# Stationarity of quantum statistical ensembles at first-order phase
transition points

Yasushi Yoneta<EMAIL_ADDRESS>Department of Basic Science, The
University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan  RIKEN Center
for Quantum Computing, 2-1 Hirosawa, Wako City, Saitama 351-0198, Japan

###### Abstract

We study the dynamics of quantum statistical ensembles at first-order phase
transition points of finite macroscopic systems. First, we show that at the
first-order phase transition point of systems with an order parameter that
does not commute with the Hamiltonian, any quantum state with a non-zero value
of the order parameter always evolves towards a macroscopically distinct state
after a sufficiently long time. From this result, we argue that stationarity
required for statistical ensembles should be interpreted as stationarity on a
sufficiently long but finite time scale. Finally, we prove that the density
matrix of the squeezed ensemble, a class of generalized statistical ensembles
proposed as the only concrete method of constructing phase coexistence states
applicable to general quantum systems, is locally stationary on time scales
diverging in the thermodynamic limit. Our results support the validity of the
squeezed ensemble from a dynamical point of view and open the door to non-
equilibrium statistical physics at the first-order phase transition point.

##  I Introduction

Understanding the dynamics of quantum many-body systems is one of the most
intriguing research topics in condensed matter physics and quantum statistical
mechanics. There, statistical ensembles are used as the initial states of the
dynamics [1]. Therefore, the statistical ensemble must correctly provide not
only static properties but also dynamic properties.

Stationarity is one of the most fundamental dynamic properties of equilibrium
states [2]. Accordingly, the density matrix given by the statistical ensemble
should also be stationary. However, stationarity is not apparent when the
equilibrium state is specified by a noncommutative set of additive observables
since these observables are, in general, not conserved quantities.
Nevertheless, away from the first-order phase transition point, one can employ
the canonical ensemble, which is strictly invariant under the time evolution.

However, the canonical ensemble cannot generally be applied to the equilibrium
state at the first-order phase transition point [3, 4]. First-order phase
transitions are characterized by a discontinuous change in the equilibrium
state as a function of intensive parameters, such as the temperature and the
magnetic field. Thus, at the first-order phase transition point, there exist
several equilibrium states for a single value of a set of the intensive
parameters: a single-phase state, which is the equilibrium state immediately
before or after the phase transition point, and phase coexistence states, in
which several phases coexist spatially in various proportions. On the other
hand, the canonical ensemble is specified one-to-one by a set of the intensive
parameters. As a result, at the first-order phase transition point, the
canonical ensemble can give only a certain (single-phase states in many cases)
or a statistical mixture of the various equilibrium states with the same value
of intensive parameters. Consequently, it has long been pointed out that in
finite systems, the equilibrium state at the first-order phase transition
point can fluctuate macroscopically in time [5, 6].

Equilibrium states at the first-order phase transition point can be uniquely
specified by a proper additive quantity, called the ‘order parameter’. Hence,
at the first-order phase transition point, the statistical ensembles specified
by additive observables including the order parameter have been employed, such
as the microcanonical and restricted ensemble. However, such ensembles are
ill-defined or ill-behaved when some of the additive observables do not
commute with each other [7]. Thus, it is difficult to theoretically give the
general form of the quantum state corresponding to the equilibrium state at
the first-order phase transition point.

Recently, this fundamental problem has been solved by extending the
generalized ensemble, called the squeezed ensemble (SE), in such a way that it
is applicable to equilibrium states specified by noncommutative additive
observables [7]. It was proved that the SE correctly gives the density matrix
for any equilibrium state at the first-order phase transition point and the
thermodynamic functions. Therefore, using the SE, one can thoroughly analyze
microscopic structures and thermodynamic properties at first-order phase
transition points of general quantum systems. However, the dynamic properties
of the SE still need to be clarified.

In this paper, we study the dynamics of the SE at the first-order phase
transition point of finite macroscopic systems. We consider the exact unitary
dynamics rather than the dissipative dynamics [8, 9]. This paper is organized
as follows. In Section II, we describe the setup and notation. In Section III,
we prove that at the first-order phase transition point with an order
parameter that does not commute with the other additive observables specifying
the equilibrium state, any quantum state with a non-zero value of the order
parameter always changes macroscopically after a sufficiently long time. We
then argue that stationarity of the equilibrium state should be interpreted as
stationarity on a sufficiently long but finite time scale, at least in the
first-order phase transition point. Finally, in Section IV, we prove that the
density matrix of the SE is locally stationary on time scales diverging in the
thermodynamic limit. From these results, we discuss that the SE is consistent
with thermodynamics.

##  II Setup

We consider a quantum spin system with short-range interactions on the
$\nu$-dimensional hypercubic lattice $\Lambda=[-n,+n]^{\nu}$ with
$N=(2n+1)^{\nu}$ sites. We assume that the equilibrium state for each $N$ can
be specified uniquely by a set of $m$ additive quantities
$(X_{{0}}=U,X_{{1}},\cdots,X_{{m-1}})$, where $U$ is the internal energy. Let
$(\hat{X}_{{N,0}}=\hat{H}_{N},\hat{X}_{{N,1}},\cdots,\hat{X}_{{N,m-1}})$ be a
set of the corresponding additive observables, where $\hat{H}_{N}$ is the
Hamiltonian. We assume that each additive observable $\hat{X}_{{N,i}}$ is
expressed as

$\displaystyle\hat{X}_{{N,i}}=\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{\nu}\text{\
s.t.}\\\
\gamma_{j}(I_{{i}})\subset\Lambda\end{subarray}}\gamma_{j}(\hat{o}_{{i}}),$
(1)

where $\gamma_{j}$ is the $j$-lattice translation for $j\in\mathbb{Z}$ and
$\hat{o}_{{i}}$ is an $N$-independent local observable with support $I_{{i}}$.
For simplicity, we assume open boundary conditions, but the arguments in this
paper easily extend to the case of general boundary conditions.

Our final purpose is to prove stationarity in the thermodynamic limit:
$N\to\infty$ while $X_{{i}}/N$ is fixed. Therefore, we introduce additive
quantities per site

$\displaystyle x_{{i}}\equiv X_{{i}}/N\qquad(i=0,1,\cdots,m-1),$ (2)

and corresponding observable

$\displaystyle\hat{x}_{{N,i}}\equiv\hat{X}_{{N,i}}/N\qquad(i=0,1,\cdots,m-1).$
(3)

For simplicity of notation, sets of $m$ physical quantities are denoted by
bold symbols like

$\displaystyle\bm{x}$ $\displaystyle=(x_{{0}},x_{{1}},\cdots,x_{{m-1}}).$ (4)

Let $s_{\mathrm{TD}}$ be the thermodynamic entropy per site. According to
thermodynamics, $s_{\mathrm{TD}}$ is a function of $\bm{x}$. The first
derivative of $s_{\mathrm{TD}}$ is called the ‘(entropic) intensive parameter’
[2]. Let $\Pi_{{i}}$ denote the intensive parameter conjugate to $X_{{i}}$,
i.e.,

$\displaystyle\Pi_{{i}}(\bm{x})\equiv\frac{\partial s_{\mathrm{TD}}}{\partial
x_{{i}}}(\bm{x}).$ (5)

Particularly, $\beta(\bm{x})\equiv\Pi_{{0}}(\bm{x})$ is the inverse
temperature of the equilibrium state specified by $\bm{x}$.

