aainstitutetext: Graduate School of China Academy of Engineering Physics , P.
R. China

# Entropy Linear Response Theory with Non-Markovian Bath

Yu Chen a<EMAIL_ADDRESS>
###### Abstract

We developed a perturbative calculation for entropy dynamics considering a
sudden coupling between a system and a bath. The theory we developed can work
in general environment without Markovian approximation. A perturbative formula
is given for bosonic environment and fermionic environment, respectively. We
find the Rényi entropy response is only related to the spectral functions of
the system and the environment, together with a specific statistical kernel
distribution function. We find a $t^{2}$ growth/decay in the short time limit
and a $t$ linear growth/decay in longer time scale for second Rényi entropy. A
non-monotonic behavior of Rényi entropy for fermionic systems is found to be
quite general when the environment’s temperature is lower. A Fourier’s law in
heat transport is obtained when two systems’ temperature are close to each
other. A consistency check is made for Sachdev-Ye-Kitaev model coupling to
free fermions, a Page curve alike dynamics is found in a process dual to black
hole evaporation. An oscillation of entanglement entropy is found for a gapped
spectrum of environment.

###### Keywords:

Rényi entropy, non-Markovian, SYK model

##  1 Introduction

Linear response theory (LRT) is essencial for studying quantum matters, it
lays the foundation for physical interpretations of the dynamical response
after a perturbative probe to the systemMahan . LRT is widely used in all
kinds of condensed matter experiments, cold atoms experiments, such as APRES,
conductivity, neutron scattering, etc. In original linear response theory, the
dynamics of physical observables are considered in response to external
sources. These perturbations are hermitian couplings to the systems. However,
when we consider a more general case for dissipative couplings, a new response
theory is needed and established as a Non-hermitian linear response theory
(NHLRT) Hui20 . A relation between dissipative dynamics DissBH  and the
correlation function in initial equilibrium state can be then established with
the help of NHLRT.

However, previous response theories are mostly focused on physical
observables, or for dissipative couplings with Markovian approximations,
entropies response theories for a general environment is still not fully
established. In this paper, we are going to establish a general perturbative
response theory for entropy with a generic bath. Nowadays, thermal entropies
as well as entanglement entropies become increasingly important in studying
eigenstate thermalizationDeutsch91 ; Srednicki94 ; Cardy04 ; Grover18 ,
information scramblingQi16 ; Ruihua17  and quantum gravityTBook . It is in the
core of holographic dualityMaldacena98  which bridges the conformal field
theory and the gravity through Ryu-Takayanagi formulaRT06 ; T07 ; Wall15 . In
addition to these theoretical interests, new technical advances are also
achieved recently in experimental measurement of Rényi entropy (RE) Greiner15
; Greiner16 ; Xinhua17 . All these advances in both the theoretical and
experimental studies make a general response theory for entropy a necessity.

Following the spirit of taking the coupling between the environment and the
system as a perturbation, we can establish a linear response theory of Rényi
entropy for a general system with a sudden coupling to a general environment.
Here we assume a general non-Markovian bath. The term non-Markovian means the
temperature and spectrum of the environment can be arbitrary. Now we summarize
our results as follows. We find that to the lowest order of coupling strength,
the Rényi entropy response is only related to the spectral functions of the
system and the environment, as well as a specific statistical kernel function.
General properties of bosonic systems and fermionic systems are discussed and
we predict a short time quadratic growth (decay) and long time linear growth
(decay).

Further, we apply the linear response theory to Sachdev-Ye-Kitaev (SYK)
modelKitaev15 ; SY ; Maldacena16 ; Kitaev17  with a free fermion bath with
arbitrary different temperatures. SYK model is recently discovered as an
example of holographic dualityKitaev17 ; Maldacena16a ; Gross17 . It is dual
to a Jackiw-Teitelboim (JT) gravityJackiw ; Teitelboim  in AdS2
spacetimeKitaev17 ; Maldacena16 . Since SYK model is a solvable model in large
N limit, quite a few results for SYK model’s entropy dynamics are obtained for
various SYK-like modelsGu17a ; Gu17b ; Balents17 ; Pengfei19 ; Yiming20 ;
Pengfei20a ; Pengfei20b ; Banerjee20 ; Pengfei20c  and for Maldacena-Qi
modelMQ  which is dual to a traversable wormhole. These entropy dynamics are
also calculated in gravity sidePenington19 ; Almheiri19 ; Maldacena20 ;
Pedraza17 ; Yiming19 . Here we try to ask, to what kind of extent could our
perturbative theory apply to a general case. Can we recover the Page curve in
“ black hole ” evaporation casePage01 ; Page02  where SYK model’s initial
temperature is higher than the environment’s temperature. A Page curve is an
entanglement entropy curve of the environment which follows a linear decay
after an initial linear growth. Even through we are calculating black hole’s
entropy, still, we wonder if we can see signals for Page curve. Eventually, we
observed this Page curve like behavior in long time dynamics. Further we can
change a gapless fermion to a gapped fermion to change the environment’s
spectrum dramatically. an interesting phenomenon is observed for RE
oscillation at some special temperature window. It seems that the entangled
information can travel back and forth between the system and the bath in this
situation, which is quite curious.

Our paper is orgarnized as follows. In the next section, we first establish
the general linear response theory for RE of the subsystem. Then we discuss
general properties of RE dynamics for bosonic systems and fermionic systems.
Then in the third section, we apply our theory to SYK model with a non-
Markovian bath and we changed the temperature and the spectrum of environment
to calculate the RE dynamics. Finally, we summarize our results in the
conclusion section.

##  2 General Entropy Response Theory

Here we consider a general situation for a sudden coupling between a system
and a bath. Before the coupling, the hamiltonian is $\hat{H}_{0}=\hat{H}_{\rm
S}+\hat{H}_{\rm E}$, and after the coupling the hamiltonian becomes

$\displaystyle\hat{H}=\hat{H}_{\rm S}+\hat{H}_{\rm E}+\hat{V}.$ (1)

Here
$\hat{V}=g\sum_{j}(\hat{O}_{j}^{\dagger}\hat{\xi}_{j}+\hat{\xi}_{j}^{\dagger}\hat{O}_{j})$,
where $\hat{O}_{j}$ is an operator acting on the system and $\hat{\xi}_{j}$ is
an operator in the environment. $j$ is the index of the operator. We can find
out that the system’s density operator evolutes like follows

$\displaystyle\hat{\rho}_{\rm S}(t)={\rm
Tr}_{E}\left(e^{-i\hat{H}t}\hat{\rho}_{0}^{\rm tot}e^{i\hat{H}t}\right)/{\rm
Tr}(\hat{\rho}_{\rm E}(t)),$ (2) $\displaystyle\hat{\rho}_{\rm E}(t)={\rm
Tr}_{S}\left(e^{-i\hat{H}t}\hat{\rho}_{0}^{\rm tot}e^{i\hat{H}t}\right)/{\rm
Tr}(\hat{\rho}_{\rm S}(t)),$ (3)

where $\hat{\rho}_{0}^{\rm tot}$ is the initial density operator, which we
assume as a product state of thermal states in the system and in the bath.

$\displaystyle\hat{\rho}_{0}^{\rm tot}=\hat{\rho}_{0}\otimes\hat{\rho}_{E}.$
(4)

Because after a unitary evolution, the trace of $\hat{\rho}_{\rm S}(t)$ and
$\hat{\rho}_{\rm E}(t)$ is invariant. Therefore we can simply replace ${\rm
Tr}_{E}(\hat{\rho}_{\rm E}(t))$ to $Z_{\rm E}={\rm Tr}_{E}(\hat{\rho}_{\rm
E}(0))$. Here we employ ${\rm Tr}_{E}$ as a trace over the environment’s
Hilbert space. ${\rm Tr}$ is short for ${\rm Tr}_{S}$ as trace over system’s
Hilbert space.

The reduced density operator of the system is evoluting in an interacting
picture as

$\displaystyle\hat{\rho}_{\rm S}(t)$ $\displaystyle=$ $\displaystyle{\rm
Tr}_{E}\left(e^{i\hat{H}t}e^{-i\hat{H}_{0}t}e^{i\hat{H_{0}}t}\hat{\rho}_{0}\otimes\hat{\rho}_{\rm
E}e^{-i\hat{H}_{0}t}e^{i\hat{H}_{0}t}e^{-i\hat{H}t}\right)/Z_{\rm E}$ (5)
$\displaystyle\equiv$ $\displaystyle{\rm Tr}_{E}\left({\cal
U}(0,t)\hat{\rho}^{\rm I}_{0}(t)\otimes\hat{\rho}_{\rm E}^{\rm I}(t){\cal
U}(t,0)\right)/Z_{\rm E}$

Here we introduce the evolution operator ${\cal U}(0,t)\equiv
e^{i\hat{H}_{0}t}e^{-i\hat{H}t}=\hat{\cal
T}_{t}\exp(-i\int_{0}^{t}\hat{V}^{\rm I}(t^{\prime})dt^{\prime})$, where
$\hat{V}^{\rm I}(t^{\prime})=e^{i\hat{H}_{0}t}\hat{V}e^{-i\hat{H}_{0}t}$ is in
interaction picture, $\hat{\cal T}_{t}$ is time-ordering operator. By
inserting $\hat{\cal U}(0,t)$ and $\hat{\cal U}(t,0)=\hat{\cal
U}(0,t)^{\dagger}$ into Eq. (5), keeping to $g^{2}$ order, we have

