# Finite-time scaling of dynamic quantum criticality

Shuai Yin<EMAIL_ADDRESS>Xizhou Qin  Chaohong Lee
<EMAIL_ADDRESS>Fan Zhong<EMAIL_ADDRESS>State Key
Laboratory of Optoelectronic Materials and Technologies, School of Physics and
Engineering, Sun Yat-sen University, Guangzhou 510275, People’s Republic of
China

###### Abstract

We develop a theory of finite-time scaling for dynamic quantum criticality by
considering the competition among an external time scale, an intrinsic
reaction time scale and an imaginary time scale arising respectively from an
external driving field, the fluctuations of the competing orders and thermal
fluctuations. Through a successful application in determining the critical
properties at zero temperature and the solution of real-time Lindblad master
equation near a quantum critical point at nonzero temperatures, we show that
finite-time scaling offers not only an amenable and systematic approach to
detect the dynamic critical properties, but also a unified framework to
understand and explore nonequilibrium dynamics of quantum criticality, which
shows specificities for open systems.

###### pacs:

64.70.Tg, 64.60.Ht, 75.10.Pq

Detecting quantum phase transitions (QPTs) and understanding their real-time
dynamics are of great importance sachdev ; Coleman ; sachdevpt ; Dziarmaga ;
polrmp . Recent experimental breakthrough in ultracold atoms Greiner  promises
new tools to study the quantum critical dynamics Zhang . In the nonequilibrium
critical dynamics of QPTs at zero temperature in which a controlling parameter
is changed with time through a critical point Dziarmaga ; polrmp , the Kibble-
Zurek mechanism (KZM), which was first introduced in cosmology by Kibble
kibble1  and then in condensed matter physics by Zurek zurek1 , has been found
to describe the dynamics of QPTs well Dziarmaga ; polrmp ; Zurek . In this
adiabatic–impulse–adiabatic approximation of the KZM, the system considered is
assumed to cease evolving in the impulse regime within which adiabaticity
breaks down due to critical slowing down binder . Yet, dynamical scaling has
been reported just within this regime deng  and confirmed in both integrable
and nonintegrable systems grandi ; kolodrubetz . In classical critical
dynamics, an explanation based on coarsening has been developed biroli ,
however in quantum phase transitions, a systematic understanding of the full
scaling behavior is still lacking.

On the other hand, natural systems and their measurements exist inevitably in
nonzero temperatures, though probably only initial thermal states need
considering in ultracold atoms Polkovnikov . Thermal effects on a quantum
critical state can give rise to a variety of exotic behavior in the famous
quantum critical regime (QCR) Chakravarty  as exhibits in a wide range of
strongly correlated systems sachdev ; Coleman ; sachdevpt ; Broun . Yet, as
both phases exhibit complex long-range quantum entanglement near the quantum
critical point and are violently excited thermally, it is a great challenge to
describe quantum critical dynamics at finite temperatures, let alone
nonequilibrium real-time effects patane ; chandran . Indeed, none of the
analytic, semiclassical, or numerical methods of condensed-matter physics
yields accurate results for dynamics in the QCR except for some special
systems in 1D sachdevpt . Even in isolated situations it is difficult to study
the time evolution of nonequilibrium systems with many degrees of freedom
Kinoshita ; Hofferberth ; rigol ; Dziarmaga ; polrmp . Therefore, systematic
approaches have to be invoked.

Figure 1: (color online) Schematic phase diagram under a sweep of $g$ near its
critical value 0. Two equilibrium phases (light grey/green) dominated by the
reaction time $\tau_{s}$ are separated by two crossover domains fanning out
from the quantum critical point $O$. One domain is the QCR (dark/blue)
controlled by the imaginary time scale $\tau_{T}$. The other is the new FTS
(grey/red) regime governed by the driving time scale $\tau_{d}$.