Below we consider the time evolution of the density matrix corresponding to
the equilibrium state. Let $\bm{\Pi}$ be a set of intensive parameters of the
equilibrium state,

$\displaystyle\bm{\Pi}=(\Pi_{{0}},\Pi_{{1}},\cdots,\Pi_{{m-1}}).$ (6)

According to thermodynamics, the equilibrium state is macroscopically
stationary under external fields coupled to $X_{{i}}$ of magnitude

$\displaystyle f_{{i}}=-\Pi_{{i}}/\Pi_{{0}}.$ (7)

On the other hand, according to quantum mechanics, such a dynamics is
generated by the Hamiltonian including the interactions with external fields
111$\hat{H}_{N}=\hat{X}_{{N,0}}$ is the Hamiltonian of the completely isolated
system and does not include the interactions with external fields.:

$\displaystyle\hat{G}_{N}(\bm{\Pi})$
$\displaystyle\equiv\hat{H}_{N}-f_{{1}}\hat{X}_{{N,1}}-\cdots-
f_{{m-1}}\hat{X}_{{N,m-1}}$
$\displaystyle=\hat{H}_{N}+\frac{\Pi_{{1}}}{\Pi_{{0}}}\hat{X}_{{N,1}}+\cdots+\frac{\Pi_{{m-1}}}{\Pi_{{0}}}\hat{X}_{{N,m-1}}.$
(8)

We rescale the time variable and the Hamiltonian as

$\displaystyle\tau$ $\displaystyle\equiv t/\Pi_{{0}},$ (9)
$\displaystyle\hat{\Gamma}_{N}(\bm{\Pi})$
$\displaystyle\equiv\Pi_{{0}}\hat{G}_{N}=\sum_{i}\Pi_{{i}}\hat{X}_{{N,i}},$
(10)

and define the Heisenberg operator as

$\displaystyle\hat{A}(\tau)$ $\displaystyle\equiv
e^{+i\hat{\Gamma}_{N}\tau}\hat{A}e^{-i\hat{\Gamma}_{N}\tau}.$ (11)

Consider the case where the equilibrium state is not at the first-order phase
transition point. In this case, the canonical Gibbs state is one of the
quantum states that correspond to that equilibrium state:

$\displaystyle\hat{\rho}_{N}^{\mathrm{c}}\propto\exp\left[-\sum_{i}\Pi_{{i}}\hat{X}_{{N,i}}\right].$
(12)

Obviously, we can write $\hat{\rho}_{N}^{\mathrm{c}}$ as

$\displaystyle\hat{\rho}_{N}^{\mathrm{c}}\propto\exp\left[-\hat{\Gamma}_{N}(\bm{\Pi})\right].$
(13)

Therefore, this density matrix is strictly invariant under the dynamics
generated by $\hat{\Gamma}_{N}(\bm{\Pi})$.

However, in the case where the equilibrium state is at the first-order phase
transition point, we cannot employ the canonical ensemble. For example, at the
first-order phase transition point due to the spontaneous symmetry breaking,
the canonical ensemble always gives a mixture of all ordered single-phase
states with the same weight [11, 12, 13, 14] and cannot give a macroscopically
definite state 222 It is sometimes possible to obtain the desired state at the
first-order phase transition point using the canonical ensemble by imposing
clever boundary conditions [30, 31, 32]. In this case, however, one would
artificially add boundary terms to additive observables $\hat{\bm{X}}_{N}$ and
construct the canonical ensemble from a set of these observables
$\hat{\bm{X}}_{N}^{\prime}$. On the other hand, since the generator of the
time evolution is determined purely from quantum mechanics, independent of
such artificial manipulations in statistical mechanics, a set of observables
appearing in the definition of $\hat{\Gamma}_{N}$ are the same as the original
set of additive observables $\hat{\bm{X}}_{N}(\neq\hat{\bm{X}}_{N}^{\prime})$.
As a result, Eq. (13) no longer holds. Therefore, the canonical Gibbs state
$\hat{\rho}_{N}^{\mathrm{c}}$ is not strictly time invariant. . In the next
section, we study the time evolution of the quantum state corresponding to the
equilibrium state at the first-order phase transition point.

######  Example (two-dimensional transverse-field Ising model).

Consider a two-dimensional system defined by the Hamiltonian

$\displaystyle\hat{H}_{N}$ $\displaystyle=-J\sum_{\langle
i,j\rangle}\hat{\sigma}_{i}^{z}\hat{\sigma}_{j}^{z}-g\sum_{i}\hat{\sigma}_{i}^{x}.$
(14)

Here, $\langle i,j\rangle$ denotes the nearest neighbors. We take the coupling
$J$ to be ferromagnetic ($J>0$). For $g$ smaller than the critical value
$g_{C}\simeq 3.044$, this model exhibits spontaneous symmetry breaking [16,
17, 18, 19, 20, 21, 22], and at low temperature there exist several
equilibrium states for a single $\beta$, which can be distinguished by the
order parameter

$\displaystyle\hat{X}_{{N,1}}$ $\displaystyle=\sum_{i}\hat{\sigma}_{i}^{z},$
(15)

i.e., the equilibrium state can be uniquely specified not by a value of the
internal energy $U$ alone, but by a set of values of $U$ and $X_{{1}}$. In
other word, this system exhibits a first-order phase transition where
$X_{{1}}$ changes discontinuously with respect to its conjugate intensive
parameter $\Pi_{{1}}$ (or longitudinal magnetic field
$f_{{1}}=-\Pi_{{1}}/\Pi_{{0}}$) at $\Pi_{{1}}=0$ (or $f_{{1}}=0$). Note that
the order parameter $\hat{X}_{{N,1}}$ do not commute with $\hat{H}_{N}$. Thus,
it is not a conserved quantity. Therefore, at the first-order phase transition
point, this system can fluctuate macroscopically in time between the
equilibrium states with the same $\beta$ and different values of
$\hat{X}_{{N,1}}$.

##  III Macroscopic non-stationarity of ordered states

In this section, we prove under several conditions that at the first-order
phase transition point with an order parameter that does not commute with the
other additive observables, any quantum state with a non-zero value of the
order parameter always changes macroscopically after a sufficiently long time.
We then argue that, at the first-order phase transition point, it is
impossible to require strict time invariance for statistical ensembles and
that stationarity of the equilibrium state should be interpreted as
stationarity on a sufficiently long but finite time scale.

