimental works Huang
_et al._ (2014); Liu  _et al._ (2014); Yin  _et al._ (2014); Liu  _et al._
(2015); Yin  _et al._ (2017); Li  _et al._ (2019); Shu  _et al._ (2023); Clark
_et al._ (2016); Gong  _et al._ (2010); Feng  _et al._ (2016); King  _et al._
(2023); Keesling  _et al._ (2019); Ebadi  _et al._ (2021); Garcia and Chepiga
(2024); King  _et al._ (2024). Moreover, the KZM and the FTS recently show
their fabulous power in state preparations and probing critical properties in
fast-developing programmable quantum devices King  _et al._ (2023); Keesling
_et al._ (2019); Ebadi  _et al._ (2021); Garcia and Chepiga (2024); King  _et
al._ (2024); Dupont and Moore (2022).

The fermionic QCP Sachdev (2011); Hertz (1976); Millis (1993); Berg  _et al._
(2019); Löhneysen  _et al._ (2007); Lee (2018a); Metlitski and Sachdev (2010);
Yerzhakov and Maciejko (2018); Esterlis  _et al._ (2021); Chubukov  _et al._
(2004), in which gapless fermionic excitations are present in vicinity of the
transition point, are ubiquitous to strongly correlated quantum materials such
as heavy-fermion systems Paschen and Si (2021); Stewart (1984); Si  _et al._
(2001); Custers  _et al._ (2003); Senthil  _et al._ (2004), cuprate and Fe-
based superconductors Keimer  _et al._ (2015); Lee  _et al._ (2006); Sachdev
(2003); Chen  _et al._ (2014); Lee (2018b); Fernandes  _et al._ (2022), and
twisted bilayer graphene Cao  _et al._ (2018a, b); Jaoui  _et al._ (2022);
Andrei and MacDonald (2020). In the aspects of equilibrium nature, many
intriguing phenomena are unveiled arising from the interplay between critical
fluctuations of order parameter and gapless fermions, including strange
metallic transport Prochaska  _et al._ (2020); Patel  _et al._ (2023); Shen
_et al._ (2020); Varma (2020); Phillips  _et al._ (2022), high-$T_{c}$
superconductivity Scalapino (2012); Lederer  _et al._ (2015); Li  _et al._
(2016a); Berg  _et al._ (2012); Marel  _et al._ (2003) and intertwined
ordering Kivelson  _et al._ (2003); Fradkin  _et al._ (2015); Schattner  _et
al._ (2016); Wang  _et al._ (2018); Li  _et al._ (2017a); Gerlach  _et al._
(2017); Varma (1999). In contrast, the nonequilibrium properties of quantum
driven dynamics in fermionic QCP are largely unaddressed.

The QCP occuring in Dirac semimetal (DSM), dubbed as Dirac QCP, represents a
typical class of fermionic QCP. The studies of Dirac QCP stem from the
research in modern high-energy physics, such as chiral symmetry in QCD and
mass generation via spontaneously symmetry breaking Gross and Neveu (1974);
Roy  _et al._ (2016); Gracey  _et al._ (2016); Gehring  _et al._ (2015);
Poland  _et al._ (2019); Gazit  _et al._ (2017); You  _et al._ (2018). From
the perspectives of statistical mechanics and condensed matter physics, Dirac
quantum criticality also attracts enormous attentions, particularly after the
experimental realization of two-dimensional Dirac fermions in graphene and
various topological insulators or semimetals. The presence of Dirac fermions
is theoretically revealed to tremendously enrich quantum critical properties,
rendering novel universality classes of quantum phase transition without
classical counterpart Ladovrechis  _et al._ (2023); Janssen and Herbut (2014);
Parisen Toldin  _et al._ (2015); Knorr (2016); Ihrig  _et al._ (2018); Boyack
_et al._ (2021); Lang and Läuchli (2019); Liu  _et al._ (2023); Li  _et al._
(2018); Zhou  _et al._ (2016); Tabatabaei  _et al._ (2022); Otsuka  _et al._
(2018); Xu and Grover (2021); Chandrasekharan and Li (2013); Wang and Meng
(2023); Yu  _et al._ (2024); Xu  _et al._ (2023); Zhu  _et al._ (2022);
Sorella and Tosatti (1992); Herbut (2006); Herbut  _et al._ (2009); Assaad and
Herbut (2013); Otsuka  _et al._ (2016); Mihaila  _et al._ (2017); Zerf  _et
al._ (2017); Grover  _et al._ (2014); Seifert  _et al._ (2020); Ray and
Janssen (2021); Corboz  _et al._ (2018); Rader and Läuchli (2018); Da Liao
_et al._ (2022); Wang  _et al._ (2014); Li  _et al._ (2015a); Liu  _et al._
(2021, 2022); Chen  _et al._ (2019); Zhang  _et al._ (2019); Roy and Das Sarma
(2016); Han  _et al._ (2018); Herbut (2023); Otsuka  _et al._ (2020); Li  _et
al._ (2017b, 2020); Torres  _et al._ (2018); Classen  _et al._ (2017); Yin and
Zuo (2020). Consequently, fathoming the non-equilibrium driven dynamics in
Dirac QCP has overarching meaning in basic theory, as well as immediate
applications in the context of detecting and exploring fermionic QCP in
experimental platform including realistic materials and quantum devices Castro
Neto  _et al._ (2009); Hasan and Kane (2010); Qi and Zhang (2011).
Particularly, whether the original KZM is still applicable in the presence of
gapless fermions remains largely elusive, constituting a fundamental question
that bears profound impact on the theory of phase transition and
nonequilibrium physics.