Time plays a fundamental role in quantum criticality owing to the interplay of
static and dynamic behaviors. Specifically, by varying the distance to the
critical point $g$ at a time rate $R$, a continuous QPT at a finite
temperature $T$ is characterized mainly by three time scales. The first one is
a reaction time $\tau_{s}$ that arises from the fluctuations of the competing
orders and blows up as $\tau_{s}\sim|g|^{-\nu z}$ with the standard critical
exponents $\nu$ and $z$ as $g$ vanishes sachdev . The second one is an
“imaginary” time scale $\tau_{T}=1/T$ (the Plank and the Boltzmann constants
have been set to 1) due to the finite $T$, since the real time is its
analytical continuation to imaginary numbers through a Feynman path integral
representation sachdev . The third one is an externally imposed driving time
scale $\tau_{d}$ that results from the driving and grows as $\tau_{d}\sim
R^{-z/r}$ with a rate exponent $r$ that is related to $z$ and other static
critical exponents Gong . It is the competition among $\tau_{s}$, $\tau_{T}$,
and $\tau_{d}$ that lead to a diversity of equilibrium and dynamic universal
phenomena near a quantum critical point.

Here we study systematically the competition among the three characteristic
time scales according to the theory of finite-time scaling (FTS) Gong . As
seen in Fig. 1, besides the usual equilibrium regimes and the QCR which are
respectively dominated by $\tau_{s}$ and $\tau_{T}$, our most important result
is that a new nonequilibrium FTS domain is created. In this domain, $\tau_{d}$
is the shortest time among the trio and thus dominates, just as the well-known
regime of finite-size scaling in which the characteristic size $L$ of the
system is shorter than its correlation length. At $T=0$, this indicates the
FTS domain overlaps just the impulse regime of the KZM for sweeping $g$. As a
consequence, although the system falls out off equilibrium, the state does not
cease evolving; rather, it evolves according to the imposed time scale
$\tau_{d}$ instead of $\tau_{s}$ with nonadiabatic excitations obeying FTS.
Therefore, FTS improves the understanding of KZM on its dark impulse regime
and produces naturally scaling forms suggested in deng ; grandi ; kolodrubetz
. In addition, FTS enables us to the study within the same framework other
driving dynamics than the KZ protocols chandran , which focus on changing non-
symmetry breaking terms like $g$. We shall show that these provides a
convenient method to determine the critical points and exponents, which were
invoked as input for scaling collapses reported in deng ; grandi ; kolodrubetz
. Similarly, at $T\neq 0$, in the FTS regime, there are now nonadiabatic
thermal excitations controlled again by $\tau_{d}$ and thus obeying again FTS.
Further, from Fig. 1, the FTS regime pushes the QCR to higher temperatures
since only then $\tau_{T}$ dominates. Consequently, FTS enables one to probe
directly the quantum critical point and its scaling behavior at nonzero
temperatures as $T$ becomes subordinate and just a perturbation. Thus, it
shows a ‘dynamic cooling’ effect that enables one to probe the zero-
temperature scaling at nonzero-temperatures while keeping $T$ subsidiary. This
offers us an extra approach to detect and study quantum criticality at finite
temperatures. Owing to its conceptual simplicity and accessability, FTS
therefore provides a unified framework not only to detect the dynamic critical
properties, but also to understand and explore the nonequilibrium dynamics of
quantum criticality both at $T=0$ and $T\neq 0$.

As another important result, we shall show that in nonequilibrium quantum
critical dynamics of open systems one must include an additional variable such
as the coupling to a heat bath to the intrinsic quantum dynamics note . This
is an important difference from the classical case and must be considered when
extending nonequilibrium quantum critical dynamics to finite temperatures
patane . We shall see that the master equation in the Lindblad form just
offers such a variable and is thus an appropriate platform to study real-time
nonequilibrium quantum criticality.