To be concrete, we consider a system that exhibits spontaneous symmetry
breaking and thus has an equilibrium state with a non-zero value of the order
parameter. Suppose that the system is macroscopic but of finite size, as is
the case for real thermodynamic systems. That is, we consider the case where
the number of sites $N$ is fixed to a certain macroscopic but finite value.
Choose a set of additive dynamical quantity per site $\bm{x}$ to be at the
first-order phase transition point and consider the time evolution of the
equilibrium state specified by that $\bm{x}$. Then, the argument $\bm{\Pi}$ of
the generator of the time evolution $\hat{\Gamma}_{N}(\bm{\Pi})$ is taken to
the set of intensive parameters at $\bm{x}$; $\bm{\Pi}=\bm{\Pi}(\bm{x})$.

Here, let us clarify the definitions of the symmetry and the order parameter
of the system. When the unitary transformation $\hat{V}$ keeps
$\hat{\Gamma}_{N}(\bm{\Pi})$ invariant, i.e.,

$\displaystyle\hat{V}^{\dagger}\hat{\Gamma}_{N}(\bm{\Pi})\hat{V}=\hat{\Gamma}_{N}(\bm{\Pi}),$
(16)

then we call $\hat{V}$ the ‘symmetry transformation at $\bm{\Pi}$’. Suppose
that the group $F$ formed by symmetric transformations at $\bm{\Pi}$ is a
compact group. Let $Z(F)$ be a center of $F$, i.e.,

$\displaystyle Z(F)\equiv\left\\{\hat{U}\in
F\middle|\hat{U}\hat{V}=\hat{V}\hat{U}\text{\ for all\ }\hat{V}\in
F\right\\}.$ (17)

Suppose that the additive observable $\hat{X}_{{N,1}}$ is an order parameter
that measures the spontaneous breaking of the symmetry for an element of
$Z(F)$, written as $\hat{R}$. That is, we consider the case where

  1. (i)

There exists a symmetry transformation $\hat{R}\in Z(F)$ such that

$\displaystyle\hat{R}^{\dagger}\hat{X}_{{N,1}}\hat{R}=-\hat{X}_{{N,1}}.$ (18)

######  Example (two-dimensional transverse-field Ising model).

To illustrate the settings described above more concretely, let us take an
example of the transverse-field Ising model defined by Eqs. (14) and (15). As
symmetries of this system at the first-order phase transition point
$\bm{\Pi}=(\beta,0)$, the following are known:

  * •

$\mathbb{Z}_{2}$ symmetry with respect to the spin rotation by $\pi$ around
the $x$-axis

$\displaystyle\hat{R}=\exp\left[i\frac{\pi}{2}\sum_{i}\hat{\sigma}_{i}^{x}\right]$
(19)

  * •

$C_{4v}$ symmetry of the square lattice

Let $F$ be the group formed by these symmetry transformations (i.e.,
$F\cong\mathbb{Z}_{2}\times C_{4v}$), then $\hat{R}\in Z(F)$ and
$\hat{R}^{\dagger}\hat{X}_{{N,1}}\hat{R}=-\hat{X}_{{N,1}}$. Therefore, it can
be confirmed that condition (i) is indeed fulfilled.

In addition, we assume that there is no accidental degeneracy in the
eigenstates of $\hat{\Gamma}_{N}$. More precisely, we assume that

  1. (ii)

For any pair of eigenstates $\ket{E_{1}}$ and $\ket{E_{2}}$ which belong to
the same eigenvalue of $\hat{\Gamma}_{N}$, there exists a symmetry
transformation $\hat{V}\in F$ such that $\braket{E_{1}}{\hat{V}}{E_{2}}\neq
0$.

Under the above conditions (i)-(ii), as proved in Appendix A, we have the
following theorem:

######  Theorem 1.

For any initial state $\hat{\rho}$, the long-time average of the expectation
value of $\hat{X}_{{N,1}}$ under the time evolution generated by
$\hat{\Gamma}_{N}(\bm{\Pi})$ is exactly zero:

$\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}d\tau\
\mathrm{Tr}\left[\hat{X}_{{N,1}}(\tau)\hat{\rho}\right]=0.$ (20)

From this theorem, it follows that at the first-order phase transition point,
any quantum state corresponding to an equilibrium state in which the order
parameter $X_{{1}}$ is non-zero always changes to a macroscopically different
state after an infinitely long time, whether it is a single-phase state or
phase coexistence state.

Since this theorem holds for any initial state, this macroscopic non-
stationarity cannot be avoided even by choosing an appropriate ensemble. On
the other hand, it has been observed experimentally that the ordered state
does exist stably even at the first-order phase transition point where the
order parameter does not commute with the Hamiltonian [23, 24]. However, the
experiment is only conducted on a finite time scale, not on a truly infinite
time scale. Therefore, to reconcile these two facts, it seems reasonable to
interpret stationarity of the equilibrium state as stationarity on a
sufficiently long but finite time scale, at least at the first-order phase
transition point.

##  IV Stationarity of the density matrix of the squeezed ensemble

As discussed in Section I, one cannot generally employ conventional ensembles
at the first-order phase transition point when the order parameter does not
commute with the other additive observables that specify the equilibrium
state. In Ref. [7], the author has solved this fundamental problem by
extending the generalized ensemble, called the squeezed ensemble (SE), in such
a way that it is applicable to equilibrium states specified by noncommutative
additive observables. We here investigate the time evolution of the density
matrix of the SE and prove that it is stationary on the time scale that
diverges in the thermodynamic limit. In this section, we consider general
first-order phase transitions, not only those due to spontaneous symmetry
breaking.

We first review the definition of the SE and its properties that will be used
in our analysis. Interested reader can find more information in Ref. [7]. Let
$\eta(\bm{x})$ be a polynomial defined up to the order of the product, with
real coefficients in $m$ noncommutative variables $\bm{x}$. Suppose that
$\eta$ satisfies the following conditions:

  1. (A)

$\eta(\hat{\bm{x}}_{N})$ is self-adjoint for all $N$.

  2. (B)

$s_{\mathrm{TD}}(\bm{x})-\eta(\bm{x})$ has the unique maximum point
$\bm{x}_{\mathrm{max}}^{\eta}$ in the thermodynamic state space and is
strongly concave in a neighborhood of $\bm{x}_{\mathrm{max}}^{\eta}$.

Using this $\eta$, the density matrix of the squeezed ensemble (SE) is defined
as

$\displaystyle\hat{\rho}_{N}^{\eta}$ $\displaystyle\equiv\frac{\displaystyle
e^{-N\eta(\hat{\bm{x}}_{N})}}{\displaystyle\mathrm{Tr}\left[e^{-N\eta(\hat{\bm{x}}_{N})}\right]}.$
(21)

As shown in Ref. [7], the density matrix $\hat{\rho}_{N}^{\eta}$ correctly
gives the equilibrium state specified by $\bm{x}=\bm{x}_{\mathrm{max}}^{\eta}$
even when $\hat{\bm{x}}_{N}$ do not commute with each other. Thus, one can
investigate microscopic structures of the equilibrium state at the first-order
phase transition point of general quantum systems by choosing $\eta$ such that
$\bm{x}_{\mathrm{max}}^{\eta}$ coincides with $\bm{x}$ in that equilibrium
state.

