# Analog de Sitter universe in quantum Hall systems with an expanding edge

Yasusada Nambu<EMAIL_ADDRESS>Department of Physics,
Graduate School of Science, Nagoya University, Chikusa, Nagoya 464-8602, Japan
Masahiro Hotta<EMAIL_ADDRESS>Department of Physics, Tohoku
University, Sendai 980-8578, Japan

(January 23, 2023)

###### Abstract

Expanding edges in quantum Hall systems can become a simulator of quantum 1+1
dimensional expanding universes. In these systems, edge exciations are
represented as a chiral scalar field in curved spacetimes. We investigate
Hawking radiation and entanglement behavior predicted by this model assuming
that the expansion law of the edge region corresponds to a de Sitter universe.
As observable quantities for the quantum field, local spatial modes associated
with detection regions are introduced using window functions for the field,
and their correlations are evaluated. We found impact of Hawking radiation
caused by the edge expansion on auto-correlation functions of the local modes,
and confirmed that entanglement death due to Hawking radiation occurs. This
behavior of entanglement is related to “quantum to classical transition” in
cosmic inflations.

##  I Introduction

Analog models of gravitational systems are useful to understand the physics of
black holes and early universe Barcelo2005 . Indeed, although quantum aspects
of black hole evaporation with Hawking radiation has been paid much attention
to theoretically, it is difficult to observe the phenomena in our universe due
to its too low temperature for astrophysical black holes. Concering
cosmological particle creations, it is hard to observe the occurence of this
effect in the early stage of the universe directly. However, considering
condensed matter systems, it is possible to design models with causal horizons
for excitations, which have similar properties as black hole horizons or
cosmological horizons for null rays in general relativity. Thus it is possible
to perform experiments of analog Hawking radiation Steinhauer2016a ;
MunozdeNova2019  and particle creations in expanding universes in
laboratories. Quantum Hall (QH) systems can become one of such analog models
Hegde2019 ; Dalui2020 ; Subramanyan2021 . A QH system emerges when a strong
perpendicular magnetic field is applied to two-dimensional electrons when the
Landau level filling factor becomes an integer or a rational fraction
Yoshioka2002 ; Tong2016 . The QH systems are typical topological materials
consisting of the bulk and edge. In the bulk, the dynamics yields a large
energy gap in its dispersion relation. In the edge, the dispersion relation of
the edge current is protected owing to the topological structure of the
system, and the edge excitations are always gapless. Thus the edge effective
theories are given by free field theories with a chiral condition, and belong
to a class of conformal field theory in 1+1 dimensional spacetime.

Most of all experiments of QH systems have been performed in a static
situation. The electrons are confined in the bulk region by a static electric
field created by the surface potential of host semiconductors of the 2D
electrons and the edge attached to the bulk remains unchanged in time.
Expanding edges were proposed in Hotta2014a  and experimental realization is
on going Kamiyama2022a ; Kamiyama2022b ; the edge expands by gradually
relaxing the external electric fields through continuous electron supply into
the bulk and the excitations moving along the edge are affected by the
expansion. Thus, it is possible to perform experiments of the quantum scalar
field in an analog expanding universe. In our previous paper Hotta2022b , we
formulated a quantum field theory in 1+1 dimensional curved spacetime to
analyze the edge dynamics. It was shown that the expanding edges can be
regarded as expanding universe simulators of two dimensional dilaton-gravity
models, and pointed out that our theoretical setup can simulate the classical
counterpart of an analog Hawking radiation with Gibbons-Hawking temperature
from the future de Sitter horizon formed in the expanding edge region.

In this paper, applying the formulation developed in our previous paper
Hotta2022b , we investigate quantum aspects of the scalar field in an
expanding edge model which reproduces an analog de Sitter universe. In
particular, we focus on particle creation in the analog de Sitter universe
(Hawking radiation), generation of quantum fluctuations by edge expansion, and
their entanglement behavior. We will show that the thermal radiation with the
Gibbons-Hawking temperature from the expanding edge region is created and that
it is detectable in a static edge region. For this purpose, instead of
introducing a specific detector model to measure the Hawking radiation, we
define local spatial modes of the scalar field using window functions and
consider their correlations. Furthermore, we will also investigate the
entanglement between two spatial regions and show that it decreases by Hawking
radiation coming from the expanding edge region. We regard this behavior as a
feature corresponding to the disappearance of quantumness of the primordial
quantum fluctuations expected in cosmic inflations Nambu2008 ; Nambu2011 ;
Matsumura2018 . Thus, our analog de Sitter model can be available as a
simulator of the early universe to explore the generation mechanism and the
feature of primordial quantum fluctuations originated by cosmic inflations.

The plan of the paper is as follows. In Sec.II, we review our setup of the
expanding edge of the QH systems and an analog de Sitter universe. In Sec.
III, we present the behavior of classical wave propagation in the expanding
edge system. In Sec.IV, we formulate the quantum treatment of edge excitations
and investigate the behavior of spatial local modes which are measurable in an
experiment of the QH system. In Sec. V, we discuss entanglement between
spatial modes. Sec. VI is devoted to summary and speculation.

##  II Expanding edge of quantum Hall system

Let us consider a massless scalar field $\varphi$ on the edge of QH systems.
Based on the effective theory of the edge excitations in QH systems
Yoshioka2002 ; Tong2016 , the edge mode is represented by a massless scalar
field $\varphi$ the wavelength of which is 100 times larger than the magnetic
length

$\ell_{B}=\sqrt{\frac{\hbar}{eB}},$ (1)

where $B$ is a perpendicular magnetic field. The edge current and the edge
charge density are given as derivatives of the scalar field. We derive the
wave equation for $\varphi$. The left moving modes and the right moving modes
of $\varphi$ obey

$\partial_{\tau}\varphi_{L}-\frac{v}{a(\tau)}\partial_{x}\varphi_{L}=0,\quad\partial_{\tau}\varphi_{R}+\frac{v}{a(\tau)}\partial_{x}\varphi_{R}=0.$
(2)

where $\tau$ denotes a time variable in a laboratory, $x$ is the comoving
coordinate along the edge and the proper length along the edge is given by
$a(\tau)\int dx$. The scale factor $a(\tau)$ represents the expansion of the
edge. Using the trapping potential $U(y)$ perpendicular to the edges of the QH
system, the propagation speed of the edge excitation $v$ is determined as

$v=\frac{c\,U^{\prime}(y)}{eB}=\frac{cE}{B},$ (3)

where $E$ is the electric field induced by $U$. This propagation speed of the
edge excitation is same as the classical drift velocity of electrons. The
solution of these equations is

$\varphi_{L}=A\left(v\int\frac{d\tau}{a}+x\right),\quad\varphi_{R}=B\left(v\int\frac{d\tau}{a}-x\right),$
(4)

where $A,B$ are arbitrary functions. The scalar field
$\varphi:=\varphi_{L}+\varphi_{R}$ obeys

$\ddot{\varphi}+\frac{\dot{a}}{a}\dot{\varphi}-\frac{v^{2}}{a^{2}}\partial_{x}^{2}\varphi=0,\quad\dot{}=\frac{\partial}{\partial\tau}.$
(5)

This is the Klein-Gordon equation $\square\varphi=0$ in a 1+1 dimensional
expanding universe the metric of which is given by

$ds^{2}=-v^{2}d\tau^{2}+a^{2}(\tau)dx^{2}.$ (6)

The propagation speed of the edge excitation $v$ plays the same role as the
speed of light $c$ in general relativity which determines causal structures of
spacetimes. It is possible to control the expansion law $a(\tau)$ by tuning
the external trapping electric field for the edge region. We can perform
experiments of quantum physics of an early universe using the analog expanding
universe by analysing QH systems with an expanding edge. From now on, we set
$v=1$ and we use $v$ as a unit of length and time in our analog spacetimes. By
introducing the conformal time $t:=\int d\tau/a$ and null coordinates
$x^{\pm}:=t\pm x$, the metric is written as the conformally flat form:

$ds^{2}=-a^{2}dx^{+}dx^{-}.$ (7)

The scalar field is represented as

$\varphi=\varphi_{L}(x^{+})+\varphi_{R}(x^{-}).$ (8)

For a given form of $a(\tau)$ which represents the expansion law of the edge
region, it is possible to identify a corresponding analog universe using the
metric (7). In the QH systems, either $\varphi_{L}$ or $\varphi_{R}$ is
allowed due to the boundary condition of QH systems.