However, directly attacking the real-time dynamics in two or higher spatial
dimension is largely hindered by the lack of reliable theoretical or numerical
methods. Specifically, quantum Monte Carlo (QMC) fails as a result of the
notorious sign problem Assaad and Evertz (2008); Troyer and Wiese (2005); Li
and Yao (2019), while the tensor-network method still needs tremendous
improvements despite remarkable progress in recent years Schmitt  _et al._
(2022); Dziarmaga and Mazur (2023). Fortunately, scaling analysis demonstrates
that both real- and imaginary-time driven dynamics share the same scaling form
De Grandi  _et al._ (2011); Grandi  _et al._ (2013); Liu  _et al._ (2013).
This inference has been verified in various systems Liu  _et al._ (2013); Shu
_et al._ (2023); Schmitt  _et al._ (2022), and particularly, substantiated by
the remarkable consistency between the imaginary-time scaling and the
annealing experiments in quantum devices Liu  _et al._ (2015); King  _et al._
(2023), bridging the gap between the QMC imaginary-time simulations without
sign problem and the real-time dynamics.

In this work, for the first time we investigate the driven critical dynamics
of two representative Dirac quantum critical points, belonging to chiral
Heisenberg and chiral Ising universality classes respectively, via the
determinant QMC method Assaad and Evertz (2008). By linearly varying the
interaction strength along the imaginary-time direction to cross the QCP from
both the DSM and Mott insulator phases, we uncover that for large driving rate
the quantum critical dynamics is governed by the driving rate. The numerical
results confirm that the whole driven process satisfies the scaling form of
FTS. Hence, our study significantly generalizes the conventional KZM theory.
Since DSM phase harbors gapless excitations, our results unambiguously show
that the prerequisite of KZM theory that the initial state is fully gapped can
be relaxed to the systems accommodating gapless Dirac fermions. In addition,
our numerical simulation achieves the critical exponents of phase transition,
whose values obtained in previous studies on equilibrium properties are still
under debate. Our approach of QMC simulation on driven critical dynamics
offers a novel theoretical tool to unravel properties of Dirac QCP. Through
the innovative generalization of quantum driven critical dynamics to Dirac-
fermion systems, our study not only leads to a great leap to the fundamental
theory of KZM and FTS, but also contributes to a feasible approach to
investigate the fermionic quantum critical phenomena in realistic platform
such as quantum materials and devices.

Dynamics in chiral Heisenberg criticality— A typical model hosting QCP
belonging to chiral Heisenberg universality class is the Hubbard model on the
half-filled honeycomb lattice with the Hamiltonian Sorella and Tosatti (1992);
Herbut (2006); Herbut  _et al._ (2009); Assaad and Herbut (2013); Otsuka  _et
al._ (2016)

$H=-t\sum_{\langle
ij\rangle,\sigma}c_{i\sigma}^{\dagger}c_{j\sigma}+U\sum_{i}\left({n_{i\uparrow}-\frac{1}{2}}\right)\left({n_{i\downarrow}-\frac{1}{2}}\right),$
(1)

in which $c_{i\sigma}^{\dagger}$ ($c_{j\sigma}$) represents the creation
(annihilation) operator of electrons with spin $\sigma$, $n_{i\sigma}\equiv
c_{i\sigma}^{\dagger}c_{i\sigma}$ is the electron number operator, $t$ is the
hopping amplitude between nearest neighbor sites and set as the energy unit in
the following. $U$ is the strength of the on-site repulsive interaction. The
model is absent from sign problem in QMC simulationWu and Zhang (2005). As
shown in Fig. 1, a critical point $U_{c}$ separates two phases. When
$U>U_{c}$, the system is in the antiferromagnetic (AFM) Mott insulator phase
in which fermions acquire a mass originating from spontaneously symmetry
breaking characterized by the finite AFM order parameter
$m^{2}=\sum_{i,j}\eta_{i}\eta_{j}\langle{S_{i}^{z}S_{j}^{z}}\rangle/L^{2d}$
with $S_{i}^{z}\equiv\frac{1}{2}\bm{c}^{\dagger}_{i}\sigma^{z}\bm{c}_{i}$,
${\bm{c}\equiv({c_{\uparrow},c_{\downarrow}}})$, and $\eta_{i}=\pm 1$ for
$i\in A(B)$ sublattice Assaad and Herbut (2013); Otsuka  _et al._ (2016).
Here, $L$ is the linear size of the system and $d$ is the spatial dimension.
In this phase, the transverse spin excitation is massless due to the presence
of the Goldstone modes. In contrast, when $U<U_{c}$, the system is in the DSM
phase with four-component massless Dirac fermion ($N_{f}=2$). At $U_{c}$, both
Dirac fermions and AFM order parameter bonsons are gapless, giving rise to
Gross-Nevue QCP belonging to chiral Heisenberg universality class Sorella and
Tosatti (1992); Herbut (2006); Herbut  _et al._ (2009); Assaad and Herbut
(2013); Otsuka  _et al._ (2016).