One can also obtain thermodynamic functions easily from the SE. In particular,
intensive parameters can be calculated using the following formula:

$\displaystyle\Pi_{{i}}^{\eta}$
$\displaystyle\equiv\lim_{N\to\infty}\frac{\partial\eta}{\partial
x_{{i}}}(\bm{x}_{N}^{\eta})=\Pi_{{i}}(\bm{x}^{\eta}),$ (22)

where
$x_{{N,i}}^{\eta}\equiv\mathrm{Tr}\left[\hat{x}_{{N,i}}\hat{\rho}_{N}^{\eta}\right]$
and $\displaystyle x_{{i}}^{\eta}\equiv\lim_{N\to\infty}x_{{N,i}}^{\eta}$.
Since $\eta$ is a known function, one can obtain $\Pi_{{i}}$ just by
calculating $\bm{x}_{N}^{\eta}$.

Below in this section we investigate the time evolution of the density matrix
of the SE. From the formula (22) for intensive parameters, in the equilibrium
state described by the SE, the set of intensive parameters are

$\displaystyle\bm{\Pi}^{\eta}=(\Pi_{{0}}^{\eta},\Pi_{{1}}^{\eta},\cdots,\Pi_{{m-1}}^{\eta}).$
(23)

Hence, according to the setup described in Section II, we consider the time
evolution generated by $\hat{\Gamma}_{N}(\bm{\Pi}^{\eta})$. Then, as proved in
Appendix B, the following holds which implies stationarity of the SE on time
scales diverging in the thermodynamic limit:

######  Theorem 2.

If there exists a positive constant $\alpha$ such that for every
$i=0,1,\cdots,m-1$ the following conditions are fullfield:

$\displaystyle\left(x_{{N,i}}^{\eta}-x_{{i}}^{\eta}\right)^{2}$
$\displaystyle=O(N^{-2\alpha}),$ (24)
$\displaystyle\mathrm{Tr}\left[\left(\hat{x}_{{N,i}}-x_{{N,i}}^{\eta}\right)^{2}\hat{\rho}_{N}^{\eta}\right]$
$\displaystyle=O(N^{-2\alpha}).$ (25)

Then for any $\epsilon>0$ and $U\subset\mathbb{Z}^{\nu}$, there exists
$T=\Theta(N^{\min[\alpha/(\nu+1),1/\nu]})$ such that for any local observable
$\hat{A}$ supported on $U$ and $|\tau|<T$ the relative time variation of the
expectation value of $\hat{A}$ is less that $\epsilon$:

$\displaystyle\left|\mathrm{Tr}\left[\hat{A}(\tau)\hat{\rho}_{N}^{\eta}\right]-\mathrm{Tr}\left[\hat{A}(0)\hat{\rho}_{N}^{\eta}\right]\right|<\epsilon\|\hat{A}\|.$
(26)

First, we would like to comment on the conditions of this theorem. Equation
(24) and (25) is saying that the finite-size effects and variance of
$\hat{x}_{{N,i}}$ in the SE decay algebraically in the thermodynamic limit,
respectively. These conditions are confirmed numerically for a specific model
with noncommutative $\hat{\bm{x}}_{N}$ (see Section IX of Ref. [7]). Note that
all conditions can be checked using only the statistical-mechanical quantities
of the SE. Therefore, even for phase coexistence states of general quantum
systems for which there is no concrete construction method other than SE, we
can easily check stationarity of the density matrix of the SE.

Next, we discuss the consequences of this theorem. Consider performing a
macroscopic experiment where the initial state is the density matrix
$\hat{\rho}_{N}^{\eta}$ of the SE. Let $\epsilon$ be the maximum value of
relative uncertainties of the measurement devices for local observables used
in the experiment. Then it follows from Theorem 2 that the time variations of
the expectation values of the local observables are within the uncertainties
of the measurements, on time scales diverging in the thermodynamic limit.
Therefore, as long as one is interested in the local observables, one cannot
observe any time variation. Since in thermodynamics we are only interested in
local observables and additive observables, which can be expressed as a sum of
local observables, stationarity in the above seance is sufficient for
stationarity of the equilibrium state. Thus, we conclude that the SE is
consistent with thermodynamics.

##  V Conclusion

In this paper, we have studied the time evolution of the quantum statistical
ensemble at the first-order phase transition point. Stationarity is one of the
most fundamental properties of equilibrium states. Therefore, the density
matrix given by the statistical ensemble should also satisfy stationarity.
However, as we have shown, at the first-order phase transition point of
systems with an order parameter that does not commute with the other additive
observables specifying the equilibrium state, any quantum state with a non-
zero value of order parameter always evolves towards a macroscopically
distinct state after a sufficiently long time. From this result, we have
argued that, at least at the first-order phase transition point, stationarity
required for statistical ensembles should be interpreted as stationarity on a
sufficiently long but finite time scale. Then We have proved that the density
matrix of the squeezed ensemble, a class of generalized ensembles proposed as
the only concrete method of constructing phase coexistence states applicable
to general quantum systems, satisfies stationarity in the above sense. These
results support the validity of the SE from a dynamical point of view and open
the door to non-equilibrium statistical physics at the first-order phase
transition point.

###### Acknowledgements.

We thank Y. Chiba and A. Shimizu for discussions. This work was supported by
The Japan Society for the Promotion of Science, KAKENHI No. 20J22982.

##  Appendix A Proof of Theorem 1

###### Proof of Theorem 1.

Since $Z(F)$ is an Abelian group, its irreducible representations are all one-
dimensional and can be expressed with real function $\theta_{\mu}$ as

$\displaystyle Z(F)\ni\hat{U}\longmapsto e^{i\theta_{\mu}(\hat{U})}\in
M(\mathbb{C}^{1}),$ (27)

where $\mu$ labels the irreducible representation [25]. Moreover, let
$(\mathcal{H}_{\nu},\sigma_{\nu})$ be the irreducible representation of $F$.
Here $\nu$ labels the irreducible representation.

Let $\mathcal{H}$ denote the Hilbert space associated to this quantum system.
The following map $f$ is a representation on $\mathcal{H}$ of the direct
product group $Z(F)\times F$:

$\displaystyle f:Z(F)\times F\ni(\hat{U},\hat{V})\longmapsto\hat{U}\hat{V}\in
F.$ (28)

In fact, for any $\hat{U}_{1},\hat{U}_{2}\in Z(F)$ and
$\hat{V}_{1},\hat{V}_{2}\in F$, since $\hat{U}_{2}$ is in the center of $F$,
it holds

$\displaystyle
f(\hat{U}_{1},\hat{V}_{1})f(\hat{U}_{2},\hat{V}_{2})=\hat{U}_{1}\hat{V}_{1}\hat{U}_{2}\hat{V}_{2}=\hat{U}_{1}\hat{U}_{2}\hat{V}_{1}\hat{V}_{2}=f(\hat{U}_{1}\hat{U}_{2},\hat{V}_{1}\hat{V}_{2}).$
(29)

That is, $f$ is indeed a group homomorphism.