In this paper, we consider an analog de Sitter universe in our setup of the QH
system. Left panel of Fig. 1 depicts a setup of our QH experiment with the
expanding edge of the QH system: the edge system is composed of an input
static region I ($L/2<x$), an expanding region II ($-L/2\leq x\leq L/2$), and
a output static region III ($x<-L/2$). The analog metric of this system is
written as

$\displaystyle ds^{2}=a^{2}(t)(-dt^{2}+dx^{2}),$ $\displaystyle
a(t)=\begin{cases}\dfrac{1}{\cos(Ht)}\theta(t)+\theta(-t)&\quad\text{for}\quad-L/2\leq
x\leq L/2\quad\text{(region II)},\\\ 1&\quad\text{for}\quad
L/2<|x|\quad\text{(region I,III)}.\end{cases}$ (9)

where $\theta(t)$ is the Heaviside function. Using the proper time
$\tau=\int_{0}^{t}dt^{\prime}a(t^{\prime})$, the metric in region II is

$ds^{2}=\begin{cases}-d\tau^{2}+\cosh^{2}(H\tau)dx^{2},&\quad\text{for}\quad\tau\geq
0.\\\ -d\tau^{2}+dx^{2},&\quad\text{for}\quad\tau<0.\end{cases}$ (10)

Thus, we assume a spacetime that is flat Minkowski for $t<0$ and de Sitter
expansion starts at $t=0$ in region II. The global structure of this spacetime
is shown in the right panel of Fig. 1 with the parameter $\pi/4<LH<\pi/2$.
There emerges formation of the future de Sitter horizon $\mathcal{H}^{+}$ in
region II.

Figure 1: Left: schematic picture of the expanding edge system. $x$ denotes a
coordinate along the edge of the QH system. Region I and III are static
Minkowski regions and the expanding region II corresponds to a de Sitter
universe. Right panel: Penrose diagram representing the present setup with the
expanding edge region II (gray region), which starts accelerated expansion at
$t=0$. A and B denote world lines of detectors which perform measurements of
edge excitations. This diagram corresponds to $\pi/4<LH<\pi/2$ case. For
parameter values not included in this range, global structure of the spacetime
becomes different (see Hotta2022b ).

It is possible to obtain a relation between spatial coordinates of region I
and region II explicitly. Null coordinates in region I and region III are
related by the following formula (see detail in Appendix):

$\displaystyle
x_{\text{I}}^{+}=\Phi[-L+\Phi^{-1}[x_{\text{III}}^{+}+\Phi[L/2]]]+L/2=:f(x_{\text{III}}^{+}),$
(11) $\displaystyle
x_{\text{III}}^{-}=\Phi[-L+\Phi^{-1}[x_{\text{I}}^{-}+L/2]]-\Phi[L/2]+L.$ (12)

where the function $\Phi$ is defined by

$\displaystyle\Phi(x)$ $\displaystyle=\int_{0}^{x}dy\,a(y)$
$\displaystyle=\begin{cases}\dfrac{1}{2H}\ln\dfrac{1+\sin Hx}{1-\sin
Hx}&\quad\text{for}\quad 0\leq x<\dfrac{\pi}{2H},\\\ x&\quad\text{for}\quad
x<0.\end{cases}$ (13)

Figure 2: Left panel: the function $\Phi(x)$. $\Phi=+\infty$ at $Hx=\pi/2$.
Right panel: the function $f(x)$.

The inverse function is

$\Phi^{-1}(x)=\begin{cases}\dfrac{1}{H}\,\mathrm{arcsin}\tanh(Hx)&\quad\text{for}\quad
x>0\\\ x&\quad\text{for}\quad x<0\end{cases},$ (14)

and for $x\rightarrow+\infty$,

$\Phi^{-1}(x)\sim\frac{\pi}{2H}-\frac{2}{H}e^{-Hx}.$ (15)

The asymptotic form of the function
$f(x^{+}_{\text{III}})=x^{+}_{\text{I}}[x^{+}_{\text{III}}]$ is

$f(x_{\text{III}}^{+})\sim\begin{cases}c_{0}-c_{1}e^{-Hx_{\text{III}}^{+}}&\quad\text{for}\quad
x_{\text{III}}^{+}\rightarrow+\infty\\\
x_{\text{III}}^{+}&\quad\text{for}\quad
x_{\text{III}}^{+}\rightarrow-\infty\end{cases}$ (16)

where the constants $c_{0}$ and $c_{1}$ are

$c_{0}=\frac{L}{2}-\ln\tan(HL/2),\quad c_{1}=\frac{2}{H\sin
HL}\sqrt{\frac{1-\sin HL/2}{1+\sin HL/2}}.$ (17)

##  III Classical simulation of Hawking radiation

We send plane waves from region I (in-region) and detect them at a point in
region III (out-region). Normalized wave modes in region I and III are

$\varphi_{k}^{\text{(I)}}=\frac{e^{-ikx_{\text{I}}^{+}}}{\sqrt{4\pi
k}}=\frac{e^{-ikf(x_{\text{III}}^{+})}}{\sqrt{4\pi
k}},\quad\varphi_{k}^{\text{(III)}}=\frac{e^{-ikx_{\text{III}}^{+}}}{\sqrt{4\pi
k}},\quad k>0.$ (18)

An input plane wave $e^{-ikx^{+}_{\text{I}}}$ in region I has the wave form
$\exp(-ikf(x^{+}_{\text{III}}))$ in region III. Distortion of plane waves due
to de Sitter expansion of region II is encoded in the function
$f(x_{\text{III}}^{+})$. Figure 3 depicts wave forms in region III. Left panel
shows real part and imaginary part of $\varphi_{k}^{(\text{I})}$ as the
function of $x^{+}_{\text{III}}$. Right panel shows snapshots of wave forms
$\mathrm{Re}~{}\varphi_{k}^{(\text{I})}$ at $t=0,3,6$. We can observe the wave
is stretched by de Sitter expansion in region II.

Figure 3: Wave forms in region III ($L=H=1,k=7$). Left panel : Real part
(blue) and imaginary part (red) of $\varphi$. Right panel: Change of the
spatial profile of waves (real part of $\varphi$) at different times
($t=0,3,6$).