$\displaystyle\hat{\rho}_{\rm S}(t)$ $\displaystyle=$ $\displaystyle{\rm
Tr}_{E}\left((1-i\int_{0}^{t}\hat{V}^{\rm
I}(t^{\prime})dt^{\prime}-\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\hat{V}^{\rm
I}(t_{1})\hat{V}^{\rm I}(t_{2}))\hat{\rho}_{0}\otimes\hat{\rho}_{\rm
E}\right.$ (6) $\displaystyle\left.(1+i\int_{0}^{t}\hat{V}^{\rm
I}(t^{\prime})dt^{\prime}-\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\hat{V}^{\rm
I}(t_{2})\hat{V}^{\rm I}(t_{1}))\right)/Z_{\rm E}.$

More explicitly,

$\displaystyle\hat{\rho}_{\rm S}(t)$ $\displaystyle=$
$\displaystyle\hat{\rho}_{0}+g^{2}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\langle\hat{\xi}^{\rm
I}_{j}(t_{1})\hat{\xi}_{\ell}^{{\rm I},{\dagger}}(t_{2})\rangle_{\rm
E}\left(\hat{\cal O}^{\rm I}_{\ell}(t_{2})\hat{\rho}_{0}\hat{\cal O}_{j}^{{\rm
I},{\dagger}}(t_{1})-\hat{\cal O}^{{\rm I},{\dagger}}_{j}(t_{1})\hat{\cal
O}^{\rm I}_{\ell}(t_{2})\hat{\rho}_{0}\right)$ $\displaystyle\hskip
12.70132pt+g^{2}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\langle\hat{\xi}_{\ell}^{{\rm
I},{\dagger}}(t_{2})\hat{\xi}_{j}^{\rm I}(t_{1})\rangle_{\rm E}\left(\hat{\cal
O}^{{\rm I},{\dagger}}_{j}(t_{1})\hat{\rho}_{0}\hat{\cal O}_{\ell}^{\rm
I}(t_{2})-\hat{\cal O}^{\rm I}_{\ell}(t_{2})\hat{\cal O}^{{\rm
I},{\dagger}}_{j}(t_{1})\hat{\rho}_{0}\right)$ $\displaystyle\hskip
12.70132pt+g^{2}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\langle\hat{\xi}^{\rm
I}_{j}(t_{2})\hat{\xi}_{\ell}^{{\rm I},{\dagger}}(t_{1})\rangle_{\rm
E}\left(\hat{\cal O}^{\rm I}_{\ell}(t_{1})\hat{\rho}_{0}\hat{\cal O}_{j}^{{\rm
I},{\dagger}}(t_{2})-\hat{\rho}_{0}\hat{\cal O}^{{\rm
I},{\dagger}}_{j}(t_{2})\hat{\cal O}^{\rm I}_{\ell}(t_{1})\right)$
$\displaystyle\hskip
12.70132pt+g^{2}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\langle\hat{\xi}_{j}^{{\rm
I},{\dagger}}(t_{2})\hat{\xi}_{\ell}^{\rm I}(t_{1})\rangle_{\rm
E}\left(\hat{\cal O}^{{\rm I},{\dagger}}_{\ell}(t_{1})\hat{\rho}_{0}\hat{\cal
O}_{j}^{\rm I}(t_{2})-\hat{\rho}_{0}\hat{\cal O}^{\rm I}_{j}(t_{2})\hat{\cal
O}^{{\rm I},{\dagger}}_{\ell}(t_{1})\right),$

where $\langle\cdot\rangle_{\rm E}={\rm Tr}_{E}(\hat{\rho}_{\rm
E}\cdot)/Z_{\rm E}$ is the environment ensemble average. Operators with upper
index ${\rm I}$ are in interacting picture. By introducing Green’s function
for $\hat{\xi}$ field and Lindbald-like operators $\hat{\cal
L}^{d,e}_{j\ell,{\cal O}}$, we can simplify the above result as

$\displaystyle\hat{\rho}_{\rm
S}(t)=\hat{\rho}_{0}\\!+\\!ig^{2}\\!\\!\iint_{0}^{t}\\!\\!\\!dt_{1}dt_{2}\left(G^{<}_{\xi,j\ell}(t_{1},t_{2})\hat{\cal
L}^{d}_{j\ell,{\cal
O}}(t_{1},t_{2})\\!+\\!G^{>}_{\xi,j\ell}(t_{1},t_{2})\hat{\cal
L}^{e}_{j\ell,{\cal O}}(t_{1},t_{2})\right)\\!\hat{\rho}_{0},$ (7)

Figure 1: Diagrammatic illustration of Eq. (7)

where

$\displaystyle G^{<}_{\xi,j\ell}(t_{1},t_{2})\equiv-i\langle\hat{\xi}_{j}^{\rm
I}(t_{1})\hat{\xi}_{\ell}^{{\rm I},{\dagger}}(t_{2})\rangle_{\rm E},\hskip
12.91663ptG^{>}_{\xi,j\ell}(t_{1},t_{2})\equiv-i\langle\hat{\xi}^{{\rm
I},{\dagger}}_{\ell}(t_{1})\hat{\xi}_{j}^{\rm I}(t_{2})\rangle_{\rm E},$ (8)

$\hat{\cal L}^{d}$ and $\hat{\cal L}^{e}$ are dissipation and enhancement
operators which are defined as

$\displaystyle\hat{\cal L}^{d}_{j\ell,{\cal O}}(t_{1},t_{2})\hat{\rho}_{0}$
$\displaystyle\equiv$ $\displaystyle\hat{\cal O}^{\rm
I}_{\ell}(t_{2})\hat{\rho}_{0}\hat{\cal O}^{{\rm
I},{\dagger}}_{j}(t_{1})\\!-\\!\theta_{12}\hat{\cal O}^{{\rm
I},{\dagger}}_{j}(t_{1})\hat{\cal O}^{\rm
I}_{\ell}(t_{2})\hat{\rho}_{0}\\!-\\!\theta_{21}\hat{\rho}_{0}\hat{\cal
O}^{{\rm I},{\dagger}}_{j}(t_{1})\hat{\cal O}^{\rm I}_{\ell}(t_{2}),$ (9)
$\displaystyle\hat{\cal L}^{e}_{j\ell,{\cal O}}(t_{1},t_{2})\hat{\rho}_{0}$
$\displaystyle\equiv$ $\displaystyle\hat{\cal O}_{\ell}^{{\rm
I},{\dagger}}(t_{2})\hat{\rho}_{0}\hat{\cal O}^{\rm
I}_{j}(t_{1})\\!-\\!\theta_{12}\hat{\cal O}^{\rm I}_{j}(t_{1})\hat{\cal
O}^{{\rm
I},{\dagger}}_{\ell}(t_{2})\hat{\rho}_{0}\\!-\\!\theta_{21}\hat{\rho}_{0}\hat{\cal
O}^{\rm I}_{j}(t_{1})\hat{\cal O}^{{\rm I},{\dagger}}_{\ell}(t_{2}).$ (10)

Here $\theta_{12}=\theta(t_{1}-t_{2})$ is heavyside step function, which is 1
when $t_{1}>t_{2}$, $-1$ when $t_{1}<t_{2}$, and it is $\frac{1}{2}$ when
$t_{1}=t_{2}$. Here we also assumed that $\langle\hat{\xi}(t)\rangle_{\rm
E}=0$. This condition is usually satisfied. Here we assume the vacuum of
environment is not a symmetry broken state of $\hat{\xi}_{j}$.

Now we introduce a diagram to describe above result. Diagrammatically, Eq. (7)
can be expressed as Fig. 1. The diagram rules are the following, we use red
dot to represent $\hat{\cal O}^{\dagger}$, red cross to represent $\hat{\cal
O}$. Here we dropped the subindices of operators $\hat{\cal O}$. Blue dot is
$\hat{\xi}^{\dagger}$ and blue cross is $\hat{\xi}$. The arrow lines are
evolution operators. The half circle represent an evolution in imaginary time
in the system — $e^{-\beta\hat{H}_{\rm S}}$ and the blue full circle is
$e^{-\beta_{\rm E}\hat{H}_{\rm E}}$. A closed circle means trace. Every dashed
line connecting blue dot(cross) and red cross(dot) carries a factor $g$. The
length of lines in x direction represents real time, representing
$e^{-i\hat{H}t}$ when the arrow is pointing at the origin of the circle and
representing $e^{i\hat{H}t}$ in outgoing direction. A closed blue curve
represents $G^{>,<}_{\xi}$. One can find, by applying these rules, the
diagrams given in Fig. 1 recover Eq. (7).

Then we are ready to calculate the Rényi entropy of the system up to $g^{2}$
order. Now we calculate the second Rényi entropy, which is $S_{\rm
RE}^{(2)}=-\log({\rm Tr}(\hat{\rho}_{\rm S}^{2}(t)))$. Here ${\rm Tr}$ is
trace over system. First we calculate ${\rm Tr}(\hat{\rho}_{\rm S}(t)^{2})$,

$\displaystyle{\rm Tr}(\hat{\rho}_{\rm S}(t)^{2})$ $\displaystyle=$
$\displaystyle{\rm
Tr}(\hat{\rho}_{0}^{2})+2ig^{2}\sum_{j,\ell}\iint_{0}^{t}\\!\\!\\!dt_{1}dt_{2}\left(G^{<}_{\xi,j\ell}(t_{1},t_{2}){\rm
Tr}(\hat{\rho}_{0}(\hat{\cal L}^{d}_{j\ell,{\cal
O}}(t_{1},t_{2})\hat{\rho}_{0}))\right.$ (11) $\displaystyle\left.\hskip
116.24963pt+G^{>}_{\xi,j\ell}(t_{1},t_{2}){\rm Tr}(\hat{\rho}_{0}(\hat{\cal
L}^{e}_{j\ell,{\cal O}}(t_{1},t_{2})\hat{\rho}_{0}))\right).$

Above expression can also displayed in diagram language as is shown in Fig. 2.