Figure 1: Sketch of the phase diagram in Dirac systems and the protocol for
driven dynamics with different initial states. The correlation time scales for
both boson (yellow curve) and fermion (violet curve) are finite in one phase
(solid) but divergent (dotted) in the other phase. Accordingly, the
prerequisite of the original KZM that a gapped initial state should exist to
protect an initial adiabatic stage, in which the transition time is larger
than (dashed line) than the correlation time, breaks down.

For the driven dynamics across the critical point, based on an adiabatic-
impulse-adiabatic scenario, the original KZM often requires a gapped initial
state to maintain an adiabatic initial evolution Dziarmaga (2010); Polkovnikov
_et al._ (2011); Polkovnikov and Gritsev (2008), since the frozen time, which
is a central notion in the KZM, lives at the edge of the adiabatic stage.
After that, the KZM predicts an impulse regime wherein the system ceases
evolving due to the critical slowing down Zurek (1985); Dziarmaga (2010);
Polkovnikov  _et al._ (2011). This oversimplified assumption was further
improved by the FTS theory, which demonstrated that a driven-induced time
scale controls the dynamics in the impulse region and the whole evolution
process can be described by the FTS forms Gong  _et al._ (2010); Feng  _et
al._ (2016); Huang  _et al._ (2014). Here we begin to explore whether the FTS
forms are still applicable in Dirac systems with gapless initial states.

Figure 2: Driven dynamics from the DSM phase of model (1). (a) Log-log plots
of $m^{2}$ versus $R$ for different $L$ driven to $U_{c}=3.85$ before (a1) and
after (a2) rescaling. Inset in (a1) shows $m^{2}\propto L^{-2}$ for $R=0.15$
(dash-dotted line). For large $R$, power fitting for $L=24$ (black solid line)
shows $m^{2}\propto R^{-0.26}$ with the exponent close to $(2\beta-d\nu)/\nu
r=-0.257$ (dash line) from Ref. Otsuka  _et al._ (2016). (b) Curves of $m^{2}$
versus $g$ for fixed $RL^{r}=5.41$ and different $L$ before (b1) and after
(b2) rescaling.

First we study the driven dynamics by varying $U$ with imaginary time $\tau$
as $U=U_{0}+R\tau$ starting from the DSM initial state with interaction
strength $U_{0}$, as illustrated in Fig. 1. We denote the distance to the
critical point as $g$ (Here $g=U-U_{c}$). When $g=0$, from Fig. 2 (a1), we
find that for large $R$, $m^{2}\propto R^{(2\beta-d\nu)/\nu r}$ in which
$\beta$ and $\nu$ are the exponents for order parameter and correlation length
respectively Otsuka  _et al._ (2016), and $r=z+1/\nu$ with $z$ the dynamic
exponent. And the exponent on $R$ is almost independent of $L$. In addition,
for fixed $R$, we find that $m^{2}\propto L^{-2}$. Combination of these
results yields the scaling relation $m^{2}\propto L^{-d}R^{(2\beta-d\nu)/\nu
r}$. In contrast, when $R$ is small, Fig. 2 (a1) shows that $m^{2}$ tends to
saturate and the usual finite-size scaling $m^{2}\propto L^{-2\beta/\nu}$ is
restored. To reconcile these rescaling relations, the scaling form must
satisfy

$m^{2}(R,L,g)=L^{-d}R^{(2\beta-d\nu)/\nu r}\mathcal{F}(RL^{r},gL^{1/\nu}),$
(2)

in which $\mathcal{F}$ is a non-singular scaling function and therein
dimensionless quantity $gL^{1/\nu}$ is also included to take account of the
off-critical-point effects. Eq. (2) is consistent with the FTS in conventional
bosonic QCP Huang  _et al._ (2014); Liu  _et al._ (2014).