In-mode and out-mode are related by the Bogoliubov transformation

$\displaystyle\varphi_{k}^{\text{(I)}}=\int_{0}^{\infty}dk^{\prime}\left[\alpha(k,k^{\prime})\,\varphi_{k^{\prime}}^{\text{(III)}}+\beta(k,k^{\prime})\,\varphi_{k^{\prime}}^{\text{(III)*}}\right],$
(19)
$\displaystyle\varphi_{k}^{\text{(III)}}=\int_{0}^{\infty}dk^{\prime}\left[\alpha^{*}(k^{\prime},k)\,\varphi_{k^{\prime}}^{\text{(I)}}-\beta(k^{\prime},k)\,\varphi_{k^{\prime}}^{\text{(I)*}}\right],$
(20)

where Bogoliubov coefficients $\alpha$ and $\beta$ are obtained from the
relation

$\frac{e^{-ik^{\prime}f(x_{\text{III}}^{+})}}{\sqrt{k^{\prime}}}=\int_{0}^{\infty}\frac{dk}{\sqrt{k}}\left[\alpha(k,k^{\prime})\,e^{-ikx_{\text{III}}^{+}}+\beta(k,k^{\prime})\,e^{ikx_{\text{III}}^{+}}\right].$
(21)

Thus,

$\alpha(k,k^{\prime})=\frac{1}{2\pi}\sqrt{\frac{k}{k^{\prime}}}\int^{\infty}_{-\infty}dy\,e^{-ik^{\prime}f(y)}\,e^{iky},\quad\beta(k,k^{\prime})=\frac{1}{2\pi}\sqrt{\frac{k}{k^{\prime}}}\int^{\infty}_{-\infty}dy\,e^{-ik^{\prime}f(y)}\,e^{-iky}.$
(22)

Using the asymptotic form (16) of $f(x)$, we have Hotta2022b

$|\beta(k,k^{\prime})|^{2}\sim\begin{cases}0&\quad\text{for}\quad
x_{\text{III}}^{+}\rightarrow-\infty,\\\ \dfrac{1}{2\pi
Hk^{\prime}}\dfrac{1}{\exp(2\pi k/H)-1}&\quad\text{for}\quad
x_{\text{III}}^{+}\rightarrow+\infty.\end{cases}$ (23)

For $x_{\text{III}}^{+}\rightarrow+\infty$, the Bogoliubov coefficient $\beta$
show the Planckian distribution with a temperature

$T_{H}=\frac{H}{2\pi}.$ (24)

This temperature coincides with the Gibbons-Hawking temperature in the de
Sitter spacetime. Thus, it is possible to detect the classical counterpart of
Hawking radiation from the cosmological horizon in a de Sitter universe by
measuring the Fourier component of wave signals in the Minkowski region III.

##  IV Quantum simulation of Hawking radiation

From now on, we consider a quantum scalar field $\hat{\varphi}$ in the
expanding edge system. Our main purpose is to investigate quantum effects of
the edge mode in the quantum Hall system, which is measurable through the
local charge density $\partial_{x^{+}}\hat{\varphi}$ of the edge excitation.

###  IV.1 Correlation functions

Let us consider the setup shown in Fig. 1. We prepare measurement points A and
B in region III. At these points, a part of the signals emitted from
$\mathcal{I}^{-}$ of region I cannot reach region III after the formation of
the future de Sitter horizon $\mathcal{H}^{+}$ in region II. Thus, the
spacetime structure is similar to that with a black hole formation by
gravitational collapse. We consider detection of quantum fluctuations of the
scalar field in region III by imposing the vacuum condition at
$\mathcal{I}^{-}$ in region I:

$\hat{a}_{k}\ket{0_{\text{I}}}=0.$ (25)

Owing to the chirality of the edge mode, we only consider a left moving scalar
field and the field operator in region III is expressed as

$\hat{\varphi}(x^{+})=\int_{0}^{\infty}\frac{dk}{\sqrt{4\pi
k}}\left[\hat{a}_{k}\,e^{-ikf(x^{+})}+\hat{a}_{k}^{\dagger}\,e^{ikf(x^{+})}\right].$
(26)

In our setup, the gauge invariant physical quantity is charge density, which
is given by the derivative of the field operator $\hat{\varphi}$:

$\hat{\Pi}(x^{+}):=\hat{\varphi}^{\prime}(x^{+})=-if^{\prime}(x^{+})\int_{0}^{\infty}dk\sqrt{\frac{k}{4\pi}}\left[\hat{a}_{k}\,e^{-ikf(x^{+})}-\hat{a}_{k}^{\dagger}\,e^{ikf(x^{+})}\right].$
(27)

We investigate quantum effects based on the field operator $\hat{\Pi}$.
Commutators between field operators are

$\displaystyle[\hat{\varphi}(x^{+}),\hat{\varphi}(y^{+})]=-\frac{i}{4}\mathrm{sign}(f(x^{+})-f(y^{+})),$
(28)
$\displaystyle[\hat{\varphi}(x^{+}),\hat{\Pi}(y^{+})]=\frac{i}{2}f^{\prime}(y^{+})\delta(f(x^{+})-f(y^{+})),$
(29)
$\displaystyle[\hat{\Pi}(x^{+}),\hat{\Pi}(y^{+})]=\frac{i}{2}f^{\prime}(x^{+})f^{\prime}(y^{+})\delta^{\prime}(f(x^{+})-f(y^{+})).$
(30)

The Wightman function for $\hat{\varphi}$ is

$\displaystyle D(x_{1}^{+},x_{2}^{+})$
$\displaystyle=\expectationvalue{\hat{\varphi}(x_{1}^{+})\hat{\varphi}(x_{2}^{+})}=\frac{1}{4\pi}\int_{\mu}^{\infty}\frac{dk}{k}e^{-ik(f(x_{1}^{+})-f(x_{2}^{+})-i\Delta
f)},$ $\displaystyle=-\frac{1}{4\pi}\log[\mu(f(x_{1}^{+})-f(x_{2}^{+})-i\Delta
f)],$ (31)

where we introduced an IR cutoff $\mu$ as the lower bound of the integral, and
a UV cutoff $\Delta f>0$ by

$\Delta
f:=f^{\prime}\left((x_{1}^{+}+x^{+}_{2})/2\right)|_{x_{1}^{+}=x_{2}^{+}}\,\epsilon,$
(32)

with the spatial cutoff length $\epsilon$ in the flat region III. In the
Minkowski phase $t<0$, $\Delta f=\epsilon$, and in the de Sitter phase $t\geq
0$, $\Delta f\sim e^{-Ht}\epsilon$ which corresponds to the comoving
wavelength in the de Sitter region II. The local spatial modes prepared in the
Minkowski region III can detect long wavelength quantum fluctuation in the de
Sitter region II. In our analysis, the scalar field $\hat{\varphi}$ is an
effective one and there exists the short-distance cutoff length $\epsilon$
below which effective treatment of the edge mode breaks down. In the QH
systems, this scale corresponds to the magnetic length $\ell_{B}$. We regard
the short-distance cutoff $\epsilon$ as this length in our analysis.

The Wightman function for $\hat{\Pi}$ is

$\displaystyle D_{\Pi}(x_{1}^{+},x_{2}^{+})$
$\displaystyle:=\expectationvalue{\hat{\Pi}(x^{+}_{1})\hat{\Pi}(x^{+}_{2})}=\partial_{x_{1}^{+}}\partial_{x_{2}^{+}}D(x_{1}^{+},x_{2}^{+})$
$\displaystyle=\frac{f^{\prime}(x^{+}_{1})f^{\prime}(x^{+}_{2})}{4\pi}\int_{0}^{\infty}dkk\,e^{-ik(f(x_{1}^{+})-f(x_{2}^{+})-i\Delta
f)}$
$\displaystyle=-\frac{1}{4\pi}\frac{f^{\prime}(x_{1}^{+})\,f^{\prime}(x_{2}^{+})}{\left(f(x_{1}^{+})-f(x_{2}^{+})-i\Delta
f\right)^{2}}.$ (33)

This quantity is independent of the IR cutoff $\mu$. Using (16), the
asymptotic behavior becomes

$D_{\Pi}(x_{1}^{+},x_{2}^{+})\sim-\frac{1}{4\pi}\begin{cases}\dfrac{1}{\left[(2/H)\sinh(H(x_{1}^{+}-x_{2}^{+})/2)-i\epsilon\right]^{2}}&\quad\text{for}\quad
x_{{1,2}}^{+}\rightarrow+\infty\\\
\dfrac{1}{(x_{1}^{+}-x_{2}^{+}-i\epsilon)^{2}}&\quad\text{for}\quad
x_{{1,2}}^{+}\rightarrow-\infty\end{cases}$ (34)