Figure 2: Perturbation digrams for ${\rm Tr}(\hat{\rho}_{\rm S}^{2}(t))$, up
to $g^{2}$ order. Diagram representation of Eq. (11)

Now we discuss two situations, the first case is $\hat{\cal O}_{j}$ and
$\hat{\xi}_{j}$ are both bosonic annihilation operators. Both in Green’s
function and the $\hat{\cal L}$ operators, there are sub-indices $j$ and
$\ell$. Here we assume $G^{>,<}_{\xi,j\ell}=G^{>,<}_{\xi,j}\delta_{j\ell}$.
Then only $\hat{\cal L}^{d,e}_{j\ell,{\cal O}}$ contributes. In this diagonal
basis, we assume $G^{>,<}_{\xi,j}$ and $\hat{\cal L}^{d,e}_{j,{\cal O}}$ are
$j$ independent, then we denote $G^{>,<}_{\xi,j}$ as$G^{>,<}_{\xi}$ and
$\hat{\cal L}^{d,e}_{j,{\cal O}}$ as $\hat{\cal L}^{d,e}_{{\cal O}}$ for
simplicity. According to the relation between the greater (lesser) Green’s
function and the spectral function, we have

$\displaystyle G_{\xi}^{>}(t_{1},t_{2})=-i\int d\omega^{\prime}{\cal
A}_{\xi}(\omega^{\prime})\frac{1}{e^{\beta_{E}\omega^{\prime}}-1}e^{i\omega^{\prime}(t_{1}-t_{2})},$
(12) $\displaystyle G_{\xi}^{<}(t_{2},t_{1})=-i\int d\omega^{\prime}{\cal
A}_{\xi}(\omega^{\prime})\frac{e^{\beta_{E}\omega^{\prime}}}{e^{\beta_{E}\omega^{\prime}}-1}e^{i\omega^{\prime}(t_{1}-t_{2})},$
(13)

where ${\cal A}_{\xi}(\omega)$ is the spectral function of $\hat{\xi}$ field,
${\cal A}_{\xi}(\omega)=-\frac{1}{\pi}{\rm Im}G^{R}_{\xi}(\omega)$ where
$G_{\xi}^{R}(t)=-i\theta(t)\langle[\hat{\xi}(t),\xi^{\dagger}(0)]\rangle$ is
the retard Green’s function of $\hat{\xi}$ field. $1/\beta_{E}$ is the
environment temperature. Here we can see, a white noise bath can be generated
in two cases. At high temperature, $\beta_{E}$ is close to zero, therefore
when ${\cal A}_{\xi}(\omega)\sim\omega$,
$G_{\xi}^{>,<}\propto\delta(t_{1}-t_{2})$. This is Ohm bath at high
temperature. At low temperature, we find $G^{>}_{\xi}\sim 0$, and a white
noise is generated by a constant density of states ${\cal A}_{\xi}\sim{\rm
const.}$. This is the other case of white noise.

On the other hand, we find ${\rm Tr}(\hat{\rho}_{0}\hat{\cal L}^{d}_{{\cal
O}}(t_{1},t_{2}))={\rm Tr}(\hat{\rho}_{0}\hat{\cal L}^{e}_{{\cal
O}}(t_{2},t_{1}))$ is a two-point Wightman Green’s function,

$\displaystyle{\rm Tr}\left(\hat{\rho}_{0}\hat{\cal
O}^{\dagger}(t_{1})\hat{\rho}_{0}\hat{\cal O}(t_{2})\right)={\rm
Tr}\left(\hat{\rho}_{0}\hat{\cal O}^{\dagger}\hat{\rho}_{0}\hat{\cal
O}(t_{2}-t_{1})\right)=\int\frac{d\omega}{2\pi}{\cal A}_{\cal
O}(\omega)\frac{e^{\beta\omega}}{e^{2\beta\omega}-1}e^{i\omega(t_{2}-t_{1})},$
(14)

where ${\cal A}_{\cal O}(\omega)$ is the spectral function of ${\cal O}$
field. The last equation in above line can be proved by Lehmann spectrum
Theorem. By inserting Eq. (12), Eq. (13) and Eq. (14) into Eq. (11), the
deviation of Renyi entropy $\delta S_{\rm RE}^{(2)}(t)\equiv S_{\rm
RE}^{(2)}(t)+\log({\rm Tr}(\hat{\rho}_{0}^{2}))$ is

$\displaystyle\delta S_{\rm RE}^{(2)}(t)=\frac{2g^{2}N_{j}}{{\rm
Tr}(\rho_{0}^{2})}\int_{0}^{t}dt_{0}d\omega
d\omega^{\prime}\frac{\sin(\omega-\omega^{\prime})t_{0}}{\omega-\omega^{\prime}}{\cal
W}_{\beta\beta_{E}}^{B}(\omega,\omega^{\prime}){\cal
A}_{\xi}(\omega^{\prime}){\cal A}_{\cal O}(\omega)$ (15)

where $N_{j}$ is the mode number, and

$\displaystyle{\cal
W}_{\beta\beta_{E}}^{B}(\omega,\omega^{\prime})=\frac{(e^{\beta\omega}-1)(e^{\beta\omega}-e^{\beta_{E}\omega^{\prime}})}{(e^{2\beta\omega}-1)(e^{\beta_{E}\omega^{\prime}}-1)}.$
(16)

At high temperature limit and ${\cal A}_{\xi}$ being Ohm bath, we have $\delta
S_{\rm RE}^{(2)}\sim\lambda_{0}t$ where $\lambda_{0}=2g^{2}/{\rm
Tr}(\rho_{0}^{2})\int d\omega(e^{\beta\omega}-1){\cal A}_{\cal
O}(\omega)/(e^{\beta\omega}+1)$ is the entropy growth rate. According to this
formula, we learn, the entropy response is only dependent on the distribution
function and the spectrum of bath and the system.

As the entropy response theory at $g^{2}N_{j}$ order is indeed an exponential
contribution in ${\rm Tr}(\hat{\rho}^{2}_{S}(t))$’s perturbation, therefore
the response theory for entropy may have larger applicable region. Actually,
we can apply a perturbation calculation to next order $g^{4}$, where we can
capture how the spectrum function of the system and the environment is changed
by interactions. But these back action effects are not considered in present
calculation.

Quite similarly, we can work out the Rényi entropy response for fermionic bath
and fermionic operator $\hat{\cal O}$, which is

$\displaystyle\delta S_{\rm RE}^{(2)}(t)=\frac{2g^{2}N_{j}}{{\rm
Tr}(\rho_{0}^{2})}\int_{0}^{t}dt_{0}d\omega
d\omega^{\prime}\frac{\sin(\omega-\omega^{\prime})t_{0}}{\omega-\omega^{\prime}}{\cal
W}_{\beta\beta_{E}}^{F}(\omega,\omega^{\prime}){\cal
A}_{\xi}(\omega^{\prime}){\cal A}_{\cal O}(\omega),$ (17)

where

$\displaystyle{\cal
W}_{\beta\beta_{E}}^{F}(\omega,\omega^{\prime})=\frac{(e^{\beta\omega}-1)(e^{\beta\omega}-e^{\beta_{E}\omega^{\prime}})}{(1+e^{2\beta\omega})(1+e^{\beta_{E}\omega^{\prime}})}.$
(18)

Notice that in fermionic case, the Markovian condition is requiring a
frequency independent spectrum together with frequency independent
distribution function. The latter condition is satisfied only for zero and
infinite temperature. Therefore there are two ways to break down Markovian
approximation, one is by finite temperature; another is by adding strong
frequency dependence in the noise spectrum of the environment.

###  2.1 General Features for Bosonic systems

In this section, we first discuss the short time and long time behavior of
entanglement entropy growth for bosonic environment. Then we derive a “
Fourier law ” in heat transport when the system and the environment have small
temperature difference.

First, we discuss the short time and long time dynamics of $\delta S_{\rm
RE}^{(2)}$. When $t$ is small, in a sense ${\cal J}t\ll 1$, ${\cal
J}^{\prime}t\ll 1$ ( ${\cal J}$ and ${\cal J}^{\prime}$ are typical energy
scale for the system and the bath respectively ), we have

$\displaystyle\frac{\sin(\omega-\omega^{\prime})t_{0}}{\omega-\omega^{\prime}}\approx
t_{0}$

For this reason, the short time dynamics is always proportional to $t^{2}$.