To confirm Eq. (2), we find that the Eq. (2) yields the scaling form
$m^{2}=L^{-2\beta/\nu}\mathcal{F}(RL^{r},0)$ with fixing $g=0$. Here, we
rescale $m^{2}$ and $R$ as $m^{2}L^{2\beta/\nu}$ and $RL^{r}$ respectively,
and reveal that the rescaled curves collapse well into a single curve, as
shown in Fig. 2 (a2). The results of data collapse not only confirm Eq. (2) at
$g=0$, but also verify the values of critical point and critical exponents.

Figure 3: Driven dynamics from the AFM phase of model (1). (a) Log-log plots
of $m^{2}$ versus $R$ for different $L$ driven to $U_{c}=3.85$ before (a1) and
after (a2) rescaling. For large $R$, power fitting (black solid line) shows
$m^{2}\propto R^{0.71}$ with the exponent close to $2\beta/\nu r=0.752$ (dash
line) from Ref. Otsuka  _et al._ (2016). (b) Curves of $m^{2}$ versus $g$ for
fixed $RL^{r}=41.5$ and different $L$ before (b1) and after (b2) rescaling.

In addition, to unravel the scaling properties in the whole driven process, we
calculate the dependence of $m^{2}$ on $g$ for an arbitrary fixed $RL^{r}$ and
present the results in Fig. 2 (b1). After rescaling $m^{2}$ and $g$ by
$L^{2\beta/\nu}$ and $L^{1/\nu}$ respectively, the curves with various $L$
collapse into each other, as displayed in Fig. 2 (b2), confirming that the
universal scaling behavior of physical observable in the whole driven process
is described by Eq. (2).

The reason for the appearance of the scaling relation $m^{2}\propto
L^{-d}R^{(2\beta-d\nu)/\nu r}$ is that for large $R$, driven induced length
scale $\xi_{R}\sim R^{-1/r}$ is smaller than $L$. Thus, the definition of
$m^{2}$ indicates that $m^{2}\propto L^{-d}$ owing to the central limit
theorem. Meanwhile, the rest part of the dimension of $m^{2}$ should be borne
by $R$, giving rise to the leading term of Eq. (2). In this case,
$\mathcal{F}(x,0)$ tends to a constant. In contrast, for small $R$,
$\xi_{R}>L$ and the conventional finite-size scaling at equilibrium
$m^{2}\propto L^{-2\beta/\nu}$ is recovered, as confirmed in Fig. 2 (a2), and
$\mathcal{F}(x,0)$ obeys the scaling form $\mathcal{F}(x,0)\sim
x^{d/r-2\beta/\nu r}$.

Figure 4: Driven dynamics from the DSM phase of model (4). (a) Log-log plots
of $m^{2}$ versus $R$ for different $L$ driven to $V_{c}=1.355$ before (a1)
and after (a2) rescaling. Inset in (a1) shows $m^{2}\propto L^{-2}$ for $R=1$
(dash-dotted line). For large $R$, power fitting for $L=24$ (black solid line)
shows $m^{2}\propto R^{-0.32}$ with the exponent close to $(2\beta-d\nu)/\nu
r=-0.31$ (dash line) from Ref. Hesselmann and Wessel (2016). (b) Curves of
$m^{2}$ versus $g$ for fixed $RL^{r}=63.34$ and different $L$ before (b1) and
after (b2) rescaling.

Next, we turn to explore the driven dynamics starting from the Mott insulator
initial state. This state has the AFM order with transverse gapless modes. For
large $R$, Fig. 3 (a1) shows that $m^{2}$ obeys $m^{2}\propto R^{2\beta/\nu
r}$ and is nearly independent of $L$. Combining this scaling relation with the
usual finite-size scaling $m^{2}\propto L^{-2\beta/\nu}$ which is restored for
small $R$, the scaling form should obey

$m^{2}(R,L,g)=R^{2\beta/\nu r}\mathcal{G}(RL^{r},gL^{1/\nu}),$ (3)

where $\mathcal{G}$ is the non-singular scaling function and $g$ is also
included therein. Eq. (3) is also accordant with the conventional FTS with
ordered initial state Gong  _et al._ (2010); Feng  _et al._ (2016); Huang  _et
al._ (2014)

We rescale curves of $m^{2}$ versus $R$ for various $L$ at $g=0$ according to
the scaling function $m^{2}(R,L,g)=L^{-2\beta/\nu}\mathcal{G}(RL^{r},0)$ and
find that the rescaled curves collapse well, which confirms Eq. (3) at $g=0$.
Furthermore, Fig. 3 (b1) shows the curves of $m^{2}$ versus $g$ for an
arbitrary fixed $RL^{r}$. The rescaled results
$(gL^{1/\nu},m^{2}L^{2\beta/\nu})$ for various $L$ collapse into a single
smooth curve, as displayed in Fig. 3 (b2), confirming that the driven process
from AFM initial state is described by Eq. (3).