First, we consider the irreducible decomposition of $(\mathcal{H},f)$. Any
irreducible representation of the direct product group $Z(F)\times F$ can be
written as a tensor product of irreducilble representations of $Z(F)$ and $F$
[25]. Thus, in the present case, any irreducible representation of $Z(F)\times
F$ can be expressed as

$\displaystyle f_{\mu\nu}:Z(F)\times F\ni(\hat{U},\hat{V})\longmapsto
e^{i\theta_{\mu}(\hat{U})}\sigma_{\nu}(\hat{V})\in\mathcal{H}_{\nu}.$ (30)

Therefore, we have the irreducible decomposition of $(\mathcal{H},f)$ as

$\displaystyle\mathcal{H}$
$\displaystyle\cong\bigoplus_{\mu,\nu}{\mathcal{H}_{\nu}}^{\oplus
n_{\mu\nu}}\cong\bigoplus_{\mu,\nu}\underbrace{\mathbb{C}^{n_{\mu\nu}}\otimes\mathcal{H}_{\nu}}_{\equiv\mathcal{K}_{\mu\nu}},$
(31) $\displaystyle f$
$\displaystyle\cong\bigoplus_{\mu,\nu}{f_{\mu\nu}}^{\oplus
n_{\mu\nu}}\cong\bigoplus_{\mu,\nu}e^{i\theta_{\mu}}\hat{1}_{\mathbb{C}^{n_{\mu\nu}}}\otimes
f_{\nu}.$ (32)

Here, $\cong$ denotes the unitary equivalence, $\oplus$ denotes the direct
sum, and $n_{\mu\nu}$ denotes the number of copies of the irreducible
representation $(\mathcal{H}_{\nu},f_{\mu\nu})$ in $(\mathcal{H},f)$. In
particular, we obtain

$\displaystyle\hat{R}=f(\hat{R},\hat{1})\cong\bigoplus_{\mu,\nu}e^{i\theta_{\mu}(\hat{R})}\hat{1}_{\mathbb{C}^{n_{\mu\nu}}}\otimes\hat{1}_{\mathcal{H}_{\nu}}.$
(33)

On the other hand, because for any $(\hat{U},\hat{V})\in Z(F)\times F$ it
holds $[\hat{\Gamma}_{N},f(\hat{U},\hat{V})]=0$ by the definition of $F$, it
follows from Schur’s lemma that there exists $\hat{\gamma}_{\mu\nu}\in
M(\mathbb{C}^{n_{\mu\nu}})$ such that

$\displaystyle\hat{\Gamma}_{N}\cong\bigoplus_{\mu\nu}\hat{\gamma}_{\mu\nu}\otimes\hat{1}_{\mathcal{H}_{\nu}}.$
(34)

Next, we diagonalize $\hat{\Gamma}_{N}$. Since $\hat{\Gamma}_{N}$ is self-
adjoint, $\hat{\gamma}_{\mu\nu}$ is also self-adjoint. Then let
$\\{\ket{i}_{\mu\nu}\\}_{i=1,\cdots,n_{\mu\nu}}$ be an orthonormal basis of
$\mathbb{C}^{n_{\mu\nu}}$ with respect to which $\hat{\gamma}_{\mu\nu}$ is
diagonal and $\gamma_{\mu\nu}^{i}$ be the (real) eigenvalue associated with
$\ket{i}_{\mu\nu}$. Furthermore, we take an arbitrary orthonormal basis of
$\mathcal{H}_{\nu}$ as
$\\{\ket{a}_{\nu}\\}_{a=1,\cdots,\dim\mathcal{H}_{\nu}}$. Using
$\ket{i}_{\mu\nu}$ and $\ket{a}_{\nu}$, we set
$\ket{E_{\mu\nu}^{ia}}(\in\mathcal{K}_{\mu\nu}\subset\mathcal{H})$ as

$\displaystyle\ket{E_{\mu\nu}^{ia}}\equiv\ket{i}_{\mu\nu}\otimes\ket{a}_{\nu}.$
(35)

Then $\\{\ket{E_{\mu\nu}^{ia}}\\}_{\mu,\nu,i,a}$ is a complete orthonomal
basis of $\mathcal{H}$. In addition, using Eqs. (33) and (34), we obtain

$\displaystyle\hat{R}\ket{E_{\mu\nu}^{ia}}$
$\displaystyle=e^{i\theta_{\mu}(\hat{R})}\ket{E_{\mu\nu}^{ia}},$ (36)
$\displaystyle\hat{\Gamma}_{N}\ket{E_{\mu\nu}^{ia}}$
$\displaystyle=\gamma_{\mu\nu}^{i}\ket{E_{\mu\nu}^{ia}}.$ (37)

Then we examine the properties of $\gamma_{\mu\nu}^{i}$. Suppose that
$\gamma_{\mu\nu}^{i}=\gamma_{\mu^{\prime}\nu^{\prime}}^{i^{\prime}}$. Then,
condition (ii) implies that there exists $\hat{V}\in F$ such that

$\displaystyle\braket{E_{\mu\nu}^{ia}}{\hat{V}}{E_{\mu^{\prime}\nu^{\prime}}^{i^{\prime}a^{\prime}}}\neq
0.$ (38)

On the other hand, using Eq. (32) and (35), we have

$\displaystyle\ket{E_{\mu\nu}^{ia}}$ $\displaystyle\in\mathcal{K}_{\mu\nu}$
(39)
$\displaystyle\hat{V}\ket{E_{\mu^{\prime}\nu^{\prime}}^{i^{\prime}a^{\prime}}}$
$\displaystyle\in\mathcal{K}_{\mu^{\prime}\nu^{\prime}}.$ (40)

Since $\mathcal{K}_{\mu\nu}\cap\mathcal{K}_{\mu^{\prime}\nu^{\prime}}=\\{0\\}$
form Eq. (31), Eq. (38) holds only when $\mu=\mu^{\prime},\nu=\nu^{\prime}$.
Furthermore, when $\mu=\mu^{\prime},\nu=\nu^{\prime}$, since we have

$\displaystyle\braket{E_{\mu\nu}^{ia}}{\hat{V}}{E_{\mu^{\prime}\nu^{\prime}}^{i^{\prime}a^{\prime}}}=\braket{i}{i^{\prime}}_{\mu\nu}\braket{a}{\sigma_{\nu}(\hat{V})}{a^{\prime}}_{\nu},$
(41)

Eq. (38) holds only when $i=i^{\prime}$. Thus, we get 333 Conversely, Eq. (43)
implies condition (ii). Suppose Eq. (43) holds. Then for any eigenstates
$\ket{E_{1}}$ and $\ket{E_{2}}$ which belong to the same eigenvalue of
$\hat{\Gamma}_{N}$, we can expand as  $\displaystyle\ket{E_{s}}$
$\displaystyle=\sum_{a}c_{s}^{a}\ket{E_{\mu\nu}^{ia}}=\ket{i}_{\mu\nu}\otimes\left(\sum_{a}c_{s}^{a}\ket{a}_{\nu}\right)=\ket{i}_{\mu\nu}\otimes\ket{\phi_{s}}_{\nu}.$
(42) On the other hand, since $(\mathcal{H}_{\nu},\sigma_{\nu})$ is an
irreducible representation of $F$, there exists $\hat{V}\in F$ such that
$\braket{\phi_{1}}{\sigma_{\nu}(\hat{V})}{\phi_{2}}\neq 0$. Therefore, we have
condition (ii).