We start with an open many-body quantum system interacting with a heat bath
patane  to study the interplay of quantum and thermal fluctuations. The state
of such a system can be described by a density matrix operator $\rho$
according to quantum statistical physics. For weak system-environment
couplings, after assuming Markovian and tracing over the bath variables, one
obtains the master equation for $\rho$ in the Lindblad form Lindblad ; attal ;
Mai ,

$\partial\rho/\partial t=-i[\mathcal{H},\rho]+c\mathcal{L}\rho,$ (1)

where $\mathcal{L}\rho=-\sum_{i=1,j\neq i}^{N_{E}}\beta_{j}(V_{i\rightarrow
j}^{{\dagger}}V_{i\rightarrow j}\rho+\rho V_{i\rightarrow
j}^{{\dagger}}V_{i\rightarrow j}-2V_{i\rightarrow j}\rho V_{i\rightarrow
j}^{{\dagger}})/2$, $c$ is the dissipation rate and measures the coupling
strength between the system and the bath, $N_{E}$ is the total number of
energy levels,
$\beta_{i}=\textrm{exp}(-E_{i}/T)/\textrm{Tr}\textrm{exp}(-\mathcal{H}/T)$
with $E_{i}$ being the $i$th eigenvalue of $\mathcal{H}$, and $V_{i\rightarrow
j}$ is the thermal jump matrix whose element at the $j$th row and $i$th column
is one or zero in the energy representation. $V_{i\rightarrow j}$ fulfills
$\beta_{i}\rho_{E}V_{i\rightarrow j}=\beta_{j}V_{i\rightarrow j}\rho_{E}$ with
the equilibrium density matrix operator
$\rho_{E}\equiv\textrm{exp}(-\mathcal{H}/T)/\textrm{Tr}\textrm{exp}(-\mathcal{H}/T)$
whose eigenvalues are $\beta_{i}$. This can be regarded as a detailed balance
condition in equilibrium. The Lindblad equation (1) is a real-time dynamical
equations which integrates both the quantum and the thermal contributions. It
has been widely used in quantum optics orszag  and relaxation processes in
open quantum systems attal ; znidaric . Although for large couplings, Eq.(1)
may be inapplicable wgwang , this equation gives a reasonable description in
the weak coupling limit, for instance, for the time-independent $\mathcal{H}$,
the steady solution of Eq. (1) is $\rho_{E}$ independent of $c$ Lindblad ;
attal ; Mai , this is consistent with the foundation of statistical mechanics
Schrodinger . In the following, we focus on the weak coupling limit and
consider the scaling properties of the Lindblad equation.

The theory of FTS Gong  takes explicitly into account the rate $R$, which
plays a role similar to $L^{-1}$ since it imposes on a system an additional
time scale that manipulates its evolution. In classical critical dynamics, the
nonequilibrium dynamic scaling can be generalized directly from the
equilibrium ones as confirmed by the renormalization-group theory Gong .
However, in the nonequilibrium quantum criticality, as pointed out, a coupling
strength must be considered as an independent scaling variable. In the weak
coupling limit, this strength can be reduced to the dissipation rate $c$.
Accordingly, for a length rescaling of factor $b$, an order parameter $M$
transforms as

$\displaystyle
M(t,g,h_{z},T,L,c,R)=b^{-\beta/\nu}M(tb^{-z},gb^{1/\nu},h_{z}b^{\beta\delta/\nu},\quad$
$\displaystyle Tb^{z},L^{-1}b,cb^{z},Rb^{r}),\quad$ (2)

where the two critical exponents $\beta$ and $\delta$ are defined as usual in
classical critical phenomena by $M\propto g^{\beta}$ in the absence of an
external probe field $h_{z}$ conjugate to $M$ and $M\propto h_{z}^{\delta}$ at
$g=0$, respectively. In the weak coupling limit, $c$ is small thus one can
expect its scaling behavior is controlled by the the fixed point corresponding
to the critical point at $c=0$, thus the dimension of $c$ is identical with
$t^{-1}$ as can be inspected from Eq. (1) Mai . This is checked latter by the
numerical solution of Eq. (1).