The appearance of $m^{2}\propto R^{2\beta/\nu r}$ reflects the fact that when
$\xi_{R}<L$ the initial ordered magnetization domain is maintained. In this
case, $\mathcal{G}(x,0)$ tends to a constant. In contrast, for small $R$ with
$\xi_{R}>L$, the usual finite size scaling is recovered, indicating
$\mathcal{G}(x,0)\sim x^{-2\beta/\nu r}$ as $x\rightarrow 0$.

Figure 5: Driven dynamics from the CDW phase of model (4). (a) Log-log plots
of $m^{2}$ versus $R$ for different $L$ driven to $V_{c}=1.355$ before (a1)
and after (a2) rescaling. For large $R$, power fitting (black solid line)
shows $m^{2}\propto R^{0.50}$ with the exponent close to $2\beta/\nu r=0.54$
from Ref. Hesselmann and Wessel (2016) (dash line). (b) Curves of $m^{2}$
versus $g$ for fixed $RL^{r}=87.97$ and different $L$ before (b1) and after
(b2) rescaling.

Dynamics in chiral Ising criticality— To further verify the FTS in Dirac
systems, we also explore the driven dynamics of Dirac QCP belonging to chiral
Ising universality class, which is realized in the interacting spinless
fermion model on the half-filled honeycomb lattice with the Hamiltonian Wang
_et al._ (2014); Li  _et al._ (2015a); Wang  _et al._ (2016); Hesselmann and
Wessel (2016); Hesselmann  _et al._ (2018):

$H=-t\sum_{\langle ij\rangle}c_{i}^{\dagger}c_{j}+V\sum_{\langle
ij\rangle}\left({n_{i}-\frac{1}{2}}\right)\left({n_{j}-\frac{1}{2}}\right),$
(4)

where $V$ measures the nearest-neighbor interaction. The model is amendable to
sign-problem-free QMC simulationLi  _et al._ (2015b, 2016b); Wei  _et al._
(2016); Wang  _et al._ (2015). It is shown that the ground state undergoes a
continuous quantum phase transition at $V_{c}\approx 1.355t$ from the DSM
phase to the charge-density-wave (CDW) insulating phase, characterized by the
order parameter
$m^{2}\equiv\sum_{i,j}\eta_{i}\eta_{j}\langle{(n_{i}-1/2)(n_{j}-1/2)}\rangle/L^{2d}$
with $\eta_{i}=\pm 1$ for $i$ $\in$ $A(B)$ sublattice Wang  _et al._ (2014);
Li  _et al._ (2015a); Hesselmann and Wessel (2016). At $V=V_{c}$, both fermion
and boson degrees of freedom are gapless, similar to model (1). However, a
difference is that in the CDW phase, the bosonic fluctuation is fully gapped
owing to the discrete symmetry breaking.

From the DSM initial state, Fig. 4 (a1) shows that fixed at $V=V_{c}$, namely
$g=0$, $m^{2}\propto L^{-d}R^{(2\beta-d\nu)/\nu r}$ for large $R$, whereas
$m^{2}\propto L^{-2\beta/\nu}$ for small $R$, similar to the results in model
(1). In addition, we find the rescaled curves of $m^{2}$ versus $R$ and $g$
collapse well, as shown in Fig. 4 (a2) and (b2), confirming that Eq. (2) gives
a universal description on the driven critical dynamics from the DSM initial
state.

From the CDW initial state, Fig. 5 (a1) shows that at $g=0$, $m^{2}\propto
R^{2\beta/\nu r}$ for large $R$, whereas $m^{2}\propto L^{-2\beta/\nu}$ for
small $R$, similar to the case of chiral Heisenberg universality class.
Moreover, Eq. (3) is verified by the data collapse of the curves
$(RL^{r},m^{2}L^{-2\beta/\nu})$ at fixed $g=0$ and
$(gL^{1/\nu},m^{2}L^{-2\beta/\nu})$ for an arbitrary $RL^{r}$, as shown in
Fig. 5 (a2) and (b2). Consequently, Eq. (3) provides a universal description
on the driven dynamics from ordered initial states, regardless of whether or
not gapless bosonic modes exist.

Discussion— Some remarks are discussed as follows. (a) A crucial prerequisite
in the original KZM is the existence of initial adiabatic stage in which the
excitations possess finite gap Dziarmaga (2010); Polkovnikov  _et al._ (2011);
Polkovnikov and Gritsev (2008). In contrast, the FTS of driven critical
dynamics still holds in Dirac systems, indicating that this requirement is not
indispensable, since in both quantum phases separated by the QCP the gapless
excitations are present: the DSM phase features gapless fermions, whereas
bosonic gapless excitation exists in the AFM Mott insulator phase where
continuous symmetry is spontaneously broken.