$\displaystyle\gamma_{\mu\nu}^{i}=\gamma_{\mu^{\prime}\nu^{\prime}}^{i^{\prime}}\Longleftrightarrow\mu=\mu^{\prime},\nu=\nu^{\prime},i=i^{\prime}.$
(43)

Finaly, we evaluate the long-time average of the expectation value of
$\hat{X}_{{N,1}}(\tau)$. Due to the completeness of
$\\{\ket{E_{\mu\nu}^{ia}}\\}_{\mu,\nu,i,a}$, we have

$\displaystyle\hat{X}_{{N,1}}(\tau)=\sum_{\begin{subarray}{c}\mu,\nu,i,a\\\
\mu^{\prime},\nu^{\prime},i^{\prime},a^{\prime}\end{subarray}}e^{+i(\gamma_{\mu\nu}^{i}-\gamma_{\mu^{\prime}\nu^{\prime}}^{i^{\prime}})\tau}\ket{E_{\mu\nu}^{ia}}\braket{E_{\mu\nu}^{ia}}{\hat{X}_{{N,1}}}{E_{\mu^{\prime}\nu^{\prime}}^{i^{\prime}a^{\prime}}}\bra{E_{\mu^{\prime}\nu^{\prime}}^{i^{\prime}a^{\prime}}}.$
(44)

Therefore, form Eq. (43), we have

$\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}d\tau\hat{X}_{{N,1}}(\tau)=\sum_{\begin{subarray}{c}\mu,\nu,i,a,a^{\prime}\end{subarray}}\ket{E_{\mu\nu}^{ia}}\braket{E_{\mu\nu}^{ia}}{\hat{X}_{{N,1}}}{E_{\mu\nu}^{ia^{\prime}}}\bra{E_{\mu\nu}^{ia^{\prime}}}.$
(45)

On the other hand, by condition (i), we have

$\displaystyle\braket{E_{\mu\nu}^{ia}}{\hat{X}_{{N,1}}}{E_{\mu\nu}^{ia^{\prime}}}$
$\displaystyle=-\braket{E_{\mu\nu}^{ia}}{\hat{R}^{\dagger}\hat{X}_{{N,1}}\hat{R}}{E_{\mu\nu}^{ia^{\prime}}}$
$\displaystyle=-\braket{E_{\mu\nu}^{ia}}{e^{-i\theta_{\mu}(\hat{R})}\hat{X}_{{N,1}}e^{+i\theta_{\mu}(\hat{R})}}{E_{\mu\nu}^{ia^{\prime}}}=-\braket{E_{\mu\nu}^{ia}}{\hat{X}_{{N,1}}}{E_{\mu\nu}^{ia^{\prime}}}.$
(46)

Therefore, we obtain

$\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}d\tau\hat{X}_{{N,1}}(\tau)=0.$
(47)

∎

##  Appendix B Proof of Theorem 2

In our arguments, the following lemma plays a key role.

######  Lemma 1 (Lieb-Robinson Bound [27, 28, 29]).

There exist positive constants $C$, $\mu$ and $v$, such that for any
observables $\hat{A}$ and $\hat{B}$ with finite supports $U\subset\Lambda$ and
$V\subset\Lambda$, respectively, and for any $\tau\in\mathbb{R}$,

$\displaystyle\left\|\left[\hat{A}(\tau),\hat{B}\right]\right\|$
$\displaystyle\leq
C\left\|\hat{A}\right\|\left\|\hat{B}\right\|\min\left[\left|U\right|,\left|V\right|\right]e^{-\mu\left(d(U,V)-v|\tau|\right)}.$
(48)

where $d(x,y)$ is the distance which is defined to be the shortest path length
that one needs to connect $x$ to $y$.

###### Proof.

This follows from Eq. (2.15) in Ref. [29]. ∎

###### Proof of Theorem 2.

First, we construct a local observable which approximates $\hat{A}(\tau)$.
Take a real number $l$ such that $1\ll l\ll L$. Here $L$ is a linear dimension
of the system. Let $\Lambda^{\mathrm{in}}\equiv\left\\{j\in\Lambda\mid
d(U,j)<l\right\\}$, then
$\left|\partial\Lambda^{\mathrm{in}}\right|=O\left(l^{\nu-1}\right)$. Since
$\hat{\Gamma}_{N}$ is additive, there exist observables
$\hat{\Gamma}_{N}^{\mathrm{in}}$, $\hat{\Gamma}_{N}^{\mathrm{out}}$ and
$\hat{\Gamma}_{N}^{\mathrm{int}}$ such that

  * •

$\hat{\Gamma}_{N}=\hat{\Gamma}_{N}^{\mathrm{in}}+\hat{\Gamma}_{N}^{\mathrm{out}}+\hat{\Gamma}_{N}^{\mathrm{int}}$,

  * •

$\mathrm{supp\ }\hat{\Gamma}_{N}^{\mathrm{in}}=\Lambda^{\mathrm{in}}$,

  * •

$\mathrm{supp\
}\hat{\Gamma}_{N}^{\mathrm{out}}=\Lambda\setminus\Lambda^{\mathrm{in}}$,

  * •

$\left\|\hat{\Gamma}_{N}^{\mathrm{int}}\right\|=O\left(l^{\nu-1}\right)$.