$\displaystyle\delta S_{\rm RE}^{(2)}(t)=\frac{1}{2}\kappa_{\rm B}t^{2}$ (19)

where coefficient is

$\displaystyle\kappa_{\rm B}\equiv\frac{2g^{2}N_{j}}{{\rm
Tr}(\rho_{0}^{2})}\int d\omega\int d\omega^{\prime}{\cal
W}_{\beta\beta_{E}}^{B}(\omega,\omega^{\prime}){\cal A}_{\cal O}(\omega){\cal
A}_{\xi}(\omega^{\prime}).$ (20)

Bosonic environment is unique because the chemical potential of Bose
distribution is negative ( although in our formula we take chemical potential
to be zero ), and the frequency of bosonic excitations are always positive.
Therefore when $\beta_{\rm E}>\beta$, and the environment has a lower
temperature, $\kappa_{\rm B}$ is usually negative. That is due to ${\cal
A}_{\cal O}(\omega)$, ${\cal A}_{\xi}(\omega)$ are positive spectral
functions. Further

$\displaystyle{\cal
W}_{\beta\beta_{E}}^{B}(\omega,\omega^{\prime})=\frac{e^{\beta\omega}-e^{\beta_{\rm
E}\omega^{\prime}}}{(e^{\beta\omega}+1)(e^{\beta_{\rm
E}\omega^{\prime}}-1)}\approx\frac{e^{\beta\omega-\beta_{\rm
E}\omega^{\prime}}-1}{e^{\beta\omega}+1}$

As $\omega$, $\omega^{\prime}$ are positive, as long as
$\beta\omega<\beta_{\rm E}\omega^{\prime}$, the factor ${\cal
W}_{\beta\beta_{E}}^{B}(\omega,\omega^{\prime})$ is negative. As long as the
low energy spectrum of the bath and the system are large, then $\kappa_{\rm
B}<0$. Later we will see it is completely different in fermionic bath case.

Now we turn to long time case. Here we assume time is long and ${\cal
J}t_{0}\sim{\cal J}^{\prime}t_{0}\sim 1$, we have

$\displaystyle\frac{\sin(\omega-\omega^{\prime})t_{0}}{\omega-\omega^{\prime}}\sim\delta(\omega-\omega^{\prime})$

Therefore the long time dynamics is approximately

$\displaystyle\delta S_{\rm RE}^{(2)}(t)=\frac{2g^{2}N_{j}}{{\rm
Tr}(\rho_{0}^{2})}t\int d\omega{\cal
W}_{\beta\beta_{E}}^{B}(\omega,\omega){\cal A}_{\xi}(\omega){\cal A}_{\cal
O}(\omega)$ (21)

The sign for the entropy growth rate in long time limit is determined by the
sign of ${\cal W}_{\beta\beta_{E}}^{B}(\omega,\omega)$. ${\cal
W}_{\beta\beta_{E}}^{B}(\omega,\omega)>0$ when $\beta_{\rm E}<\beta$. In this
situation the environment’s temperature is higher than the system. ${\cal
W}_{\beta\beta_{E}}^{B}(\omega,\omega)<0$ when $\beta_{\rm E}>\beta$. In this
situation, the environment has a lower temperature than the system.

Now we are going to derive a Fourier heat transport formula from our Eq. (21).
Now we assume two temperature $T=1/\beta$ and $T_{\rm E}=1/\beta_{\rm E}$ are
close. Here we have taken $k_{\rm B}=1$. Further we assume $\beta_{\rm E}$ is
small, then we have

$\displaystyle{\cal
W}_{\beta\beta_{E}}^{B}(\omega,\omega)=\frac{1}{e^{\beta\omega}+1}\frac{e^{(\beta-\beta_{\rm
E})\omega}-1}{1-e^{-\beta_{\rm E}\omega}}\approx\frac{T_{\rm
E}-T}{T}\frac{1}{(e^{\beta\omega}+1)}$ (22)

Insert Eq. (22) into Eq. (21), we have

$\displaystyle T\delta S_{\rm RE}^{(2)}(t)=\frac{2g^{2}N_{j}}{{\rm
Tr}(\rho_{0}^{2})}\left(\int_{0}^{\infty}d\omega\frac{{\cal A}_{\cal
O}(\omega){\cal A}_{\xi}(\omega)}{e^{\beta\omega}+1}\right)(T_{\rm E}-T)t$
(23)

If we can define $dQ\equiv T\delta S_{\rm RE}^{(2)}(t)$ as some kind of heat,
therefore above equation shows a Fourier law of heat transport. The heat
transport rate is proportional to the temperature difference between the
system and the bath. If we define $F_{\rm Q}=dQ/dt/(T_{\rm E}-T)$, then the
transport coefficient $F_{\rm Q}$ is predicted as

$\displaystyle F_{Q}=\frac{2g^{2}N_{j}}{{\rm
Tr}(\rho_{0}^{2})}\int_{0}^{\infty}d\omega\frac{{\cal A}_{\cal O}(\omega){\cal
A}_{\xi}(\omega)}{e^{\beta\omega}+1}.$ (24)

###  2.2 General Features for Fermionic systems

In this section, we discuss the short time and long time entropy growth for
fermionic bath. Meanwhile we will give the “ heat transport ” coefficient for
fermionic bath.

First we discuss the short time behavior of $\delta S_{\rm RE}^{(2)}(t)$.
Similar to the case of bosonic system, we have

$\displaystyle\delta S_{\rm RE}^{(2)}(t)=\frac{1}{2}\kappa_{\rm F}t^{2},$ (25)

where

$\displaystyle\kappa_{\rm F}=\frac{2g^{2}N_{j}}{{\rm Tr}(\rho_{0}^{2})}\int
d\omega\int d\omega^{\prime}{\cal
W}_{\beta\beta_{E}}^{F}(\omega,\omega^{\prime}){\cal A}_{\cal O}(\omega){\cal
A}_{\xi}(\omega^{\prime}).$ (26)

The only difference with Eq. (20) is the distribution function ${\cal
W}_{\beta\beta_{E}}^{B}$ is replaced by ${\cal W}_{\beta\beta_{E}}^{F}$. For
fermions, negative frequency is meaningful because of the presence of Fermi
level. Here Fermi level is taken to be zero. At low temperature,

$\displaystyle{\cal
W}_{\beta\beta_{E}}^{F}(\omega,\omega^{\prime})\approx\frac{(e^{\beta\omega}-1)(e^{\beta\omega-\beta_{\rm
E}\omega^{\prime}}-1)}{e^{2\beta\omega}+1}$ (27)

It is clear that when $\omega<0$, $\omega^{\prime}>0$, ${\cal
W}_{\beta\beta_{E}}^{F}(\omega,\omega^{\prime})>0$. Therefore $\kappa_{\rm F}$
is usually positive rather than being negative. This is very different from
boson case.

Now we consider the long time behavior of $\delta S_{\rm RE}^{(2)}(t)$. In
long time limit, we have

$\displaystyle\delta S_{\rm RE}^{(2)}(t)=\frac{2g^{2}N_{j}}{{\rm
Tr}(\rho_{0}^{2})}t\int d\omega{\cal
W}_{\beta\beta_{E}}^{F}(\omega,\omega){\cal A}_{\xi}(\omega){\cal A}_{\cal
O}(\omega).$ (28)

One can prove that for $\beta_{\rm E}>\beta$, ${\cal
W}_{\beta\beta_{E}}^{F}(\omega,\omega)<0$ for any $\omega$. Therefore if the
environment has a larger initial temperature, $\delta S_{\rm RE}^{(2)}(t)$
experienced a linear growth and if the environment temperature is lower than
the system, $\delta S_{\rm RE}^{(2)}(t)$ is decreasing linearly in long time
evolution. However, as we state previously, the short time dynamics of low
temperature fermionic bath is an entropy growth, therefore the entropy
dynamics could be non-monotonic. Later in the next section, we will show this
case in a specific example. Here we stress that the kink in $\delta S_{\rm
RE}^{(2)}(t)$ could be very general in fermionic case, not necessarily as an
interpretation of gravity duality.

Finally, we discuss the situation when the system and the bath’s temperature
difference is small. Again a Fourier-alike law can be presented,

$\displaystyle{\cal
W}_{\beta\beta_{E}}^{F}(\omega,\omega)=\frac{e^{\beta\omega}-1}{e^{2\beta\omega}+1}\frac{e^{(\beta-\beta_{\rm
E})\omega}-1}{1+e^{-\beta_{\rm
E}\omega}}\approx\frac{(e^{\beta\omega}-1)\omega\beta\beta_{\rm
E}}{(e^{2\beta\omega}+1)(1+e^{-\beta_{\rm E}\omega})}(T_{\rm E}-T)$

Again we can get a “ Fourier law ” in “ heat transport ” if we define $T\delta
S_{\rm RE}^{(2)}(t)$ as “ heat flow ” — $dQ$. Again we define $F_{\rm
Q}=dQ/dt/(T_{\rm E}-T)$, then

$\displaystyle F_{\rm Q}=\frac{2g^{2}N_{j}}{{\rm
Tr}(\rho_{0}^{2})}\int_{0}^{\infty}d\omega\frac{{\cal A}_{\cal O}(\omega){\cal
A}_{\xi}(\omega)(e^{\beta\omega}-1)\omega\beta_{\rm
E}}{(e^{2\beta\omega}+1)(1+e^{-\beta_{\rm E}\omega})}.$ (29)

One can observe that the temperature dependence in $F_{\rm Q}$ is very
different from bosonic bath case.