To understand the underlying reasons of the novel numerical results, we notice
that the DSM phase has the vanishing density of states at charge neutral
point. Moreover, the Pauli principle of fermions further restricts the
excitations in the vicinity of Dirac points. Accordingly, the excitations
under slow external driving are strongly suppressed in DSM phase. Hence, it is
speculated that the theories of KZM and FTS in driven dynamics are generally
obeyed in Dirac QCP with the initial state of DSM. Besides, for driven
dynamics from the AFM initial state, since we focus on the dynamic scaling of
the order parameter magnitude, the influence of gapless angular excitation is
tiny since its characteristic time scale is much longer than that of
longitudinal mode Weinberg and Sandvik (2017), similar to the case in
deconfined quantum criticality Shu  _et al._ (2023).

(b) Furthermore, our work provides an effective approach to assess fermionic
critical properties from the perspective of nonequilibirum driven dynamics.
Large discrepancy on critical exponents of chiral Heisenberg universality
class still exists even under extensive studies Assaad and Herbut (2013);
Otsuka  _et al._ (2016). Here, by means of FTS analysis we reveal that the
critical exponents for large $R$ in Figs. 2 and 3 are consistent with the ones
from Ref. Otsuka  _et al._ (2016), in which results of larger system size and
careful finite-size corrections are included. For the finite-size scaling of
equilibrium QCP, the critical exponents are extracted by the scaling analysis
with varying interaction strength and system size. In the framework of driven
dynamics, the driven rate $R$ constitutes a new tuning parameter, enabling the
achievement of critical exponents with fixing $g=0$ or $L=L_{\rm max}$
according to Eqs. (2) and (3), which offers a possible way to mitigating sub-
leading corrections arising from finite $g$ or small system size.
Consequently, the reliable values of critical exponents are accessible with
relatively small system size.

Summary— In summary, we investigate the driven dynamics of QCP in two
representative interacting Dirac-fermion systems, belonging to chiral
Heisenberg and chiral Ising universality classes respectively, through sign-
problem-free QMC simulation. Driving the system from both DSM and Mott
insulator phase as the initial state, we discover varieties of interesting
nonequilibrium scaling behaviors. Furthermore, we confirm that these scaling
behaviors can be unified by the full scaling form of the FTS theory. Through
this work we not only successfully generalize the KZM and FTS to critical
systems with joint fluctuations of gapless fermions and bosons, but also
extend the application conditions of the celebrated theory of KZM.
Additionally, we demonstrate that the nonequilibrium scaling form is capable
of determining the critical exponents in Dirac QCP, providing an effective
method to deciphering quantum critical properties in terms of driven dynamics.
Our work possesses many intriguing generalizations, for example, it is quite
instructive to consider other kinds of driven protocols, such as driving from
the critical point and from the thermal equilibrium state Chandran  _et al._
(2012); Deng  _et al._ (2009); De Grandi  _et al._ (2010); Huang  _et al._
(2016); Yin  _et al._ (2016). From the experimental perspectives, the Moiré
systems of graphene and heterostructures offer tunable platforms to observe
the fermionic QCP and the associated driven critical dynamicsKennes  _et al._
(2021); Li  _et al._ (2022). Moreover, as aforementioned, the Kibble-Zurek
driven dynamics is experimentally observed in cold-atom systemsClark  _et al._
(2016); Keesling  _et al._ (2019). Owing to the recent developments of cold
atoms based quantum simulator of fermions Jördens  _et al._ (2008); Mazurenko
_et al._ (2017); Venu  _et al._ (2023), it is promising to detect driven
dynamics in Dirac QCP and verify the generalized KZM and FTS as discussed in
our study in these platforms.

Acknowledgments— We would like to thank A. W. Sandvik for helpful discussions.
Z. Zeng, Y. K. Yu, Z. X. Li and S. Yin are supported by the National Natural
Science Foundation of China (Grants No. 12075324 and No. 12222515). Z. X. Li
is supported by the NSFC under Grant No. 12347107. S. Yin is also supported by
the Science and Technology Projects in Guangdong Province (Grants No.
211193863020). The calculations reported were partly performed on resources
provided by the Guangdong Provincial Key Laboratory of Magnetoelectric Physics
and Devices, No. 2022B1212010008.

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Supplementary Materials for
Finite-time Scaling beyond the Kibble-Zurek Prerequisite: Driven Critical
Dynamics in Strongly Interacting Dirac Systems

##  I IMAGINARY-TIME DRIVEN DYNAMICS

In the main text, we explored the driven dynamics of the Hubbard model (1) and
the $t$-$V$ model (4). Here, we elucidate the theory of imaginary-time driven
dynamics.

For model (1) and (4), we linearly vary the interaction strength $U~{}(V)$
with imaginary time variable $\tau$ at the rate $R$ as

$U=U_{0}\pm R\tau,$ (S1) $V=V_{0}\pm R\tau,$ (S2)

where the $+(-)$ represents driving from disordered (ordered) initial state at
$U_{0}~{}(V_{0})$. In our numerical simulation, for disordered initial state,
we set $U_{0}=0,~{}V_{0}=0$, while for ordered initial state we set
$U_{0}=11.85,~{}V_{0}=2.5$ far from the critical point.