With Eq. (2), one can describe in a unified framework different kinds of
driven dynamics via changing $g$, $h_{z}$ or $T$ and readily define different
regimes and their crossovers. Taking $g=Rt$ for instance, neglecting $h_{z}$,
suppressing one independent variable, and choosing $b$ such that $Rb^{r}$
becomes a constant, one finds an FTS scaling form

$\displaystyle M(g,T,L,c,R)=R^{\beta/\nu r}f_{1}(gR^{-1/\nu
r},TR^{-z/r},\quad$ $\displaystyle L^{-1}R^{-1/r},cR^{-z/r}),\ $ (3)

where $r=z+1/\nu$ obtained from $g=Rt$ and its rescaling Gong  and the
function $f_{i}$ with an integer $i$ denotes a scaling function. FTS dominates
when $|g|R^{-1/\nu r}\ll 1$, $TR^{-z/r}\ll 1$, $L^{-1}R^{-1/r}\ll 1$, and
$cR^{-z/r}\ll 1$. The first gives $\tau_{d}\sim R^{-z/r}\ll|g|^{-\nu
z}\sim\tau_{s}$, the second $\tau_{d}\ll 1/T=\tau_{T}$ as they ought to be.
Crossovers to other regimes occur near $|\hat{g}|\sim R^{1/\nu r}$ and
$\hat{T}\sim R^{z/r}$ as depicted in Fig. 1 and similar ones for $L$ and $c$.
The first gives $|\hat{t}|\sim R^{-\nu z/(1+\nu z)}$ because
$\hat{g}=R\hat{t}$. This is just the scaling of the KZM upon identifying
$\hat{t}$ with the freeze-out time instant Dziarmaga ; polrmp ; Zurek  for a
closed system $c=0$ in the thermodynamic limit ($L\rightarrow\infty$) and at
$T=0$.

Several remarks are in order here. (a) Equation (3) is different from the
similar scaling form for finite temperatures in chandran  because $c$ must be
included to introduce the thermal fluctuation in the nonequilibrium situation.
(b) To return to the equilibrium scaling form at finite-temperatures sachdev ,
the scaling function $f_{i}$ must satisfy a constraint of $\partial
f_{i}/\partial c=0$ for $R=0$. (c) Beside recovering the full scaling forms of
finite-size for closed system in grandi ; kolodrubetz  by fixing $c=0$ and
$T=0$, the nonequilibrium dissipation scaling for spontaneous emissions in
zero-temperature open quantum systems can also be studied by fixing $T=0$ in
Eq. (3). (d) Note that $c$ should be small in the weak coupling limit and thus
the regime dominated by $c$ may be inaccessible.

Instead of sweeping $g$, when $h_{z}=R_{z}t$, one obtains similarly the order
parameter

$\displaystyle M_{h}=R_{z}^{\beta/\nu r_{z}}f_{2}(gR_{z}^{-1/\nu
r_{z}},h_{z}R_{z}^{-\beta\delta/\nu r_{z}},\qquad\quad$ $\displaystyle
TR_{z}^{-z/r_{z}},L^{-1}R_{z}^{-1/r_{z}},cR^{-z/r_{z}})\\!\\!\\!$ (4)

with $r_{z}=z+\beta\delta/\nu$. Different regimes and their crossovers can
also be readily defined. Different from sweeping $g$ through the critical
point as the ordinary KZM protocols deng ; grandi ; kolodrubetz ; chandran ,
here we fix $g$ and change the symmetry breaking field $h_{z}$. This provides
a method to determine the critical point from distinct critical behaviors for
$g=0$ and $g\neq 0$, a method which we shall utilize below and may also be
realizable experimentally. Note that in this protocol, the form of $\tau_{d}$
remains remarkably if $R$ and $r$ are replaced with their counterparts.
However, in addition to the fixed $\tau_{s}$ for the fixed $g$, there exists
another reaction time diverging with $|h_{z}|^{-\nu z/\beta\delta}$. These
result in new competitions but act only as corrections in the FTS regime,
showing an advantage of FTS.