For $x_{1}^{+},x_{2}^{+}\rightarrow+\infty$, $D_{\Pi}$ has the same behavior
as that of a thermal state with the Gibbons-Hawking temperature $T_{H}$.
Correlation functions are

$\displaystyle\expectationvalue{\left\\{\hat{\varphi}(x^{+}),\hat{\varphi}(y^{+})\right\\}}=\frac{1}{2\pi}\int_{\mu}^{\infty}\frac{dk}{k}\cos(k(f(x^{+})-f(y^{+})))e^{-k\Delta
f},$ (35)
$\displaystyle\expectationvalue{\left\\{\hat{\varphi}(x^{+}),\hat{\Pi}(y^{+})\right\\}}=\frac{f^{\prime}(y^{+})}{2\pi}\int_{0}^{\infty}dk\sin(k(f(x^{+})-f(y^{+})))e^{-k\Delta
f},$ (36)
$\displaystyle\expectationvalue{\left\\{\hat{\Pi}(x^{+}),\hat{\Pi}(y^{+})\right\\}}=\frac{f^{\prime}(x^{+})f^{\prime}(y^{+})}{2\pi}\int_{0}^{\infty}dkk\cos(k(f(x^{+})-f(y^{+})))e^{-k\Delta
f}.$ (37)

###  IV.2 Correlation of local spatial mode

We consider the measurement of the field $\hat{\Pi}(x^{+})$ at $x_{\text{A}}$
and $x_{\text{B}}$ in region III. This measurement process can be represented
by the interaction between the field operator $\hat{\Pi}$ and the canonical
variables $(\hat{Q}_{D},\hat{P}_{D})$ of the measurement apparatus. In the
present analysis, we do not specify details of the apparatus. The interaction
Hamiltonian between the field operator and the apparatus is

$H_{\text{int}}=\sum_{j=\text{A,B}}\lambda_{j}(t)g_{j}(\hat{Q}_{D},\hat{P}_{D})\otimes\int
dx\,w_{j}(x)\hat{\Pi}(t+x),$ (38)

where $g_{j}(\hat{Q}_{D},\hat{P}_{D})$ is a function of canonical variables of
the measurement apparatus, $w_{j}(x)$ is a window function defining a spatial
local mode of the field at $x_{\text{A,B}}$, and $\lambda_{j}(t)$ is a
switching function of the interaction. After acting on the apparatus state,
this interaction causes a change of the “reading” of the apparatus depending
on the state of the quantum field $\hat{\Pi}$ at $x_{\text{A,B}}$. In the
present analysis, we do not introduce details of measurement protocols but
just pay attention to the behavior of the local mode of the quantum field
introduced by the spatial window function $w_{\text{A,B}}(x)$.

For the purpose of observing spatial correlations of the field, we define a
canonical pair of variables corresponding to the local spatial mode of the
field at $x_{A}$ and $x_{B}$:

$\hat{Q}_{j}(t)=\int dx\,w_{Q}(x-x_{j})\hat{\Pi}(t+x),\quad\hat{P}_{j}(t)=\int
dx\,w_{P}(x-x_{j})\hat{\Pi}(t+x),\quad j=\text{A,B}.$ (39)

We assume the window functions $w_{P,Q}(x)$ have non-zero values only in a
compact spatial region $x\in[-\ell/2,\ell/2]$. Requiring these variables to be
canonical pairs, equal time commutators between these variables should be

$\displaystyle[\hat{Q}_{j},\hat{P}_{k}]=\frac{i}{2}\int
dx\,w_{Q}(x-x_{j})w_{P}^{\prime}(x-x_{k})\equiv i\delta_{jk},\quad
j,k=\text{A,B},$ (40) $\displaystyle[\hat{Q}_{j},\hat{Q}_{k}]=\frac{i}{2}\int
dx\,w_{Q}(x-x_{j})w_{Q}^{\prime}(x-x_{k})\equiv 0,$ (41)
$\displaystyle[\hat{P}_{j},\hat{P}_{k}]=\frac{i}{2}\int
dx\,w_{P}(x-x_{j})w_{P}^{\prime}(x-x_{k})\equiv 0.$ (42)

These equations provide conditions for window functions and are independent of
the state of the quantum field. Thus, the local spatial modes
$(\hat{Q}_{\text{A}},\hat{P}_{\text{A}})$,
$(\hat{Q}_{\text{B}},\hat{P}_{\text{B}})$ associated with spatial regions A
and B can be introduced by using suitably chosen window functions $w_{Q}(x)$
and $w_{P}(x)$ irrespective of the states the quantum field (Fig. 4). Regions
A and B are assumed to have no overlap and their separation is $dx$. Locality
of the spatial modes is guaranteed if we adopt window functions with compact
support. The center of each region is assumed to be

$x_{\text{A}}=-\frac{L}{2}-\frac{3\ell}{2}-dx,\quad
x_{\text{B}}=-\frac{L}{2}-\frac{\ell}{2},\quad
x_{\text{B}}-x_{\text{A}}=\ell+dx.$ (43)

Figure 4: Setup of defining spatial region A and B. The center of each region
is $x_{\text{A}}=-L/2-3\ell/2-dx$ and $x_{\text{B}}=-L/2-\ell/2$.
$x_{B}-x_{A}=\ell+dx$.

We choose the following window functions in our analysis:

$w_{Q}(x)=\frac{2}{\sqrt{\pi}}\cos\left(\frac{\pi x}{\ell}\right),\quad
w_{P}(x)=\frac{2}{\sqrt{\pi}}\sin\left(\frac{\pi x}{\ell}\right),\quad
x\in[-\ell/2,\ell/2].$ (44)

Figure 5: Spatial profile of adopted window functions.

These window functions satisfy the condition (40) and the window functions
defining the bipartite state for canonical variables
$(\hat{Q}_{\text{A}},\hat{P}_{\text{A}},\hat{Q}_{\text{B}},\hat{P}_{\text{B}})$
do not have spatial overlap for $x_{\text{B}}-x_{\text{A}}\geq\ell$. Equal
time correlations of these canonical variables are

$\displaystyle\langle\hat{Q}_{\text{A}}\hat{Q}_{\text{B}}+\hat{Q}_{\text{B}}\hat{Q}_{\text{A}}\rangle=\int
dxdy\,w_{Q}(x-x_{\text{A}})w_{Q}(y-x_{\text{B}})\expectationvalue{\left\\{\hat{\Pi}(t+x),\hat{\Pi}(t+y)\right\\}},$
(45)
$\displaystyle\langle\hat{P}_{\text{A}}\hat{P}_{\text{B}}+\hat{P}_{\text{B}}\hat{P}_{\text{A}}\rangle=\int
dxdy\,w_{P}(x-x_{\text{A}})w_{P}(y-x_{\text{B}})\expectationvalue{\left\\{\hat{\Pi}(t+x),\hat{\Pi}(t+y)\right\\}},$
(46)
$\displaystyle\langle\hat{Q}_{\text{A}}\hat{P}_{\text{B}}+\hat{P}_{\text{B}}\hat{Q}_{\text{A}}\rangle=\int
dxdy\,w_{Q}(x-x_{\text{A}})w_{P}(y-x_{\text{B}})\expectationvalue{\left\\{\hat{\Pi}(t+x),\hat{\Pi}(t+y)\right\\}}.$
(47)

As the bipartite state $\rho_{\text{AB}}$ defined by these canonical variables
is Gaussian, the state is determined by the covariance matrix