The wave function $|\psi(\tau)\rangle$ obeys the imaginary-time Schrödinger
equation De Grandi  _et al._ (2011); Grandi  _et al._ (2013); Liu  _et al._
(2013)

$-\frac{\partial}{\partial\tau}|\psi(\tau)\rangle=H(\tau)|\psi(\tau)\rangle.$
(S3)

The formal solution of Eq. (S3) is given by
$|\psi(\tau)\rangle=U(\tau,0)|\psi(0)\rangle$ in which time evolution operator

$U(\tau,0)=\mathrm{T}~{}{\rm
exp}\left[-\int_{0}^{\tau}d\tau^{\prime}H(\tau^{\prime})\right],$ (S4)

with $\mathrm{T}$ being the time-ordering operator in imaginary-time
direction. For the left vector
$\bra{\psi(\tau)}=\bra{\psi(\tau)}U^{\dagger}(\tau,0)$ with
$U^{\dagger}(\tau,0)=\overline{\mathrm{T}}~{}{\rm
exp}\left[-\int_{0}^{\tau}d\tau^{\prime}H(\tau^{\prime})\right],$ the Hermite
conjugate simply changes the time-ordering operator $\mathrm{T}$ to an anti-
time-ordering operator $\overline{\mathrm{T}}$. Since the model (1) and (4) is
sign-problem-free and imaginary-time evolution does not induce additional sign
problem, the imaginary-time dynamics of the model can be simulated by the
determinant Quantum Monte-Carlo (DQMC) method.

##  II Determinant quantum Monte Carlo

We implement DQMC simulation Assaad and Evertz (2008) to explore the
imaginary-time driven dynamics of our models. Starting with a trial wave
function $\ket{\psi_{T}}$, we project the system onto the desired initial
state $\ket{\psi(0)}$ with a sufficiently long projection time $\tau_{0}$.
That is $|\psi(0)\rangle=\mathrm{e}^{-\tau_{0}H_{0}}|\psi_{T}\rangle$ with
$H_{0}$ the Hamiltonian of $|\psi(0)\rangle$ corresponding to $V_{0}$. Then we
follow Eq. (S2) to vary $V$ and observe the scaling behavior of observables at
final state, at which the expectation value of observables is given by

$\braket{O(\tau)}=\frac{\bra{\psi(\tau)}O\ket{\psi(\tau)}}{\left\langle\psi(\tau)\middle|\psi(\tau)\right\rangle\
}.$ (S5)

We write the Hamiltonian of model (1) and model (4) as

$H=H_{t}+H_{V},$ (S6)

where $H_{t}$ is the kinetic energy and $H_{V}$ is the interaction term. In
DQMC, one usually employs Trotter decomposition to discretize imaginary time
and discrete Hubbard-Stratonovich (HS) transformation to split the two-body
fermion-fermion coupling term into a one-body fermion-auxiliary field
coupling.

The Trotter decomposition in driven dynamics is expressed as

$\mathrm{T}~{}{\rm
exp}\left[-\int_{0}^{\tau}d\tau^{\prime}H(\tau^{\prime})\right]=\lim_{\Delta\tau\to
0}\prod^{N_{\tau}}_{n=1}\left[\mathrm{e}^{-\Delta\tau
H_{t}}\mathrm{e}^{-\Delta\tau H_{V}(\tau_{n})}\right],$ (S7)

where the imaginary-time slice $\Delta\tau=\tau/N_{\tau}$ and
$\tau_{n}=n\Delta\tau$.

We apply the following HS transformation to model (1):

$\mathrm{e}^{-\frac{\Delta\tau
U(\tau_{n})}{2}\left(n_{i\uparrow}+n_{i\downarrow}-1\right)^{2}}=\sum_{l_{i,\tau_{n}}=\pm
1,\pm 2}\gamma(l_{i,\tau_{n}})\mathrm{e}^{\mathrm{i}\sqrt{\frac{\Delta\tau
U(\tau_{n})}{2}}\eta(l_{i,\tau_{n}})\left(n_{i\uparrow}+n_{i\downarrow}-1\right)},$
(S8)

and for model (4):

$\mathrm{e}^{-\frac{\Delta\tau
V(\tau_{n})}{2}\left(n_{i}+n_{j}-1\right)^{2}}=\sum_{l_{i,\tau_{n}}=\pm 1,\pm
2}\gamma(l_{i,\tau_{n}})\mathrm{e}^{\mathrm{i}\sqrt{\frac{\Delta\tau
V(\tau_{n})}{2}}\eta(l_{i,\tau_{n}})\left(n_{i}+n_{j}-1\right)},$ (S9)

where the four-component space-time auxiliary fields $\gamma$ and $\eta$ take
the following values:

$\gamma(\pm 1)=1+\sqrt{6}/3,\ \gamma(\pm 2)=1-\sqrt{6}/3,$ (S10) $\eta(\pm
1)=\pm\sqrt{2\left(3-\sqrt{6}\right)},\ \eta(\pm
2)=\pm\sqrt{2\left(3+\sqrt{6}\right)}.$ (S11)

After the aforementioned operations, the imaginary-time evolution operator can
be expressed in quadratic form of fermion operators:

$U(\tau,0)=\sum_{l}A_{l,\tau_{n}}\prod^{N_{\tau}}_{n=1}~{}\mathrm{e}^{\vec{c}^{\dagger}K\vec{c}}~{}\mathrm{e}^{\vec{c}^{\dagger}W_{l,\tau_{n}}\vec{c}},$
(S12)

where $K$ and $W_{l,\tau_{n}}$ are the quadratic coefficient matrices of
hopping term and interaction term respectively, and $A_{l,\tau_{n}}$ is the
coefficient. Then the fermionic degrees of freedom can be integrated out and
we obtain

$\bra{\psi(0)}U^{\dagger}(\tau,0)U(\tau,0)\ket{\psi(0)}=\sum_{l}A_{l,\tau_{n}}~{}\mathrm{det}\left[P^{\dagger}B_{l}(2\tau,0)P\right],$
(S13)

where $P$ represents the initial state with fermions operators intergrated out
and

$B_{l}(2\tau,0)\equiv\prod^{2N_{\tau}}_{n=1}~{}\mathrm{e}^{K}~{}\mathrm{e}^{W_{l,\tau_{n}}}.$
(S14)

Now we can perform matrix operations directly on a computer and sample over
the space-time configurations to evaluate the expectation value of physical
observables. The subsequent DQMC process is the same as that for equilibrium
state approach Assaad and Evertz (2008).

##  III RESULTS OF DIMENSIONLESS QUANTITY

In the main text, we only present dynamic results of the square of the order
parameter $m^{2}$. Here, we supplement the results of a dimensionless quantity
correlation ratio defined as Kaul (2015):

$R_{S}\equiv{1-S(\Delta\bm{q})}/{S(\bm{0})},$ (S15)

where $S({\bm{q}})$ is the structure factor and $\Delta\bm{q}$ is the minimal
lattice momentum.

Figure S1: Log-log plots of $R_{S}$ versus $R$ for different $L$ driven to
$U_{c}=3.85$, for model (1). (a) displays the results driven from the DSM
phase before (a1) and after (a2) rescaling. (b) displays the results driven
from the AFM phase before (b1) and after (b2) rescaling. Figure S2: Log-log
plots of $R_{S}$ versus $R$ for different $L$ driven to $V_{c}=1.355$, for
model (4). (a) displays the results driven from the DSM phase before (a1) and
after (a2) rescaling. (b) displays the results driven from the CDW phase
before (b1) and after (b2) rescaling.

For model (1), $S({\bm{q}})$ is the antiferromagnetic structure factor:

$S({\bm{q}})_{\rm
AFM}=\frac{1}{L^{2d}}\sum_{i,j}\mathrm{e}^{\mathrm{i}\bm{q}\cdot({\bm{r}_{i}-\bm{r}_{j}})}\langle{S_{i}^{z}S_{j}^{z}}\rangle,$
(S16)

where the staggered magnetization operator
$S_{i}^{z}\equiv\vec{c}^{\dagger}_{i,A}\sigma^{z}\vec{c}_{i,A}-\vec{c}^{\dagger}_{i,B}\sigma^{z}\vec{c}_{i,B}$
with $i$ characterizes cells and
${\vec{c}^{\dagger}\equiv({c_{\uparrow}^{\dagger},c_{\downarrow}^{\dagger}}})$.

For model (4), $S({\bm{q}})$ is the charge-density-wave structure factor:

$S({\bm{q}})_{\rm
CDW}=\frac{1}{L^{2d}}\sum_{i,j}\mathrm{e}^{\mathrm{i}\bm{q}\cdot({\bm{r}_{i}-\bm{r}_{j}})}\langle{n_{i}n_{j}}\rangle,$
(S17)

where $n_{i}\equiv n_{i,A}-n_{i,B}$ with $n_{i,A}$ being the electron number
operator defined as $n_{i,A}\equiv c^{\dagger}_{i,A}c_{i,A}$.

According to the scaling analysis in the main text, as a dimensionless
variable, $R_{S}$ in the driven process obeys the following dynamic scaling
form:

$R_{S}({g,R,L})=f(RL^{r},gL^{1/\nu}),$ (S18)

which indicates that (S18) at $g=0$ can be converted to $R_{S}=f(RL^{r})$.

As shown in Fig. S1(a2)(b2) and Fig. S2(a2)(b2), we find that the rescaled
curves all collapse well. These results not only confirm (S18) at $g=0$, but
also support the conclusions we draw in the main text.