##  3 Renyi Entropy Response in SYK model

In this section, we apply the general Rényi entropy response theory to
Sachdev-Ye-Kitaev (SYK) model with an non-Markovian environment. We assume the
system is SYK4 whose hamiltonian is,

$\displaystyle\hat{H}_{\rm
S}=\sum_{jklm}J_{jklm}\hat{\chi}_{j}\hat{\chi}_{k}\hat{\chi}_{l}\hat{\chi}_{m},$
(30)

where $\hat{\chi}_{j}$ is Majorana fermion operator. $J_{jklm}$ are random,
and we have

$\displaystyle\overline{J_{jklm}}=0,\hskip
17.22217pt\overline{J_{jklm}^{2}}=\frac{3!}{N^{3}}{\cal J}^{2},$ (31)

where $\overline{\cdot}$ is disorder average, $N$ is the total modes number.
Throughout our discussion we are in conformal limit of SYK model, so that
$N\gg\beta{\cal J}\gg 1$.

Here we are going to consider two kinds of baths. One bath is a SYK2 model
with zero chemical potential. In this model, when the coupling strength is
large, then the spectral function around zero frequency is a constant. This
kind of bath has a Markovian limit. Another bath is two SYK2 model with
opposite on site energies, which is more like a gapped system and is far away
from Markovian environment. Here we stress that non-Markovian effect comes
from both the frequency dependence in spectral function of the environment as
well as the the distribution function kernel ${\cal W}^{F}$. Therefore we
separate these two factors with a constant spectral function and a strongly
frequency dependent spectral fucntion. In the former, the non-Markovian effect
comes from the distribution function, while in the latter, the non-Markovian
effect comes from the spectral function.

###  3.1 Heating and Evaporation with Markovian-like Environment

Here we assume the environment is random free fermions, SYK2, whose
hamiltonian is

$\displaystyle\hat{H}_{\rm
E}=\sum_{\alpha=1}^{M}\sum_{jk=1}^{N}J^{\prime}_{jk}\hat{\psi}_{j,\alpha}^{\dagger}\hat{\psi}_{k,\alpha},$
(32)

where $\hat{\psi}_{j}$ are complex fermion annihilation operators.
$J^{\prime}_{jk}$ is random, and it satisfies

$\displaystyle\overline{J^{\prime}_{jk}}=0,\hskip
17.22217pt\overline{(J_{jk}^{\prime})^{2}}=\frac{1}{N}({\cal J}^{\prime})^{2}$
(33)

The interaction between the system and the bath is $\hat{V}$,

$\displaystyle\hat{V}=g\sum_{j=1}^{N}\sum_{\alpha=1}^{M}(\hat{\chi}_{j}\hat{\psi}^{\dagger}_{j,\alpha}+\hat{\psi}_{j,\alpha}\hat{\chi}_{j})$
(34)

One advantage is that SYK model is known to have a gravitaional duality. SYK
model’s low energy effective action can be described by a Schwarzian theory,
which is dual to the Jackiw-Teitelboim (JT) gravity theory in AdS2
spacetimeKitaev17 ; Maldacena16 . The present situation of sudden coupling
between the SYK4 and SYK2 model can be viewed as an black hole evaporation
problem if the environment’s temperature is much lower than the black hole. In
gravitational side, it is known that evaporation experience an entanglement
entropy growth and drop, which is well-known as Page curve. The entanglement
entropy calculation based on RT formula is done recentlyPenington19 ;
Almheiri19 ; Maldacena20  towards an understanding of Page curve. Therefore,
according to holographic principle, we expect similar non-monotonic behavior
in SYK4’s entanglement entropy dynamics.

Now we apply the perturbation theory to calculate the Rényi entropy dynamics
after a sudden coupling between SYK4 and SYK2 by $\hat{V}$ to see how the
second Rényi entropy response to the sudden coupling between these two
systems.

From our formula, we find the most important information for Rényi entropy
response is the spectrum of the system and the spectrum of the environment.
These spectrum can be obtained by calculating retard Green’s function.
Compared with the general theory, we find $\hat{\cal O}_{j}=\hat{\chi}_{j}$
and $\hat{\xi}_{j}=\hat{\psi}_{j,\alpha}$. Then the spectral function for
$\hat{\cal O}_{j}$ field is

$\displaystyle{\cal A}_{\cal
O}(\omega)=\frac{b}{\sqrt{\beta}}\frac{\Gamma(\frac{1}{4}-i\beta\omega/2\pi)}{\Gamma(\frac{3}{4}-i\beta\omega/2\pi)},$
(35)

Figure 3: Heating of the “ Black Hole ”. The figure shows the Rényi entropy
response of SYK4 with ${\cal J}=4$ and $\beta=10$ coupling to SYK2 model with
${\cal J}^{\prime}=8$ and $\beta_{\rm E}=2$. A short time $t^{2}$ growth and a
long time $t$ linear growth are shown.

where $b=(1/4\pi{\cal J}^{2})^{1/4}$. The spectral function of the environment
is

$\displaystyle{\cal A}_{\xi}(\omega)=\sqrt{({\cal
J}^{\prime})^{2}-\omega^{2}}.$ (36)

According to the entropy response theory

$\displaystyle\delta S_{\rm RE}^{(2)}(t)=\frac{4g^{2}NM}{{\rm
Tr}(\hat{\rho}_{0}^{2})}\int_{0}^{t/2}dt_{0}\int_{-{\cal J^{\prime}}}^{\cal
J^{\prime}}\\!\\!\\!d\omega^{\prime}\int_{-{\cal J}}^{\cal
J}\\!\\!\\!d\omega\frac{\sin(\omega-\omega^{\prime})t_{0}}{\omega-\omega^{\prime}}{\cal
W}_{\beta\beta_{E}}^{F}(\omega,\omega^{\prime}){\cal A}_{\cal O}(\omega){\cal
A}_{\xi}(\omega^{\prime})$ (37)

In extremely short time,
$\sin(\omega-\omega^{\prime})t_{0}\sim(\omega-\omega^{\prime})t_{0}$,
therefore we have

$\displaystyle\delta S_{\rm RE}^{(2)}(t)=\frac{1}{2}\kappa_{F}t^{2},$ (38)

where $\kappa_{F}=8g^{2}NM/{\rm Tr}(\hat{\rho}_{0}^{2})\int_{-{\cal
J^{\prime}}}^{\cal J^{\prime}}d\omega^{\prime}\int_{-{\cal J}}^{\cal
J}d\omega{\cal W}_{\beta\beta_{E}}^{F}(\omega,\omega^{\prime}){\cal A}_{\cal
O}(\omega){\cal A}_{\xi}(\omega^{\prime})$. In longer time,
$\sin(\omega-\omega^{\prime})t_{0}/(\omega-\omega^{\prime})=\delta(\omega-\omega^{\prime})$,
therefore

$\displaystyle\delta S_{\rm RE}^{(2)}(t)=\lambda t,$ (39)

where $\lambda$ is the entropy rate, whose value is

$\displaystyle\lambda=\frac{4g^{2}NM}{{\rm Tr}(\hat{\rho}_{0}^{2})}\int_{-{\rm
min}({\cal J^{\prime}},{\cal J})}^{{\rm min}({\cal J^{\prime}},{\cal
J})}\\!\\!\\!d\omega{\cal W}_{\beta\beta_{E}}^{F}(\omega,\omega){\cal A}_{\cal
O}(\omega){\cal A}_{\xi}(\omega).$ (40)

Here $\min(a,b)$ takes the smaller one in $a$ and $b$. Here $\lambda$ can be
negative. The general picture for the entanglement entropy growth is a
quadratic growth at early time and then crossover to a linear growth ( or
decrease ) at late time. As the late time behavior can be a decrease of
entropy, therefore we can see the signal of evaporation in our perturbative
calculation even if it is not early time behavior.

Here we stress that for
$\sin(\omega-\omega^{\prime})t_{0}/(\omega-\omega^{\prime})\propto\delta(\omega-\omega^{\prime})$,
we need a really large $t_{0}$. Indeed this approximation is better satisfied
when ${\rm min}({\cal J}^{\prime},{\cal J})t_{0}$ is really large. Therefore
before that time scale, there are always oscillations in $\delta S^{(2)}_{\rm
RE}(t)$, but these oscillations are slow. In long time limit, we find only if
$\beta=\beta_{\rm E}$, we have $\delta S^{(2)}_{\rm RE}=0$. However, in some
period of time, $\delta S^{(2)}_{\rm RE}\sim 0$ for quite remarkable time
scale, which may seems like prethermalization. In this section, we only focus
on the situation where $\beta$ and $\beta_{\rm E}$ has large difference.

Figure 4: Evaporation of "Black Hole", In (a) we show the spectral function of
the system ( SYK4 model , $\beta=2$, ${\cal J}=2$ ) and the environment ( SYK2
model with $\beta=20$, ${\cal J}^{\prime}=2$ ); In (b), we show the Rényi
entropy response of the sudden coupling between SYK model and the environment.
As the environment is at low temperature, it is an evaporation process. We can
see a linear growth follows a quadratic growth, then a sharp linear drop
follows. The shape of the curve is alike the Page curve in black hole
evaporation. $g^{2}NM$ is tuned for showing the figures, but these values are
not important at this level.

In the following, we present the numerical calculation for mainly two kinds of
process. First of all, we discuss the situation of heating where the system is
at low temperature and the environment is at high temperature. As is shown in
Fig. 3, we couple a SYK4 model with ${\cal J}=4$, $\beta=10$ to a SYK2 model
with ${\cal J}^{\prime}=8$ and $\beta_{\rm E}=2$. This situation mimic a Black
Hole coupling to a hot environment. Just as our analysis in the last section,
the Rényi entropy experienced a quadratic growth in the short time limit and
then crossover to a linear growth in longer time scale.