Now, we define an observable on $\Lambda^{\mathrm{in}}$ as

$\displaystyle\tilde{A}(\tau)$ $\displaystyle\equiv
e^{+i(\hat{\Gamma}_{N}-\hat{\Gamma}_{N}^{\mathrm{int}})\tau}\hat{A}e^{-i(\hat{\Gamma}_{N}-\hat{\Gamma}_{N}^{\mathrm{int}})\tau}=e^{+i\hat{\Gamma}_{N}^{\mathrm{in}}\tau}\hat{A}e^{-i\hat{\Gamma}_{N}^{\mathrm{in}}\tau}.$
(49)

Applying Lemma 1, we have

$\displaystyle\left\|\hat{A}(\tau)-\tilde{A}(\tau)\right\|$
$\displaystyle=\left\|e^{+i\hat{\Gamma}_{N}\tau}\hat{A}e^{-i\hat{\Gamma}_{N}\tau}-e^{+i\left(\hat{\Gamma}_{N}-\hat{\Gamma}_{N}^{\mathrm{int}}\right)\tau}\hat{A}e^{-i\left(\hat{\Gamma}_{N}-\hat{\Gamma}_{N}^{\mathrm{int}}\right)\tau}\right\|$
$\displaystyle=\left\|\int_{0}^{\tau}\frac{d}{d\sigma}\left(e^{+i\left(\hat{\Gamma}_{N}-\hat{\Gamma}_{N}^{\mathrm{int}}\right)(\tau-\sigma)}e^{+i\hat{\Gamma}_{N}\sigma}\hat{A}e^{-i\hat{\Gamma}_{N}\sigma}e^{-i\left(\hat{\Gamma}_{N}-\hat{\Gamma}_{N}^{\mathrm{int}}\right)(\tau-\sigma)}\right)d\sigma\right\|$
$\displaystyle=\left\|\int_{0}^{\tau}e^{+i\left(\hat{\Gamma}_{N}-\hat{\Gamma}_{N}^{\mathrm{int}}\right)\sigma}\left[\hat{\Gamma}_{N}^{\mathrm{int}},e^{+i\hat{\Gamma}_{N}\sigma}\hat{A}e^{-i\hat{\Gamma}_{N}\sigma}\right]e^{-i\left(\hat{\Gamma}_{N}-\hat{\Gamma}_{N}^{\mathrm{int}}\right)\sigma}d\sigma\right\|$
$\displaystyle\leq\int_{0}^{|\tau|}\left\|\left[\hat{\Gamma}_{N}^{\mathrm{int}},\hat{A}(\sigma)\right]\right\|d\sigma$
$\displaystyle\leq\frac{C}{\mu
v}\left\|\hat{\Gamma}_{N}^{\mathrm{int}}\right\|\left\|\hat{A}\right\|\left|U\right|e^{-\mu\left(l-v|\tau|\right)}$
$\displaystyle=O\left(l^{\nu-1}e^{-\mu\left(l-v|\tau|\right)}\right).$ (50)

Thus, we see that $\tilde{A}(\tau)$ is an observable on
$\Lambda^{\mathrm{in}}$ which approximates $\hat{A}(\tau)$ with an error of
$O\left(l^{\nu-1}e^{-\mu\left(l-v|\tau|\right)}\right)$.

Now we evaluate the time derivative of the expectation value of
$\hat{A}(\tau)$ in the SE:

$\displaystyle-i\frac{d}{d\tau}\mathrm{Tr}\left[\hat{A}(\tau)\hat{\rho}_{N}^{\eta}\right]$
$\displaystyle=\mathrm{Tr}\left[\left[\hat{\Gamma}_{N},\hat{A}(\tau)\right]\hat{\rho}_{N}^{\eta}\right].$
(51)

Decomposing $\hat{A}(\tau)$ into its local term $\tilde{A}(\tau)$ and nonlocal
term $\hat{A}(\tau)-\tilde{A}(\tau)$, we have

$\displaystyle-i\frac{d}{d\tau}\mathrm{Tr}\left[\hat{A}(\tau)\hat{\rho}_{N}^{\eta}\right]$
$\displaystyle=\underbrace{\mathrm{Tr}\left[\left[\hat{\Gamma}_{N},\tilde{A}(\tau)\right]\hat{\rho}_{N}^{\eta}\right]}_{\equiv(\ast
1)}+\underbrace{\mathrm{Tr}\left[\left[\hat{\Gamma}_{N},\hat{A}(\tau)-\tilde{A}(\tau)\right]\hat{\rho}_{N}^{\eta}\right]}_{\equiv(\ast
2)}.$ (52)

First, we evaluate $(\ast 1)$ in Eq. (52). Since $\eta$ is a polynomial,
expanding in power series around $\bm{x}^{\eta}$, $\eta(\hat{\bm{x}}_{N})$ can
be written as a finite sum as

$\displaystyle\eta(\hat{\bm{x}}_{N})=\hat{\Gamma}_{N}(\bm{\Pi}^{\eta})/N-\sum_{K(\geq
2),\\{i_{k}\\}}c_{i_{1}\cdots
i_{K}}\prod_{k=1}^{K}\left(\hat{x}_{{N,i_{k}}}-x_{{i_{k}}}^{\eta}\right)+\mathrm{const.},$
(53)

where we have used Eqs. (10) and (22). Substituting this into $(\ast 1)$, we
have

$\displaystyle(\ast 1)$ $\displaystyle=\sum_{K,\\{i_{k}\\}}c_{i_{1}\cdots
i_{K}}N\mathrm{Tr}\left[\left[\prod_{k=1}^{K}\left(\hat{x}_{{N,i_{k}}}-x_{{i_{k}}}^{\eta}\right),\tilde{A}(\tau)\right]\hat{\rho}_{N}^{\eta}\right]$
(54) $\displaystyle=\sum_{K,\\{i_{k}\\}}c_{i_{1}\cdots
i_{K}}\underbrace{\sum_{k^{\prime}=1}^{K}\mathrm{Tr}\left[\prod_{k=1}^{k^{\prime}-1}\left(\hat{x}_{{N,i_{k}}}-x_{{i_{k}}}^{\eta}\right)\hat{q}_{{N,i_{k^{\prime}}}}\prod_{k=k^{\prime}+1}^{K}\left(\hat{x}_{{N,i_{k}}}-x_{{i_{k}}}^{\eta}\right)\hat{\rho}_{N}^{\eta}\right]}_{\equiv(\ast\ast)}.$
(55)

Here we set

$\displaystyle\hat{q}_{{N,i}}$
$\displaystyle\equiv\left[\hat{X}_{{N,i}},\tilde{A}(\tau)\right]=\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{\nu}\text{\
s.t.}\\\ \Lambda^{\mathrm{in}}\ \cap\ \mathrm{supp\
}\gamma_{j}(\hat{o}_{{i}})\neq\emptyset\end{subarray}}\left[\gamma_{j}(\hat{o}_{{i}}),\tilde{A}(\tau)\right],$
(56)

where in the last equality we have used the fact that $\tilde{A}(\tau)$ is
supported on $\Lambda^{\mathrm{in}}$. Then, by the triangle inequality and the
unitary invariance of the operator norm, we have

$\displaystyle\left\|\hat{q}_{{N,i}}\right\|\leq\sum_{\begin{subarray}{c}j\in\mathbb{Z}^{\nu}\text{\
s.t.}\\\ \Lambda^{\mathrm{in}}\ \cap\ \mathrm{supp\
}\gamma_{j}(\hat{o}_{{i}})\neq\emptyset\end{subarray}}2\left\|\hat{o}_{{i}}\right\|\left\|\hat{A}\right\|=O(l^{\nu}).$
(57)