Now we show that FTS can provide methods to detect quantum critical properties
such as the critical point and critical exponents. For simplicity, we consider
$T=0$ and $c=0$ in the thermodynamic limit $L\rightarrow\infty$. According to
Eq. (4), at $h_{z}=0$, $M_{h}$ reduces to

$M_{0}(g,R_{z})=R_{z}^{\beta/\nu r_{z}}f_{3}(gR_{z}^{-1/\nu r_{z}}),$ (5)

while the field at $M_{h}=0$, denoted by $h_{z0}$, scales as

$h_{z0}(g,R_{z})=R_{z}^{\beta\delta/\nu r_{z}}f_{4}(gR_{z}^{-1/\nu r_{z}}).$
(6)

Differentiating $M_{h}$ with respect to $h_{z}$ in Eq. (4), one obtains the
susceptibility at zero field,

$\chi(g,R_{z})=R_{z}^{\beta(1-\delta)/\nu r_{z}}f_{5}(gR_{z}^{-1/\nu r_{z}}).$
(7)

To fix the critical point, we can define a cumulant

$C(g,R_{z})\equiv M_{0}/(h_{z0}\chi)=f_{6}(gR_{z}^{-1/\nu r_{z}})$ (8)

similar to the Binder cumulant in finite-size scaling binder . As $C$ is a
function of only one independent variable, its curves for different $R_{z}$
intersect at the critical point $g=0$ at which $C$ becomes a constant
$f_{6}(0)$ independent on $R_{z}$. This gives the critical point with which
all the critical exponents can then be estimated. For example, $\beta/\nu
r_{z}$ and $\beta\delta/\nu r_{z}$ can be estimated respectively from Eqs. (5)
and (6) by fitting $M_{0}$ and $h_{z0}$ for a series of $R_{z}$ at $g=0$.
Similarly, from Eq. (3) at $c=0$, $T=0$, and $L\rightarrow\infty$, $\beta/\nu
r$ can be estimated by fitting $M$ for a series of $R$ at $g=0$. From these
three exponent ratios and the scaling law sachdev  $\beta(\delta+1)=(d+z)\nu$
with the space dimensionality $d$, one can determine all the critical
exponents.

As an example of the FTS method to determine critical properties, we consider
the one-dimensional (1D) transverse-field Ising model whose Hamiltonian is
sachdev

$\mathcal{H}=-h_{x}\sum\limits_{n=1}^{N}\sigma_{n}^{x}-\sum\limits_{n=1}^{N-1}\sigma_{n}^{z}\sigma_{n+1}^{z},$
(9)

and has been realized in CoNb2O6 experimentally Coldea , where
$\sigma_{n}^{x}$ and $\sigma_{n}^{z}$ are the Pauli matrices, $h_{x}$ is the
transverse field, and the Ising coupling has been set to unity as our energy
unit. The model exhibits a continuous QPT from a ferromagnetic phase to a
quantum paramagnetic phase at a critical point $h_{xc}$ (and so
$g=h_{x}-h_{xc}$) at $T=0$ sachdev . The order parameter is the magnetization
$M=\sum_{n=1}^{N}\langle\sigma_{n}^{z}\rangle/N$ for the $N$ spins with the
angle brackets denoting the quantum and/or thermal average. As a method to
probe the transition, we add to $\mathcal{H}$ a symmetry-breaking term
$-h_{z}\sum_{n=1}^{N}\sigma_{n}^{z}$.

We illustrate our approach at $T=0$ and $c=0$ at which Eq. (1) is same to
Schödinger’s equation and some exact results are available for comparison. We
solve the model using the time-evolving block-decimation algorithm Vidal ,
which is capable of treating large system sizes. We determine the critical
point in Fig. 2 and apply it purposely to determine the critical exponents in
Fig. 3. The good agreement of the results collected in Table 1 shows the power
of FTS.

Figure 2:  (color online) Estimation of quantum critical point. Curves of the
cumulant $C$ for different $R_{z}$ intersect at the critical point $h_{xc}$ or
$g=0$. Owing to possible errors from the truncation of the singular values in
the Schmidt decomposition Vidal , however, the intersections are slightly
scattered as shown in the inset. Nevertheless, the average of all the
intersections is $h_{xc}^{N}=0.999(2)$, a good estimate of the exact value
$h_{xc}=1$. We choose a lattice size of $L=2000$, which has been checked to
produce a negligible size effect.