Second, we study a SYK4 model with ${\cal J}=2$, $\beta=2$ coupling to a SYK2
model with ${\cal J}^{\prime}=2$ and $\beta_{\rm E}=20$. The environment is at
low temperature, hence it describes a “ Black Hole ” evaporation process. A
Page curve alike behavior is expected and we find that even if the method is
perturbative, the non-monotonic behavior can be captured as is shown in Fig.
4.

###  3.2 Heating and Evaporation with non-Markovian Environment

Here we assume the environment’s hamiltonian is

$\displaystyle\hat{H}_{\rm
E}=\sum_{\alpha=1}^{M}\sum_{jk=1}^{N}(J^{\prime}_{jk,\alpha
L}-\mu)\hat{\psi}_{j,\alpha L}^{\dagger}\hat{\psi}_{k,\alpha
L}+\sum_{\alpha=1}^{M}\sum_{jk=1}^{N}(J^{\prime}_{jk,\alpha
R}+\mu)\hat{\psi}_{j,\alpha R}^{\dagger}\hat{\psi}_{k,\alpha R},$ (41)

where

$\displaystyle\overline{J^{\prime}_{jk,\alpha L(R)}}=0,\hskip
17.22217pt\overline{(J_{jk,\alpha L(R)}^{\prime})^{2}}=\frac{1}{N}({\cal
J}^{\prime})^{2}.$ (42)

Figure 5: Here we take ${\cal J}=4$, $\beta=10$, ${\cal J}^{\prime}=0.8$. In
(a) we give the spectral function of the system and the environment. The
dashed line is for $\mu=0.$, dot-dashed line is for $\mu=1.2$ and the solid
line is for $\mu=2$. The gap of the spectrum is increasing from $0$ to 1.2. In
(b) we show the Rényi entropy response for $\beta_{\rm E}=2$, $\mu=0.$ as
dashed line; $\mu=1.2$ case as dot-dashed line and $\mu=2.$ as solid line.

The interactions between the system and the environment is

$\displaystyle\hat{V}=g\sum_{j=1}^{N}\sum_{\alpha=1}^{M}\sum_{\sigma=L,R}(\hat{\chi}_{j}\hat{\psi}^{\dagger}_{j,\alpha\sigma}+\hat{\chi}_{j}\hat{\psi}_{j,\alpha\sigma})$
(43)

For our problem, this model change is equivalent to a simple change in
environment’s spectral function.

$\displaystyle{\cal A}_{\xi}(\omega)=\frac{1}{\pi{\cal
J^{\prime}}^{2}}\left(\sqrt{{\cal J}^{\prime,2}-(\omega-\mu)^{2}}+\sqrt{{\cal
J}^{\prime,2}-(\omega+\mu)^{2}}\right)$ (44)

Here we assume $\mu>{\cal J}^{\prime}$, such that we have a gapped system.
Here we take ${\cal J}^{\prime}=0.5$, $\mu=1$. Again we apply Eq. (15), then
we show the numerical results in Fig. 5 . We compared the results with the
same parameter ${\cal J}^{\prime}$ with previous spectral function. For a case
where $\beta_{\rm E}<\beta$, the presence of a gap can speed up the
entanglement entropy decreasing.

Figure 6: In (a) we show the RE response for environment with gapless
spectrum, $\beta=10$, ${\cal J}=4$, $\beta_{\rm E}=4.5$, 8 and 12 respectively
with solid line, dashed line and dot-dashed line, ${\cal J}^{\prime}=0.8$; In
(b) we show an oscillation behavior of $\delta S_{\rm RE}^{(2)}$ when
$\beta_{\rm E}\approx\beta$. Here $\beta$ is fixed as $10$. Here the solid
line has a temperature $\beta_{\rm E}=4.5$. The dashed line is at the
temperature $\beta_{\rm E}=8$ and the dot-dashed line is at the environment
temperature $\beta_{\rm E}=12$.

A very non-trivial and interesting dynamical behavior is observed that the
entropy experiences an oscillation before a late time linear growth. This
situation happens when the environment temperature is close to the system’s
temperature. We find the gap in the spectrum can result non-trivial memory
effect in entropy dynamics where the heat flow direction can change at least
twice. Deeper reasons for the extra oscillation behavior in RE dynamics is
remain to be further understood. By comparing the behavior of the gapless
spectrum and the gapped spectrum under similar circumstance, we find the
gapped spectrum bring the oscillation of RE to shorter time scale.

##  4 Conclusion

In summary, we construct a perturbative method for calculating the Rényi
entropy for a sudden coupling between a system and an environment. On
condition that the coupling between two systems are linear coupling, we
obtained general formula for Rényi entropy response for bosonic systems and
fermionic systems respectively. One can find the entropy response is related
to the spectral function of the system and the environment as well as a
distribution kernel function ${\cal W}^{B,F}$. We find the short time behavior
of Rényi entropy response follows a $t^{2}$ law, and later it follows a $t$
linear law. Fermi statistics can result a non-monotonic behavior in Rényi
entropy dynamics which may have an gravitational explanation as a Page curve
in black evaporation. Further, among two reasons for a bath being non-
Markovian, we find the distribution kernel plays the vital role in the
behavior of entanglement entropy dynamics, while the spetral function of
environment controls the detail behavior of the entropy dynamics. Further
understanding of these perturbative results in gravity language, as well as
the back action of the environment to the sudden coupling of the system are
left for further study.

At the closing stage of this paper, we notice a similar paper on this topic as
arXiv: 2011.09622 by Dadras and KitaevKitaev20 .

###### Acknowledgements.

We thank Pengfei Zhang, Yiming Chen and Hui Zhai for discussions. This work is
supported by NSFC under Grant No. 11734010, Beijing Natural Science Foundation
(Z180013).