$V_{\text{AB}}=\begin{bmatrix}a_{1}&a_{3}&c_{1}&c_{3}\\\
a_{3}&a_{2}&c_{4}&c_{2}\\\ c_{1}&c_{4}&b_{1}&b_{3}\\\
c_{3}&c_{2}&b_{3}&b_{2}\end{bmatrix},$ (48)

where its components are defined by

$\displaystyle c_{1}$
$\displaystyle=\frac{1}{2}\langle\hat{Q}_{\text{A}}\hat{Q}_{\text{B}}+\hat{Q}_{\text{B}}\hat{Q}_{\text{A}}\rangle,\quad
c_{2}=\frac{1}{2}\langle\hat{P}_{\text{A}}\hat{P}_{\text{B}}+\hat{P}_{\text{B}}\hat{P}_{\text{A}}\rangle,\quad
c_{3}=\frac{1}{2}\langle\hat{Q}_{\text{A}}\hat{P}_{\text{B}}+\hat{P}_{\text{B}}\hat{Q}_{\text{A}}\rangle,\quad
c_{4}=\frac{1}{2}\langle\hat{Q}_{\text{B}}\hat{P}_{\text{A}}+\hat{P}_{\text{A}}\hat{Q}_{\text{B}}\rangle,$
(49) $\displaystyle a_{1}$
$\displaystyle=\langle\hat{Q}^{2}_{\text{A}}\rangle,\quad
a_{2}=\langle\hat{P}^{2}_{\text{A}}\rangle,\quad
a_{3}=\frac{1}{2}\langle\hat{Q}_{\text{A}}\hat{P}_{\text{A}}+\hat{P}_{\text{A}}\hat{Q}_{\text{A}}\rangle,\quad
b_{j}=a_{j}(\text{A}\rightarrow\text{B}).$ (50)

We first show temporal behavior of auto-correlation functions of the local
spatial mode in region III. The behavior of the auto-correlation functions
$a_{1,2,3}(t)$ with different region size $\ell=1,2$ are shown in Fig. 6. We
can observe a signature of de Sitter expansion in region II as a change of
correlations around $0<t<2$.

Figure 6: Behavior of the auto-correlation functions with different region
size $\ell=1$ (blue lines) and $\ell=2$ (red lines). ($H=L=1,\epsilon=0.01$)

These quantities are measurable as output signals of the detector in our QH
experiment. To obtain a qualitative understanding of the behavior of the auto-
correlation functions, we evaluate
$a_{1}(t)=\langle\hat{Q}_{\text{A}}^{2}\rangle$ analytically with a window
function $w_{Q}(x)=w_{0}\,\theta(\ell/2+x)\theta(\ell/2-x)$ where $w_{0}$ is a
normalization constant the value of which is unspecified:

$\displaystyle\langle\hat{Q}_{\text{A}}^{2}\rangle$
$\displaystyle=\frac{1}{4\pi}\int_{0}^{\infty}dkk\left|\int
dx\,w_{Q}(x)f^{\prime}(t+x_{\text{A}}+x)e^{ikf(t+x_{\text{A}}+x)}\right|^{2}e^{-\Delta
f\,k}$
$\displaystyle=\frac{2w_{0}^{2}}{\pi}\int_{0}^{\infty}\frac{dk}{k}\sin^{2}\left[\frac{k}{2}\left(f(t+x_{\text{A}}+\ell/2)-f(t+x_{\text{A}}-\ell/2)\right)\right]e^{-\Delta
f\,k}$
$\displaystyle=\frac{w_{0}^{2}}{4\pi}\ln\left[1+\frac{(f(t+x_{\text{A}}+\ell/2)-f(t+x_{\text{A}}-\ell/2))^{2}}{(f^{\prime}(t+x_{\text{A}})\epsilon)^{2}}\right].$
(51)

Using the asymptotic form (16) of $f(x)$,

$\langle\hat{Q}_{\text{A}}^{2}\rangle\sim\begin{cases}\dfrac{w_{0}^{2}}{4\pi}\ln\left[1+\left(\dfrac{\ell}{\epsilon}\right)^{2}\right]\approx\dfrac{w_{0}^{2}}{2\pi}\ln\left(\dfrac{\ell}{\epsilon}\right)&\quad\text{for}\quad
t\rightarrow-\infty,\\\
\dfrac{w_{0}^{2}}{4\pi}\ln\left[1+\left(\dfrac{\ell}{\epsilon}\right)^{2}\left(\dfrac{\sinh(H\ell/2)}{(H\ell/2)}\right)^{2}\right]\approx\dfrac{w_{0}^{2}}{2\pi}\ln\left(\dfrac{\ell}{\epsilon}\right)+\dfrac{w_{0}^{2}}{2\pi}\ln\left(\dfrac{\sinh(H\ell/2)}{(H\ell/2)}\right)&\quad\text{for}\quad
t\rightarrow+\infty,\end{cases}$ (52)

where we assume that the region size of the detection region is far larger
than the UV cutoff and $\ell/\epsilon\gg 1$. The difference of the auto-
correlation between $t=-\infty$ and $t=+\infty$ is

$a_{1}(+\infty)-a_{1}(-\infty)\sim\frac{w_{0}^{2}}{2\pi}\ln\left[\frac{\sinh(H\ell/2)}{(H\ell/2)}\right].$
(53)

This quantity is independent of the UV cutoff and its amount depends only on
$H\ell$. Hence we expect that it reflects the signature of Hawking radiation
from the de Sitter region. For further understanding of the behavior of
$a_{1}=\langle\hat{Q}_{\text{A}}^{2}\rangle$ we pay attention to its
$\ell$-dependence. We expand $a_{1}(\ell)$ as

$a_{1}(\ell)=\int_{0}^{\infty}dK\,\tilde{a}_{1}(K)e^{iK\ell}.$ (54)

Then, the power spectrum for $a_{1}$ is obtained as

$P(K)=K|\tilde{a}_{1}(K)|,$ (55)

which represents the power of detected signals with wave number $K=2\pi/\ell$
corresponding to the size $\ell$ of the detection region. We see that the
power spectrum shows the Planckian distribution with the temperature
$T_{H}=H/(2\pi)$. Using the asymptotic form of $a_{1}$ in Eq. (52), the
Fourier component of $a_{1}(\ell)$ is

$\tilde{a}_{1}(K)\sim\begin{cases}-\dfrac{i}{k}\left(\gamma+\ln(K\epsilon)-i\pi/2\right)&\quad\text{for}\quad
t\rightarrow-\infty,\\\
\dfrac{H}{2K^{2}}\left[1-i\dfrac{2K}{H}\left(\gamma+\psi(-iK/H)\right)\right]&\quad\text{for}\quad
t\rightarrow+\infty,\end{cases}$ (56)

where $\gamma$ is Euler’s constant and $\psi(x)=\Gamma^{\prime}(x)/\Gamma(x)$
is the poly Gamma function. For $t\rightarrow-\infty$, the power spectrum of
the signal is

$P(K)\sim-\ln(K\epsilon).$ (57)

For $t\rightarrow+\infty$, the power spectrum of the signal is

$\displaystyle P(K)$ $\displaystyle\sim\dfrac{\pi}{e^{2\pi
K/H}-1}\quad\text{for}\quad K<H.$ (58)

Therefore, for long wavelength modes larger than the de Sitter horizon size
$H^{-1}$, the power spectrum observed in region III shows the Planckian
distribution with temperature $T_{H}=H/(2\pi)$ originated from Hawking
radiation in the de Sitter region II. Comparing (57) and (58), the power
$P(t=+\infty)$ is larger than $P(t=-\infty)$ in the long wavelength region
$K<H$ due to the Hawking radiation. This enhancement or amplification of the
power for long wavelength fluctuations larger than the de Sitter horizon
$H^{-1}$ has the same physical origin as that of the generation of primordial
quantum fluctuations in cosmic inflation.