Figure 3:  (color online) Estimation of critical exponents. Our solutions with $h_{xc}^{N}=0.999$, $T=0$, and $L=2000$ yield $\beta/\nu r_{z}=0.0436$, $\beta\delta/\nu r_{z}=0.651$, and $\beta/\nu r=0.0622$ from power-law fits according to the scaling forms (5), (6) and (3), respectively. We then obtain all the critical exponents listed in Table 1 with their exact results for comparison. As the statistical errors of the fits are tiny, we fit data at $h_{xc}=0.997$ and $1.001$ and the largest difference in each exponent is used as an estimate of the error given also in Table 1. Table 1: Critical point and exponents for the 1D transverse-field Ising model | $h_{xc}$ | $\beta$ | $\delta$ | $\nu$ | $z$
---|---|---|---|---|---
Numerical | 0.999(2) | 0.125(11) | 14.9(6) | 0.98(4) | 1.01(3)
Exact sachdev  | 1 | 0.125 | 15 | 1 | 1

Figure 4:  (color online) Nonequilibrium scaling at nonzero temperatures. (a)
Data of $M_{h}$ versus $T$ plotted in the inset for the three different sets
of $R_{z}$, $c$, and $L$ so choosing as to fix the value of
$L^{-1}R_{z}^{-1/r_{z}}$ and $cR_{z}^{-z/r_{z}}$ collapse as expected onto a
single curve for the fixed $LR_{z}^{1/r_{z}}=1.166$ and
$cR_{z}^{-z/r_{z}}=3.603$ according to the FTS (4) at $g=0$ ($h_{x}=0.999$),
$h_{z}=0$. (b) If, instead of $cR_{z}^{-z/r_{z}}$, we fix all $c=0.7$, the
value for $L=6$, and keep others, the rescaled curves then do _not_ collapse.

Having successfully demonstrated FTS at $T=0$, we now turn to $T\neq 0$ at
which most experiments operate. To examine the general nonequilibrium FTS (4)
for $T\neq 0$, we solve numerically Eq. (1) for the Hamiltonian (9) along with
the field $h_{z}$ by a finite difference method to second order with periodic
boundary conditions. We find that $M_{h}$ can now saturate correctly with the
thermal fluctuations. Moreover, Fig. 4 shows clearly the validity of the FTS
form (4). Further, upon comparing (a) with (b) in Fig. 4, it is obvious that
$c$ must enter into the scaling forms with a scaling dimension $z$. Note that
although here we only solve directly Eq. (1) for small lattices, the results
show that it is suitable for describing the nonequilibrium behavior at finite
temperatures near the quantum critical point. Moreover, the rapidly developing
numerical renormalization-group methods cirac , for example, seem quite
promising to solve the equation for larger lattice sizes cirac2 .

In conclusion, FTS not only provides a unified understanding of the driving
dynamics in general and lights up the dark impulse regime of KZM at zero
temperature in particular, but also sheds light on the QCR at nonzero
temperatures by establishing its own regime. It offers a powerful unified
approach amenable to both numerics and experiments to study equilibrium and
nonequilibrium dynamics of quantum criticality. We have shown that in the
latter in open systems one must include the dissipation rate as an independent
scaling variable and the Lindblad equation can be a valuable framework for
such studies. Although we have studied a simple model for illustration, our
approach should be applicable to more complex systems as well. In addition, as
our results indicate that the classical theory of FTS with proper
modifications can well describe quantum criticality, new physics may be in
action Coleman  if it is violated.

We thank Junhong An, Peter Drummond, and Xiwen Guan for their valuable
comments and discussions. Y.S. and F.Z. were supported by NNSFC (10625420) and
FRFCUC. C.L. was supported by NNSFC (11075223), NBRPC (2012CB821300
(2012CB821305)), NCETPC (NCET-10-0850). We acknowledge use of some source
codes for TEBD from http://physics.mines.edu/downloads/software/tebd/.

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