Since $K\geq 2$, applying the Cauchy-Schwartz inequality

$\displaystyle\left|\mathrm{Tr}\left[\hat{A}\hat{B}^{\dagger}\hat{\rho}_{N}^{\eta}\right]\right|^{2}\leq\mathrm{Tr}\left[\hat{A}\hat{A}^{\dagger}\hat{\rho}_{N}^{\eta}\right]\mathrm{Tr}\left[\hat{B}\hat{B}^{\dagger}\hat{\rho}_{N}^{\eta}\right],$
(58)

we have

$\displaystyle|(\ast\ast)|$
$\displaystyle\leq\left|\mathrm{Tr}\left[\hat{q}_{{N,i_{1}}}\prod_{k=2}^{K}\left(\hat{x}_{{N,i_{k}}}-x_{{i_{k}}}^{\eta}\right)\hat{\rho}_{N}^{\eta}\right]\right|$
$\displaystyle+\left|\mathrm{Tr}\left[\left(\hat{x}_{{N,i_{1}}}-x_{{i_{1}}}^{\eta}\right)\hat{q}_{{N,i_{2}}}\prod_{k=3}^{K}\left(\hat{x}_{{N,i_{k}}}-x_{{i_{k}}}^{\eta}\right)\hat{\rho}_{N}^{\eta}\right]\right|$
$\displaystyle+\cdots$
$\displaystyle+\left|\mathrm{Tr}\left[\prod_{k=1}^{K-1}\left(\hat{x}_{{N,i_{k}}}-x_{{i_{k}}}^{\eta}\right)\hat{q}_{{N,i_{K}}}\hat{\rho}_{N}^{\eta}\right]\right|$
$\displaystyle\leq\sqrt{\mathrm{Tr}\left[\hat{q}_{{N,i_{1}}}\prod_{k=2}^{K-1}\left(\hat{x}_{{N,i_{k}}}-x_{{i_{k}}}^{\eta}\right)\prod_{k=K-1}^{2}\left(\hat{x}_{{N,i_{k}}}-x_{{i_{k}}}^{\eta}\right)\hat{q}_{{N,i_{1}}}^{\dagger}\hat{\rho}_{N}^{\eta}\right]\mathrm{Tr}\left[\left(\hat{x}_{{N,i_{K}}}-x_{{i_{K}}}^{\eta}\right)^{2}\hat{\rho}_{N}^{\eta}\right]}$
$\displaystyle+\sqrt{\mathrm{Tr}\left[\left(\hat{x}_{{N,i_{1}}}-x_{{i_{1}}}^{\eta}\right)^{2}\hat{\rho}_{N}^{\eta}\right]\mathrm{Tr}\left[\prod_{k=K}^{3}\left(\hat{x}_{{N,i_{k}}}-x_{{i_{k}}}^{\eta}\right)\hat{q}_{{N,i_{2}}}^{\dagger}\hat{q}_{{N,i_{2}}}\prod_{k=3}^{K}\left(\hat{x}_{{N,i_{k}}}-x_{{i_{k}}}^{\eta}\right)\hat{\rho}_{N}^{\eta}\right]}$
$\displaystyle+\cdots$
$\displaystyle+\sqrt{\mathrm{Tr}\left[\left(\hat{x}_{{N,i_{1}}}-x_{{i_{1}}}^{\eta}\right)^{2}\hat{\rho}_{N}^{\eta}\right]\mathrm{Tr}\left[\hat{q}_{{N,i_{K}}}^{\dagger}\prod_{k=K-1}^{2}\left(\hat{x}_{{N,i_{k}}}-x_{{i_{k}}}^{\eta}\right)\prod_{k=2}^{K-1}\left(\hat{x}_{{N,i_{k}}}-x_{{i_{k}}}^{\eta}\right)\hat{q}_{{N,i_{K}}}\hat{\rho}_{N}^{\eta}\right]}.$
(59)

Furthermore,

$\displaystyle\mathrm{Tr}\left[\left(\hat{x}_{{N,i}}-x_{{i}}^{\eta}\right)^{2}\hat{\rho}_{N}^{\eta}\right]=\mathrm{Tr}\left[\left(\hat{x}_{{N,i}}-x_{{N,i}}^{\eta}\right)^{2}\hat{\rho}_{N}^{\eta}\right]+\left(x_{{N,i}}^{\eta}-x_{{i}}^{\eta}\right)^{2}.$
(60)

Therefore, under the conditions of the theorem (Eqs. (24)-(25)), we obtain

$\displaystyle|(\ast 1)|$ $\displaystyle=O\left(N^{-\alpha}l^{\nu}\right).$
(61)

Next, we evaluate $(\ast 2)$ in Eq. (52). Using Hölder’s inequality and Eq.
(50), we obtain

$\displaystyle|(\ast 2)|$
$\displaystyle\leq\left\|\left[\hat{\Gamma}_{N},\hat{A}(\tau)-\tilde{A}(\tau)\right]\hat{\rho}_{N}^{\eta}\right\|_{1}$
$\displaystyle\leq
2\left\|\hat{\rho}_{N}^{\eta}\right\|_{1}\left\|\hat{\Gamma}_{N}\right\|\left\|\hat{A}(\tau)-\tilde{A}(\tau)\right\|=O\left(Nl^{\nu-1}e^{-\mu\left(l-v|\tau|\right)}\right).$
(62)

From the above, the time derivative of the expectation value in the SE can be
evaluated as

$\displaystyle\left|\frac{d}{d\tau}\mathrm{Tr}\left[\hat{A}(\tau)\hat{\rho}_{N}^{\eta}\right]\right|$
$\displaystyle=O\left(N^{-\alpha}l^{\nu}\right)+O\left(Nl^{\nu-1}e^{-\mu\left(l-v|\tau|\right)}\right).$
(63)

Using the triangle inequality for integrals, we have

$\displaystyle\left|\mathrm{Tr}\left[\hat{A}(\tau)\hat{\rho}_{N}^{\eta}\right]-\mathrm{Tr}\left[\hat{A}(0)\hat{\rho}_{N}^{\eta}\right]\right|$
$\displaystyle=O\left(|\tau|N^{-\alpha}l^{\nu}\right)+O\left(|\tau|Nl^{\nu-1}e^{-\mu\left(l-v|\tau|\right)}\right).$
(64)

Then, letting $l=v|\tau|+\frac{2}{\mu}\log N$, we obtain

$\displaystyle\left|\mathrm{Tr}\left[\hat{A}(\tau)\hat{\rho}_{N}^{\eta}\right]-\mathrm{Tr}\left[\hat{A}(0)\hat{\rho}_{N}^{\eta}\right]\right|=\begin{cases}\displaystyle
O\left(|\tau|N^{-\alpha}(\log N)^{\nu}\right)&(v|\tau|\lesssim\log L)\\\ \\\
\displaystyle O\left(|\tau|^{\nu+1}N^{-\alpha}\right)&(\log L\ll v|\tau|\ll
L)\end{cases}.$ (65)

Therefore, we find that Eq. (26) holds for
$|\tau|<T=\Theta(N^{\min[\alpha/(\nu+1),1/\nu]})$. ∎

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