To understand the behavior of Hawking radiation in more detail, we consider
the covariance matrix of a single mode
$(\hat{Q}_{\text{A}},\hat{P}_{\text{A}})$

$V_{\text{A}}=\begin{bmatrix}a_{1}&a_{3}\\\ a_{3}&a_{2}\end{bmatrix},$ (59)

and determinant of this matrix, which is the square of the symplectic
eigenvalue $\nu^{2}$ of the reduced state $\rho_{\text{A}}$. The physical
condition of the state requires $\nu^{2}\geq 1/4$ and $\nu^{2}=1/4$
corresponds to a pure state. As we are considering a single sub-region A of
the entire space, the state $\rho_{\text{A}}$ is mixed and the mixedness
represents the amount of entanglement between A and its complement
$\overline{\text{A}}$. Temporal behavior of $\nu^{2}$ (Fig. 7) shows Hawking
radiation causes an increase of mixedness of the mode A and enhances
entanglement between A and $\overline{\text{A}}$. The mode A and
$\overline{\text{A}}$ constitute a pure two mode squeezed state and its amount
of squeezing and entanglement between A and $\overline{\text{A}}$ increases
due to the Hawking radiation created by the rapid accelerated expansion of the
background space.

Figure 7: Evolution of the symplectic eigenvalue of the state
$\rho_{\text{A}}$. $\nu^{2}-1/4$ represents the mixedness of this state, which
also represents the amount of entanglement between A and
$\overline{\text{A}}$.

##  V Entanglement of spatial modes

We investigate behavior of entanglement between spatial regions A and B in
region III (Fig. 4) using associated spatial local modes
$(\hat{Q}_{\text{A}},\hat{P}_{\text{A}})$ and
$(\hat{Q}_{\text{B}},\hat{P}_{\text{B}})$. We can evaluate entanglement
negativity from the symplectic eigenvalues of the covariance matrix
$V_{\text{AB}}$; the covariance matrix $V_{\text{AB}}$ has two symplectic
eigenvalues $\nu_{\pm}\geq 1/2$, and the partially transposed covariance
matrix $\tilde{V}_{\text{AB}}$ has two symplectic eigenvalues
$\tilde{\nu}_{\pm}$. Based on the positivity criterion of the partially
transposed covariance matrix for bipartite Gaussian states Peres1996 ;
Horodecki1997 ; Simon2000 , a measure of entanglement between A and B is given
by the logarithmic negativity defined as Vidal2002a

$E_{N}:=-\mathrm{min}[\log_{2}(2\tilde{\nu}_{-}),0].$ (60)

For $E_{N}>0$, the bipartite state $\rho_{\text{AB}}$ is entangled and the
logarithmic negativity represents the amount of entanglement between A and B.

###  V.1 Minkowski case

We first show the behavior of entanglement for the case that region II is
static and there is no expanding edge region (Minkowski case). We confirm that
the detection of entanglement between A and B is possible using local modes
defined by window functions (44). The left panel of Fig. 8 shows the
symplectic eigenvalues as a function of separation $dx$ of two regions, and
the right panel of Fig. 8 shows the negativity as a function of $dx$. There
exists a critical separation and below this separation entanglement between A
and B can be detected.

Figure 8: Left panel: separation dependence of symplectic eigenvalues
$\nu_{-}$(blue), $\tilde{\nu}_{-}$(red). Positivity of the bipartite state
$\nu_{-}\geq 1/2$ is preserved for $dx\geq 0$. For $\tilde{\nu}_{-}<1/2$, A
and B are entangled. Right panel: separation dependence of logarithmic
negativity for Minkowski case. The bipartite state $\rho_{\text{AB}}$ becomes
separable for large separation ($\epsilon=0.01,\ell=1,L=1$). The critical
separation depends on the values of the UV cutoff $\epsilon$.

The left panel of Fig. 9 shows the dependence of the cutoff parameter
$\ell/\epsilon$ on negativity with $dx=0$. For large values of the ratio
$\ell/\epsilon$, the bipartite system AB becomes separable and local modes
cannot detect entanglement of the scalar field. It is possible to understand
this behavior from the viewpoint of entanglement monogamy Hiroshima2007 . Let
us focus on the entanglement entropy of region A which is a subsystem of the
entire spatial region. The entanglement entropy for a single Gaussian mode
$(\hat{Q}_{\text{A}},\hat{P}_{\text{A}})$ is given by Holevo2001

$S_{\text{A}}=(\nu+1/2)\log_{2}(\nu+1/2)-(\nu-1/2)\log_{2}(\nu-1/2)$ (61)

with the symplectic eigenvalue $\nu$ of the covariance matrix $V_{\text{A}}$
for the mode A. As is shown in the right panel of Fig. 9, $S_{\text{A}}$
behaves $\propto\log(\ell/\epsilon)$,111This behavior is confirmed numerically
and we do not derive this relation analytically. which is the typical scaling
behavior of entanglement entropy of the massless scalar field in the 1+1
dimensional case Bombelli1986 ; Srednicki1993 . This behavior implies that
entanglement between region A and its complement becomes larger as
$\ell/\epsilon$ increases. Concerning entanglement between A and B, because A
and its complement, and B and its complement become strongly entangled as
$\ell/\epsilon$ becomes larger, owing to the monogamy property of multipartite
entanglement Bengtsson2016 ; Hiroshima2007 , the entanglement between A and B
should become smaller and the entanglement between A and B vanishes above a
some critical value of $\ell/\epsilon$.

Figure 9: Left panel: $\ell/\epsilon$ dependence of negativity with $dx=0$.
Right panel: $\ell/\epsilon$ dependence of entanglement entropy for a single
region A. For $\ell/\epsilon\gg 1$, the entanglement entropy behaves as
$S_{A}\propto\log(\ell/\epsilon)$.

###  V.2 De Sitter case

We move on to the expanding edge case which mimics a de Sitter universe and
consider entanglement between adjacent regions A and B in region III under the
influence of Hawking radiation from the de Sitter region II. Figure 10 shows
the evolution of entanglement between A and B with different sizes of A and B
with $dx=0$. As we can observe from Fig. 10, following the transient change of
negativity during $0<t<3$, which is determined by the shape of the window
function, the negativity becomes asymptotically constant. The final amount of
entanglement is reduced compared to the initial Minkowski value. The reduction
of entanglement depends on the size of spatial regions $\ell$. For
$\ell=1.0-1.4$, a non-zero value of negativity survives at $t=6$. On the other
hand, for sufficiently large region size $\ell\gg H^{-1}$, which corresponds
to detection of long wavelength super horizon fluctuation in the de Sitter
universe, the entanglement between A and B becomes zero after arrival of the
Hawking radiation. This behavior of “entanglement death” is the same as that
confirmed in inflationary models Nambu2008 ; Nambu2011 ; Matsumura2018  and is
responsible for the emergence of classical behavior from quantum fluctuations.
Thus, using our setup of the QH experiment, it is possible to simulate
“classical to quantum transition” of primordial quantum fluctuations in a
laboratory.

Figure 10: Evolution of negativity between regions A and B for the de Sitter
case with different spatial region size $\ell$ ($dx=0,\epsilon=0.01,H=L=1$).
Due to Hawking radiation from the de Sitter region, negativity decreases
around $t=0\sim 2$. The final value of negativity becomes smaller than the
initial negativity in the Minkowski region. For $\ell=1.8,2.0$, the final
value of negativity becomes zero and entanglement death occurs. For
$\ell=1.6$, both death and revival of entanglement are observed.