## References

  * (1) G. D. Mahan, _Many Particle Physics_. New York and London: Plenum Press (1981).
  * (2) L. Pan, X. Chen, Y. Chen and H. Zhai, _Non-Hermitian Linear Response Theory_ , _Nat. Phys._ 16 (2020) 767-771.
  * (3) R. Bouganne, M. B. Aguilera, A. Ghermaoui, Beugnon, J. and Gerbier, F. _Anomalous decay of coherence in a dissipative many-body system_ , _Nat. Phys._ 16 (2020) 21-25.
  * (4) J.M. Deutsch, _Quantum statistical mechanics in a closed system_ , _Phys. Rev. A_ 43 (1991) 2046.
  * (5) M. Srednicki, _Chaos and quantum thermalization_ , _Phys. Rev. E_ 50 (1994) 888.
  * (6) P. Calabrese and J. Cardy, _Entanglement entropy and quantum field theory_ , _J. Stat. Mech._ 06 (2004) P06002 [hep-th/0405152].
  * (7) J.R. Garrison and T. Grover, _Does a single eigenstate encode the full Hamiltonian?_ , _Phys. Rev. X_ 8 (2018) 021026 [arXiv:1503.00729].
  * (8) P. Hosur, X.L. Qi, D.A. Roberts and B. Yoshida, _Chaos in quantum channels_ , _JHEP_ 02 (2016) 004 [arXiv:1511.04021].
  * (9) R. Fan et al., _Out-of-time-order correlation for many-body localization_ , _Sci. Bull._ 62 (2017) 707 [arXiv:1608.01914].
  * (10) M. Rangamani, and T. Takayanagi, _Holographic Entanglement Entropy_ , Springer (2017).
  * (11) J. Maldacena, _The large-N limit of superconformal Field theories and supergravity_ , _Int. J. Theor. Phys._ 38 (1999) 1113.
  * (12) S. Ryu and T. Takayanagi, _Holographic derivation of entanglement entropy from AdS/CFT_ , _Phys. Rev. Lett._ 96 (2006) 181602 [hep-th/0603001].
  * (13) V.E. Hubeny, M. Rangamani and T. Takayanagi, _A Covariant holographic entanglement entropy proposal_ , _JHEP_ 07 (2007) 062 [arXiv:0705.0016].
  * (14) N. Engelhardt and A. C. Wall, _Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime_ , _JHEP_ 01 (2015) 073 [arXiv:1408.3203].
  * (15) R. Islam et al., _Measuring entanglement entropy in a quantum many-body system_ , _Nature_ 528 (2015) 77.
  * (16) A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, _Quantum thermalization through entanglement in an isolated many-body system_ , _Science_ 353 (2016) 6301.
  * (17) J. Li et al., _Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator_ , _Phys. Rev. X_ 7 (2017) 031011 [arXiv:1609.01246].
  * (18) A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. Choi, V. Khemani, J. Léonard, and M. Greiner, _Probing entanglement in a many-body-localized system_ , _Science_ 364 (2019) 254-260.
  * (19) T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, C. F. Roos, _Probing Rényi entanglement entropy via randomized measurements_ , _Science_ 364 (2019) 260-263.
  * (20) A. Kitaev, _Hidden correlations in the Hawking radiation and thermal noise_ , _talk given at the 2015 Breakthrough Prize Fundamental Physics Symposium_ , November 10, San Francisco U.S.A. (2015).
  * (21) S. Sachdev and J. Ye, _Gapless spin fluid ground state in a random, quantum Heisenberg magnet_ , _Phys. Rev. Lett._ 70 (1993) 3339 [cond-mat/9212030].
  * (22) J. Maldacena and D. Stanford, _Remarks on the Sachdev-Ye-Kitaev model_ , _Phys. Rev. D_ 94 (2016) 106002 [arXiv:1604.07818].
  * (23) A. Kitaev and S.J. Suh, _The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual_ , _JHEP_ 05 (2018) 183 [arXiv:1711.08467].
  * (24) J. Maldacena, D. Stanford and Z. Yang, _Conformal symmetry and its breaking in two-dimensional nearly Anti-de Sitter space_ , _Prog. Theor. Phys._ (2016) 12C104 [arXiv:1606.01857].
  * (25) D. J. Gross, V. Rosenhaus, _The Bulk Dual of SYK: Cubic Couplings_ , _JHEP_ 05 (2017) 092 [arXiv:1702.08016]
  * (26) R. Jackiw, _Lower dimensional gravity_ , _Nucl. Phy. B_ 252 (1985) 343-356.
  * (27) C. Teitelboim, _Gravitation and hamiltonian structure in two spacetime dimensions_ , _Phys. Lett._ 126B (1983) 41-45.
  * (28) Y. Gu, A. Lucas and X.-L. Qi, _Spread of entanglement in a Sachdev-Ye-Kitaev chain_ , _JHEP_ 09 (2017) 120.
  * (29) Y. Huang and Y. Gu, _Eigenstate entanglement in the Sachdev-Ye-Kitaev model_ , _Phys. Rev. D_ 100 (2017) 041901 [arXiv:1709.09160].
  * (30) C. Liu, X. Chen and L. Balents, _Quantum entanglement of the Sachdev-Ye-Kitaev models_ , _Phys. Rev. B_ 97 (2018) 245126 [arXiv:1709.06259].
  * (31) P. Zhang, _Evaporation dynamics of the Sachdev-Ye-Kitaev model_ , _Phys. Rev. B_ 100 (2019) 245104 [arXiv:1909.10637].
  * (32) Y. Chen, X.-L. Qi and P. Zhang, _Replica wormhole and information retrieval in the SYK model coupled to Majorana chains_ , _JHEP_ 06 (2020) 121 [arXiv:2003.13147].
  * (33) P. Zhang, C. Liu and X. Chen, _Subsystem Rényi entropy of thermal ensembles for SYK-like models_ , _SciPost Phys._ 8 (2020) 094 [arXiv:2003.09766].
  * (34) X.-L. Qi, and P. Zhang, _The coupled SYK model at finite temperature_ , _JHEP_ 05 (2020) 129. [arXiv: 2003.03916].
  * (35) A. Haldar, S. Bera and S. Banerjee, _Renyi entanglement entropy of Fermi liquids and non-Fermi liquids: Sachdev-Ye-Kitaev model and dynamical mean field theories_ , _Phys. Rev. Research_ 2 (2020) 033505 [arXiv:2004.04751].
  * (36) P. Zhang, _Entanglement entropy and its quench dynamics for pure states of the Sachdev-Ye-Kitaev model_ , _JHEP_ 06 (2020) 143. [arXiv: 2004.05339]
  * (37) J. Maldacena, X.-L. Qi, _Eternal traversable wormhole_ , arXiv: 1804.00491 (2018).
  * (38) G. Penington, _Entanglement wedge reconstruction and the information paradox_ , arXiv:1905.08255 (2019).
  * (39) A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, _The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole_ , _JHEP_ 12 (2019) 063 [arXiv:1905.08762].
  * (40) A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, _The Page curve of Hawking radiation from semiclassical geometry_ , _JHEP_ 03 (2020) 149 [arXiv:1908.10996].
  * (41) Sagar F. Lokhande, Gerben W.J. Oling and Juan F. Pedraza, _Linear response of entanglement entropy from holography_ , _JHEP_ 10 (2017) 104 [arXiv: 1705.10324].
  * (42) Y. Chen, and P. Zhang, _Entanglement Entropy of Two Coupled SYK Models and Eternal Traversable Wormhole_ , _JHEP_ 07 (2019) 033 [arXiv: 1903.10532].
  * (43) D.N. Page, _Information in black hole radiation_ , _Phys. Rev. Lett._ 71 (1993) 3743 [hep-th/9306083].
  * (44) D.N. Page, _Time Dependence of Hawking Radiation Entropy_ , _JCAP_ 09 (2013) 028 [arXiv:1301.4995].
  * (45) P. Dadras, A. Kitaev, _Perturbative calculations of entanglement entropy_ , arXiv: 2011.09622 (2020).


11institutetext: TU Wien, Faculty of Informatics, Research Unit E-commerce

# Statute-enhanced lexical retrieval of court cases for COLIEE 2022

Tobias Fink  Gabor Recski  Wojciech Kusa  Allan Hanbury

###### Abstract

We discuss our experiments for COLIEE Task 1, a court case retrieval
competition using cases from the Federal Court of Canada. During experiments
on the training data we observe that passage level retrieval with rank fusion
outperforms document level retrieval. By explicitly adding extracted statute
information to the queries and documents we can further improve the results.
We submit two passage level runs to the competition, which achieve high recall
but low precision.

###### Keywords:

Information Retrieval Information Extraction Legal Domain.

##  1 Introduction

In the legal domain, court cases play an unique role as they often contain the
last say on a particular legal subject. This is especially true in Common Law
systems, such as the legal systems of North America, where court cases play a
large role in shaping the law. While statutes are the foundation of the legal
system, it is often necessary to look through precedent court cases for
detailed information that is not available in statutes to reach a decision.
However, not only are court cases long and difficult to read, the number of
potentially relevant court cases is ever increasing. As such, the need for
development of automated methods for retrieval of legal information to aid
legal experts is equally increasing.

The Competition on Legal Information Extraction/Entailment
(COLIEE)111https://sites.ualberta.ca/~rabelo/COLIEE2022/ evaluates legal
information retrieval (IR) systems for a variety of legal retrieval tasks. We
participate in the COLIEE 2022 Task 1, which deals with Canadian law precedent
retrieval (notice cases). We experiment with lexical methods for retrieval,
focusing on ways of improving established methods with domain-specific fine-
tuning. Considering that statutes are still the foundation of the legal
system, we add statute information to the search to focus the models on
information that is typically defining relevancy in the legal domain. Although
not all cases contain statute information, we observe that making use of this
information will overall improve retrieval performance.

##  2 Task Description

In Task 1 of the COLIEE 2022, the goal is to retrieve supporting court cases
(notice cases) for new court cases (query cases). Notice cases can be
understood as precedent cases that are highly relevant for a query case. Each
query case is supported by at least one notice case. For this task, cases from
the Federal Court of Canada are used for both query and notice cases. A
training collection as well as a test collection is provided (see Table 1),
both having their own respective query cases, which are part of the
collection. The training collection provides labels for relevant notice cases
for each query, while the test collection only provides query cases without
labels. Cases have been edited to have references to other cases removed and
replaced by placeholder tokens. The task is to retrieve notice cases from the
test collection using the queries of the test collection. The performance is
measured using F1 score.

Table 1: Training and test collection statistics. Tokens per document and notice cases are per query. | Training | Test
---|---|---
Total Cases | 4415 | 1563
Query Cases | 898 | 300
Max # of tokens | 90567 | 61065
Median # of tokens | 3658 | 3573
Mean # of tokens | 4778 | 4979
Max # of notice cases | 34 | N/A
Median # of notice cases | 3 | N/A
Mean # of notice cases | 4.68 | N/A

The length of the query documents makes the task challenging in a few ways.
Not only are many IR methods better suited for shorter queries, due to the
length of the documents, the relationship between query cases and notice cases
is also difficult to understand without expert knowledge.

##  3 Method

We approach this task with the assumption that there is a topical overlap
between query and notice cases, but that not all parts of a query case are
equally important. It has been shown in past legal retrieval workshops (see
AILA [4, 5], COLIEE [7, 6, 1]) that lexical methods, such as BM25 or IR
language models (LM), yield competitive results, even when compared to newer
neural network based approaches. We build on top of these lexical methods and
adapt them to the legal collections of this task.

###  3.1 Document-Level

First, we experiment with using the models out-of-the-box. We preprocess the
training collection by removing special characters and tokens with two
characters or less and index the documents using Elasticsearch. Numbers are
also removed, except when they are part of a statute section citation. During
indexing, the text is also lowercased, stemmed and stopwords removed,
including the task-specific placeholder tokens. To convert case documents to
queries, we try to extract the most informative terms from the case. As a
naive approach for this term extraction, we calculate the TF-IDF score for
each token in the query case and then use the top $T$ tokens with the highest
score as query terms. We compare the performance of the Elasticsearch
implementations of BM25 and the LM Jelinek Mercer similarity [8], which
calculate a score $s$ for each document. For each query, 100 documents are
retrieved and the precision, recall and F1 scores calculated for each rank.
Although query cases are part of the collection, they are skipped during
retrieval. We perform a random search to find the best hyperparameters for
BM25 ($k1$, $b$) and LM Jelinek Mercer ($\lambda$) as well as $T$. While
searching for the best hyperparameters, we only use the first 700 queries of
the training collection (training set). We determine the best cutoff rank $k$
using the F1 micro-averages for each rank. We use the remaining 198 queries
(dev set) to evaluate the best hyperparameters and cutoff rank $k$.