Figure 11 shows the region size dependence of the negativity at $t=6$. For
$\ell\geq 1.65$, the negativity becomes zero and the two regions A and B
become separable. The quantum correlation between the two regions is lost for
large scales compared to the de Sitter horizon length $H^{-1}$. For these
large scales, spatial correlations between A and B exists as classical
correlations. Therefore, the long wavelength Hawking radiation can be treated
as classical stochastic fluctuations and we can confirm the classicality of
Hawking radiation originated from zero point quantum fluctuations of the
scalar field.

Figure 11: Region size dependence of negativity at $t=6$
($dx=0,L=H=1,\epsilon=0.01$). A and B becomes separable for large scales
$\ell\geq 1.65H^{-1}$.

##  VI Summary and Speculation

We considered the analog de Sitter universe realized by the expanding edge of
a QH system. We investigated the behavior of the chiral massless scalar field
corresponding to an edge excitation, and discussed the detection of Hawking
radiation from the de Sitter region. In our setup of the expanding edge
system, the spacetime structure is similar to that of a black hole formation
via gravitational collapse; the future event horizon is formed and Hawking
radiation with thermal spectrum from the vicinity of the future event horizon
is expected. The entanglement between spatial regions A and B in the flat
region is also evaluated and we found that Hawking radiation from the de
Sitter region reduces pre-existent entanglement before the arrival of Hawking
radiation, and for sufficient large size of detection regions compared to the
Hubble length in de Sitter region, the two regions becomes separable and only
classical correlation survives. This behavior is same as that appearing in
cosmic inflation. To conduct the UV divergence of the quantum scalar field, we
introduced a UV cutoff as the scale at which the effective field treatment of
the edge excitation breaks down. The correlation functions of the local
spatial modes also contain this cutoff dependence and it is possible to
examine the impact of the cutoff on Hawking radiation using our experiment,
which is related to the trans-Planckian problem in black hole evaporation and
cosmic inflations. It is important to investigate how the effective theory for
the edge excitation breaks down below the cutoff length since the deviation
from the effective theory may introduce corrections to the massless Klein-
Gordon equation adopted in this paper. Beside applying the expanding edge of
the QH system as a simulator of the quantum cosmology, it is also possible to
explore physics of fundamental aspects of quantum mechanics and quantum field
theory because this system can provide the squeezed vacuum state by
amplification of the vacuum fluctuations in the expanding edge region. Thus,
investigation of the violation of macro realism (the Legget-Garg inequality
Emary2014a ) with the quantum field and the realization of the quantum energy
teleportation Hotta2014a  are possible.

In this paper, the Hall edges are described by quantum field theory in curved
space in the long wave length regime compared to the magnetic length
$\ell_{B}$. As seen in the above analysis, the edge can be regarded as a fixed
1+1 dimensional universe. It may be interesting to point out a possibility
that the same system can be described by different effective theories of
quantum gravity. For instance, let us consider a static QH system confined in
a circle edge. The edge is regarded as a closed 1+1 dimensional universe, the
spacetime curvature curvature of which vanishes. Since the electrons located
at the edge are in a quantum state with position fluctuation, the edge
fluctuates quantum mechanically. This yields quantum superposition of edge
configurations with different edge lengths. In this sense, quantum universes
with different sizes are quantum mechanically superposed. This suggests a
realization of quantum gravity at the QH edge. Though the precise model for
the static quantum universe has not yet been specified, the classical action
may be given by the following dilaton gravity model:

$S=\int d^{2}x\sqrt{-g(x)}\left(\Phi(x)R(x)+\frac{\Lambda}{\ell_{B}}\right),$
(62)

where $\Lambda$ is a positive constant, $\Phi(x)$ is a real scalar field
referred to as dilaton field, and $R(x)$ is the scalar curvature of the 1+1
dimensional universe. Taking the variation of $S$ with respect to $\Phi(x)$
yields $R(x)=0$ as the equation of motion. Thus, the classical action is
evaluated as

$S_{\text{QG}}=\frac{\Lambda}{\ell_{B}}\int d^{2}x\sqrt{-g(x)}.$ (63)

By using a static configuration of $\varphi$ independent of $t$, let us
parameterize the edge in the $x$-$y$ plane as

$(x,y)=\left(x,\ell_{B}\varphi(x)\right),$ (64)

where the edge fluctuation occurs in the $y$ direction. The induced metric for
the edge is given by

$ds^{2}=-dt^{2}+dx^{2}+dy^{2}=-dt^{2}+h(x)dx^{2},$ (65)

where $h(x)=1+\ell_{B}^{2}(\partial_{x}\varphi(x))^{2}$. Then the value of
$S_{\text{QG}}$ is computed as

$S_{\text{QG}}=\frac{\Lambda}{\ell_{B}}\int
d^{2}x\sqrt{1+\ell_{B}^{2}(\partial_{x}\varphi(x))^{2}}.$ (66)

It may be worth noting that the above value of $S_{\text{QG}}$ can be
reproduced by the classical action of the field theory of QH edges:

$S_{\text{QH}}=\int
d^{2}x\left(N(x)\frac{\Lambda^{2}}{4}+\frac{1}{N}\left((\partial_{x}\varphi)^{2}+\frac{1}{\ell_{B}^{2}}\right)\right),$
(67)

where $N(x)$ is a lapse function. When we take $N(x)=1$, $S_{\text{QH}}$
yields the action of the quantum field theory for the edges. By taking the
variation of $S_{\text{QH}}$ with respect to $N(x)$, we get

$N(x)=\frac{2}{\Lambda\ell_{B}}\sqrt{1+\ell_{B}^{2}(\partial_{x}\varphi)^{2}}.$
(68)

It turns out that substitution of the above $N(x)$ into $S_{\text{QH}}$
reproduces the value of $S_{\text{QG}}$ in (66). Though the correct relation
between the theories with $S_{\text{QG}}$ and $S_{\text{QH}}$ remains vague at
present, it may be interesting to explore the correspondence and quantum
gravity effective theory for the QH edges. In this case, the circular edge
corresponds to a closed universe. The interpretation of the wave functions of
the quantum closed universe, which satisfy the Wheeler-DeWitt equation, can be
developed from the viewpoint of the many-body wave funtions of the QH systems.

###### Acknowledgements.

We would like to thank A. Matsumura, Y. Osawa, Y. Sugiyama and K. Yamamoto for
providing their valuable insight on the subject. This research was supported
in part by a Grant-in-Aid for Scientific Research, Grant No. 21H05188 (M.H.),
No. 21H05182 (M.H.), No. JP19K03838 (M.H.), No. 19K03866 (Y.N.) and No.
22H05257 (Y.N.) from the Ministry of Education, Culture, Sports, Science, and
Technology (MEXT), Japan.

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##  Appendix A Derivation of coordinate transformation

We review coordinate transformation in Hotta2022b . The metric is

$\displaystyle\text{region I: }~{}L/2\leq x_{\text{I}},\quad
ds^{2}=-dx_{\text{I}}^{+}dx_{\text{I}}^{-}$ $\displaystyle\text{region II:
}~{}-L/2\leq x\leq L/2,\quad ds^{2}=-e^{2\Theta(t)}dx^{+}dx^{-}$
$\displaystyle\text{region III: }~{}x_{\text{III}}\leq-L/2,\quad
ds^{2}=-dx_{\text{III}}^{+}dx_{\text{III}}^{-}$

where $e^{\Theta(t)}$ is the scale factor in region II. We consider the
coordinate transformation of the form
$x^{+}=x^{+}[x^{+}_{\text{I,III}}],x^{-}=x^{-}[x^{-}_{\text{I,III}}]$, which
keeps conformal invariance.

Region II and III

We look for coordinates $x^{+}_{\text{III}}[x^{+}],x^{-}_{\text{III}}[x^{-}]$
which cover region II and III. The matching point is
$x^{1}=x^{1}_{\text{III}}=(x^{+}_{\text{III}}-x^{-}_{\text{III}})/2=-L/2$. The
matching condition is

$x^{+}_{\text{III}}[t-L/2]-x^{-}_{\text{III}}[t+L/2]=-L.$ (69)

By taking the derivative with respect to $t$, we obtain

$\frac{dx^{+}_{\text{III}}}{dx^{+}}[t-L/2]=\frac{dx^{-}_{\text{III}}}{dx^{-}}[t+L/2].$
(70)

The matching of the metric between region II and III (at
$x=x_{\text{III}}=-L/2$) is

$e^{2\Theta(t)}\frac{dx^{+}}{dx^{+}_{\text{III}}}\frac{dx^{-}}{dx^{-}_{\text{III}}}=1\quad\therefore\quad\frac{dx^{+}_{\text{III}}}{dx^{+}}[t-L/2]=e^{\Theta(t)}.$
(71)

By shifting the argument of the functions,

$\frac{dx^{+}_{\text{III}}}{dx^{+}}[x^{+}]=e^{\Theta(x^{+}+L/2)},$ (72)

and we obtain

$x^{+}_{\text{III}}[x^{+}]=\int_{0}^{x^{+}}dye^{\Theta(y+L/2)}=\int_{L/2}^{x^{+}+L/2}dy\,e^{\Theta(y)}=\Phi[x^{+}+L/2]-\Phi[L/2],$
(73)

where we have fixed the integration constant such that
$x^{+}_{\text{III}}\propto x^{+}$ for constant $\Theta$ and the function
$\Phi$ is introduced by

$\Phi[x]=\int_{0}^{x}dye^{\Theta(y)}.$ (74)

The coordinate function $x^{-}_{\text{III}}(x^{-})$ can be derived from (69)
as

$\displaystyle x_{\text{III}}^{-}[x^{-}]$
$\displaystyle=x^{+}_{\text{III}}[x^{-}-L]+L=\int_{0}^{x^{-}-L}dy\,e^{\Theta(y+L/2)}+L$
$\displaystyle=\Phi[x^{-}-L/2]-\Phi[L/2]+L.$ (75)

From (73) and (75),

$\displaystyle dx^{+}_{\text{III}}\,dx^{-}_{\text{III}}$
$\displaystyle=e^{\Theta(x^{+}+L/2)}e^{\Theta(x^{-}-L/2)}dx^{+}dx^{-},$ (76)

and the metric in region II can be written as

$\displaystyle ds^{2}_{\text{II}}$ $\displaystyle=-e^{2\Theta(t)}dx^{+}dx^{-}$
$\displaystyle=-\exp\left[2\Theta(t)-\Theta(x^{+}+L/2)-\Theta(x^{-}-L/2)\right]dx^{+}_{\text{III}}\,dx^{-}_{\text{III}}.$
(77)

At the boundary $x^{1}=-L/2$, $x^{+}=t-L/2$ and $x^{-}=t+L/2$. Thus, the
metric becomes the flat form
$ds^{2}_{\text{II}}=-dx^{+}_{\text{III}}\,dx^{-}_{\text{III}}$, and can be
extended to the flat region III using coordinates $x^{\pm}_{\text{III}}$.

Region I and II

The condition of the matching at $x^{1}=L/2$ yields

$x^{+}_{\text{I}}[t+L/2]-x^{-}_{\text{I}}[t-L/2]=L.$ (78)

The coordinate function is

$x_{\text{I}}^{+}[x^{+}]=\int_{0}^{x^{+}}dy\,e^{\Theta(y-L/2)}=\Phi[x^{+}-L/2]-\Phi[-L/2],$
(79)

In the similar way, we obtain

$\displaystyle x_{\text{I}}^{-}[x^{-}]$
$\displaystyle=x^{+}_{\text{I}}[x^{-}+L]-L=\int_{0}^{x^{-}+L}dy\,e^{\Theta(y-L/2)}-L$
$\displaystyle=\Phi[x^{-}+L/2]-\Phi[-L/2]-L.$ (80)

From (79) and (80),

$\displaystyle dx^{+}_{\text{I}}\,dx^{-}_{\text{I}}$
$\displaystyle=e^{\Theta(x^{+}-L/2)}e^{\Theta(x^{-}+L/2)}dx^{+}dx^{-},$ (81)

and the metric in region II becomes

$\displaystyle ds^{2}_{\text{II}}$ $\displaystyle=-e^{2\Theta(t)}dx^{+}dx^{-}$
$\displaystyle=-\exp\left[2\Theta(t)-\Theta(x^{+}-L/2)-\Theta(x^{-}+L/2)\right]dx^{+}_{\text{I}}\,dx^{-}_{\text{I}}.$
(82)

At the boundary $x^{1}=L/2$, $x^{+}=t+L/2$ and $x^{-}=t-L/2$. Thus, the metric
becomes the flat form
$ds^{2}_{\text{II}}=-dx^{+}_{\text{I}}\,dx^{-}_{\text{I}}$, and can be
extended to the flat region III using coordinates $x^{\pm}_{\text{I}}$.

Relation between $x_{\text{I}}$ and $x_{\text{III}}$

Using the function $\Phi$,

$\displaystyle x_{\text{III}}^{+}=\Phi(y^{+}+L/2)-\Phi(L/2),\quad
x_{\text{I}}^{+}=\Phi(y^{+}-L/2)-\Phi(-L/2),$ (83) $\displaystyle
x_{\text{III}}^{-}=\Phi(y^{-}-L/2)-\Phi(L/2)+L,\quad
x_{\text{I}}^{-}=\Phi(y^{-}+L/2)-\Phi(-L/2)-L.$ (84)

By eliminating $y^{\pm}$, we obtain a connection formula between
$x_{\text{I}}$ and $x_{\text{III}}$:

$\displaystyle
x_{\text{I}}^{+}=\Phi[-L+\Phi^{-1}[x_{\text{III}}^{+}+\Phi[L/2]]]-\Phi[-L/2]=:f(x^{+}_{\text{III}}),$
(85) $\displaystyle
x_{\text{III}}^{-}=\Phi[-L+\Phi^{-1}[x_{\text{I}}^{-}+L+\Phi[-L/2]]]-\Phi[L/2]+L.$
(86)

##  Appendix B De Sitter case : global chart

We consider the global de Sitter spacetime. The conformal factor is given by

$e^{\Theta(t)}=\frac{1}{\cos(Ht)},$ (87)

and the function $\Phi$ is

$\Phi(x)=\int_{0}^{x}dye^{\Theta(y)}=\frac{1}{2H}\ln\frac{1+\sin Hx}{1-\sin
Hx},\quad\Phi^{-1}(x)=\frac{1}{H}\mathrm{arcsin}\tanh Hx.$ (88)

Figure 12: The function $\Phi(x)$ and
$x_{\text{I}}^{+}=f(x^{+}_{\text{III}})$.

For $\pi/4<LH<\pi/2$, from the behavior of the function $x_{\text{I}}^{+}$, we
can obtain the global structure as shown in Fig. 13.

Figure 13: Penrose diagram for the global de Sitter case with
$\pi/4<LH<\pi/2$.

Figure 14 shows a wave form of $\exp(-ikf(x^{+}_{\text{III}}))$. As
$x_{\text{III}}$ approaches a finite negative value, the wave length becomes
zero, which reflects a blue-shift of waves by the past horizon
$\mathcal{H}^{-}$. On the other hand, for large value of $x_{\text{III}}^{+}$,
the wave length becomes infinite which reflects a red-shift of waves by the
future horizon $\mathcal{H}^{+}$.

Figure 14: Wave form of $\exp(-ikf(x^{+}_{\text{III}}))$. Real part (blue) and
imaginary part (red) the wave are shown.