###  3.2 Passage-Level

Next, we experiment by changing the way how queries are created from query
cases and change how documents are retrieved by using passage level retrieval.
The information on where a passage starts and ends is already present in the
case files and just needs to be utilized. Similar to the lexical baseline of
[2], we split each case $c$ in the collection $C$ into passages
$p_{1},...,p_{n_{c}}$ and index the passages instead of the whole case, using
the same preprocessing method as before. Now, the score $s$ is calculated for
each passage instead of each document. A query case $q$ is also split into
passage queries $pq_{1},...,pq_{n_{q}}$ which retrieve a set of passage level
rankings $R$ with $|R|=n_{q}$. We aggregate the passage level rankings to case
level using Reciprocal Rank Fusion, a method of aggregation that outperforms
other methods, such as Condorcet Fuse or CombMNZ [3]:

$RRFscore(c\in C)=\sum_{r\in R}\frac{1}{k_{rrf}+r(c)}*p_{b}$ (1)

We set $k_{rrf}=60$, the same value as in [3]. We also add a passage boost
factor $p_{b}$ that is set to 1 for now. For this passage level ranking
approach, we again perform the same method as before to find the best
hyperparameters for BM25 ($k1$, $b$) and LM Jelinek Mercer ($\lambda$) as well
as $T$ and $k$, using the same training set / dev set split of queries for
evaluation. For all further experiments, we the values of the hyperparameters
are fixed to the best result of this random search (excluding cutoff rank
$k$).

###  3.3 Statute Field

Finally, we experiment with adding additional domain knowledge to the search
by extracting statute sections mentioned in the case documents and adding them
to the documents explicitly. For this purpose, we scrape the titles of
Canadian rules, regulations, orders and acts from the Canadian Justice Law
Website222https://laws-lois.justice.gc.ca/eng/. This scraped list of titles
also contains parts that would not be typically found in statute citations
(e.g. text fragments that the law has been repealed). Consequently, we clean
the titles by only considering text up to the first mention of regulations,
order, act or rules. Further, since some statutes are only mentioned as
acronyms, we create acronym candidates for each statute by taking the first
upper-case letter of each token in the title. We identify the statutes of a
case based on mentions of titles and generated title acronyms in the text.
Additionally, we use regular expressions to detect statute section numbers in
the text. We map statutes to section numbers by counting the number of
passages in a case where a section number co-occurs with a statute mention,
and then assigning the most frequently co-occurring statute to a section
number.

These extracted statute sections are then added to the case passages and
indexed as an additional statute-section field in Elasticsearch. We combine
the original passage query with the extracted statute sections of the passage
using a compound query. The score for the statute field $s_{statute}$ is
calculated using BM25 and added to the overall score for each passage (the
Elasticsearch default for a compound query), resulting in a new total score
$s_{total}$:

$s_{total}=s+s_{statute}*s_{b}$ (2)

To further control the influence of the statute-section field on the
similarity calculation, we adjust the weight of the similarity score of the
statute-section field with the factor $s_{b}$ (using the ElasticSearch boost
functionality). We assume that query passages that mention statute-sections
are more likely to contain information that is of particular importance for a
case. If the number of statute-sections $s_{n}$ that are present in the query
passage is at least 1, we now set the earlier introduced passage boost factor
$p_{b}$ to the hyperparameter $P_{b}$:

$p_{b}=\begin{cases}P_{b},&\text{if }s_{n}\geq 1\\\
1,&\text{otherwise}\end{cases}$ (3)

The best values for $P_{b}$ and $s_{b}$ are determined using a random search
and $k$ is determined as before.

##  4 Results

Table 2: Results for our experiments using the training set and dev set queries. | Training Set | Dev Set
---|---|---
Method | Precision | Recall | F1 | Precision | Recall | F1
Document level BM25 ($k=7$) | 0.1193 | 0.1944 | 0.1479 | 0.0871 | 0.1825 | 0.1179
Document level LM ($k=8$) | 0.1200 | 0.2200 | 0.1553 | 0.0802 | 0.1892 | 0.1127
Passage level BM25 ($k=8$) | 0.1214 | 0.2226 | 0.1571 | 0.0898 | 0.2116 | 0.1261
Passage level LM ($k=8$) | 0.1210 | 0.2218 | 0.1565 | 0.0993 | 0.2341 | 0.1395
Passage level LM + Statute Field ($k=7$) | 0.1282 | 0.2090 | 0.1589 | 0.1073 | 0.2249 | 0.1453
Table 3: Excerpt of the task 1 ranking showing selected runs and our results. Rank | Team | Run | F1 Score | Precision | Recall
---|---|---|---|---|---
1 | UA | pp_0.65_10_3.csv | 0.3715 | 0.4111 | 0.3389
2 | UA | pp_0.7_9_2.csv | 0.3710 | 0.4967 | 0.2961
3 | siat | siatrun1.txt | 0.3691 | 0.3005 | 0.4782
7 | LeiBi | run_bm25.txt | 0.2923 | 0.3000 | 0.2850
15 | TUWBR | TUWBR_LM_law | 0.2367 | 0.1895 | 0.3151
17 | TUWBR | TUWBR_LM | 0.2206 | 0.1683 | 0.3199

The results of the experiments on the training set and dev set are shown in
Table 2. On document level, BM25 achieved the highest F1 using the parameters
$T=200$, $k1~{}=1.09$, $b~{}=0.99$ while the LM Jelinek Mercer achieved the
highest F1 using $T=200$, $\lambda~{}=0.64$, $b~{}=0.99$. On passage level,
BM25 achieved the highest F1 using the parameters $T=100$, $k1~{}=0.66$,
$b~{}=0.59$ while the LM Jelinek Mercer achieved the highest F1 without
limiting $T$ and $\lambda~{}=0.56$. This means LM uses every token of a
passage as query, but with duplicate tokens removed. In our experiments, all
passage level methods with rank fusion outperform document level retrieval
methods. For this reason we did not continue experimenting on document level.
While passage level BM25 achieved a higher F1 score on our training set, the
LM model performed better on the dev set. The best overall F1 score was
achieved by the LM model with inclusion of the statute field. Especially the
dev set performance could be improved by adding this information.

For the task submission, we submitted two runs, using the Passage level LM
setup as run TUWBR_LM and using the Passage level LM + Statute Field setup as
run TUWBR_LM_LAW. The results for our submitted runs and a selection of top
scoring runs for the task are shown in Table 3. Our methods achieve a high
level of recall but perform poorly regarding precision. However, our runs are
only situated in the bottom half of the F1 score sorted ranking.

One weakness of our method is certainly that our naive term extraction
approach was insufficient. Further, we were only able to produce a ranking of
court cases and determined relevancy based on a fixed cutoff value (rank 7 or
8). Since most query cases cite fewer notice cases than our cutoff value, our
precision is low. However, extracting statute-section information produced
positive results. If we compare our two runs, we can see that adding statute
information can yield a higher precision while recall is only reduced
minimally. We expect that results can be improved further with better
strategies for utilizing this information.

##  5 Conclusion

For the COLIEE 2022 Task 1 case retrieval, we used passage-level LMs to
retrieve notice cases for case queries. Our methods achieved a high recall but
low precision. We showed that a simple method making use of statute-section
mentions in passages can achieve a higher precision with only a minor decrease
in recall. Overall, low precision remained a problem for our methods and they
were outperformed by other methods in the competition. We expect that our
lexical approach could still be improved by different query term extraction
strategies.

Acknowledgments. Project partly supported by BRISE-Vienna (UIA04-081), a
European Union Urban Innovative Actions project.

## References

  * [1] Althammer, S., Askari, A., Verberne, S., Hanbury, A.: DoSSIER@ COLIEE 2021: Leveraging dense retrieval and summarization-based re-ranking for case law retrieval. arXiv preprint arXiv:2108.03937 (2021)
  * [2] Althammer, S., Hofstätter, S., Sertkan, M., Verberne, S., Hanbury, A.: PARM: A Paragraph Aggregation Retrieval Model for Dense Document-to-Document Retrieval. arXiv preprint arXiv:2201.01614 (2022)
  * [3] Cormack, G.V., Clarke, C.L., Buettcher, S.: Reciprocal rank fusion outperforms condorcet and individual rank learning methods. In: Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval. pp. 758–759 (2009)
  * [4] Leburu-Dingalo, T., Motlogelwa, N.P., Thuma, E., Modongo, M.: UB at FIRE 2020 Precedent and Statute Retrieval. In: FIRE (Working Notes). pp. 12–17 (2020)
  * [5] Liu, L., Liu, L., Han, Z.: Query Revaluation Method For Legal Information Retrieval. In: FIRE (Working Notes). pp. 18–21 (2020)
  * [6] Ma, Y., Shao, Y., Liu, B., Liu, Y., Zhang, M., Ma, S.: Retrieving Legal Cases from a Large-scale Candidate Corpus (2021)
  * [7] Rosa, G.M., Rodrigues, R.C., Lotufo, R., Nogueira, R.: Yes, BM25 is a Strong Baseline for Legal Case Retrieval. arXiv preprint arXiv:2105.05686 (2021)
  * [8] Zhai, C., Lafferty, J.: A study of smoothing methods for language models applied to ad hoc information retrieval. In: ACM SIGIR Forum. vol. 51, pp. 268–276. ACM New York, NY, USA (2017), issue: 2