# Expanding Edges of Quantum Hall Systems in a Cosmology Language
– Hawking Radiation from de Sitter Horizon in Edge Modes

Masahiro Hotta<EMAIL_ADDRESS>Department of Physics, Tohoku
University, Sendai 980-8578, Japan  Yasusada Nambu  Graduate School of
Science, Nagoya University, Nagoya 464-8601, Japan
<EMAIL_ADDRESS>Yuuki Sugiyama  Department of Physics,
Kyushu University, 744 Motooka, Nishi-Ku, Fukuoka 819-0395, Japan
<EMAIL_ADDRESS>Kazuhiro Yamamoto  Department of Physics,
Kyushu University, 744 Motooka, Nishi-Ku, Fukuoka 819-0395, Japan  Research
Center for Advanced Particle Physics, Kyushu University, 744 Motooka, Nishi-
ku, Fukuoka 819-0395, Japan<EMAIL_ADDRESS>Go Yusa
<EMAIL_ADDRESS>Department of Physics, Tohoku University, Sendai 980-8578,
Japan  Center for Spintronics Research Network, Tohoku University, Sendai
980-8577, Japan

###### Abstract

Expanding edge experiments are promising to open new physics windows of
quantum Hall systems. In a static edge, the edge excitation, which is
described by free fields decoupled with the bulk dynamics, is gapless, and the
dynamics preserve conformal symmetry. When the edge expands, such properties
need not be preserved. We formulate a quantum field theory in 1+1 dimensional
curved spacetimes to analyze the edge dynamics. We propose methods to address
the following questions using edge waveforms from the expanding region: Does
the conformal symmetry survive? Is the nonlinear interaction of the edge
excitations induced by edge expansion? Do the edge excitations interact with
the bulk excitations? We additionally show that the expanding edges can be
regarded as expanding universe simulators of two-dimensional dilaton-gravity
models, including the Jackiw-Teitelboim gravity model. As an application, we
point out that our theoretical setup might simulate emission of analog Hawking
radiation with the Gibbons-Hawking temperature from the future de Sitter
horizon formed in the expanding edge region.

##  I Introduction

How did our universe develop from its earliest moments? To answer this major
question of quantum cosmology, great efforts have been made both in theory and
in the attempts of verifying the theory by observations of the cosmic
background radiation and black holes. An ideal scenario would be the
reproduction of the origin and evolution of the universe or producing black
holes in a laboratory and thereby experimentally verifying the theory through
controlling key parameters. Regarding this, analog experiments have been
performed in various systems [1, 2, 3, 4].

Quantum Hall (QH) systems are unique and promising. A QH system [5, 6] emerges
when a strong perpendicular magnetic field ($B$) is applied to two-dimensional
(2D) electrons when the Landau level filling factor, $\nu=2\pi\hbar
n_{e}/(eB)$, becomes an integer or a rational fraction, where $\hbar$ and $e$
are the reduced Planck constant and the elementary charge, respectively.

The theoretical study of QH systems has contributed to modern understanding of
condensed matter physics. Notably, the QH systems are regarded as typical
topological materials consisting of the bulk and edge. The dynamics in the
bulk yield a large energy gap in its dispersion relations. In the bulk, there
exist various gapped excitations such as magneto-rotons [7, 8, 9, 10] and
anyonic quasiparticles [11, 12, 13]. The dispersion relation of the edge
currents are firmly protected owing to the topological structure of the
systems, and the edge excitations are always in gapless modes. The effective
theory of the bulk is given by a topological field theory referred to as the
Chern-Simons gauge theory [14, 15, 16, 17, 18]. In the topological field
theory, no local dynamics appears, and properties of the system are stable
under local perturbations in the bulk. The bulk action is not invariant under
the gauge transformation at the edge. This unsatisfactory gauge-variant
behavior of the bulk is compensated by adding a gauge-variant edge action with
quantum anomaly. Hence, this predicts the edge dynamics [19]. It is possible
to express the same edge degrees of freedom by both a fermionic field and
bosonic field via the statistics transmutation in one-dimensional space. It is
known that the bosonic field corresponds to the charge density of the edge
current, and can be directly observed by measuring the voltage deviation in
the edge experiments [10, 20, 21, 22, 23, 24, 25]. The edge effective theories
are given by free field theories with a chiral condition. Here, the chiral
condition implies that the excitations move in one direction along the edge
and do not return to the upstream region. The theories belong to a class of
conformal field theory (CFT) in 1+1 spacetime dimensions, which preserves an
infinite dimensional conformal symmetry, referred to as the Virasoro symmetry
[26]. The total gauge symmetry of the bulk-edge composite system ensures the
gapless property of the edge excitations.

Until now, all experiments of QH systems have been performed in a static
situation, except for local edge excitations [10, 20, 21, 22, 23, 24]. The
electrons are confined in the bulk region by the static electric field created
by surface potential of host semiconductors of the 2D electrons. Thus, the
edge attached to the bulk remains unchanged in time. In reference [27],
expanding edges were proposed. The edge expands by gradually relaxing the
external electric fields through continuous electron supply into the bulk. The
excitations moving along the edge are affected by the expansion. In such
situations, it is quite nontrivial that the conformal symmetry still holds.
Recall that the static conformal symmetry is protected by topological
properties of QH systems which are insensitive to microscopic details. It is
known that such topologically robust phenomena are not limited to static
systems but may also emerge in some periodically-driven quantum systems [28].
From this viewpoint, it can be expected that the conformal symmetry still
survives in the expanding QH edges. On the other hand, there exists a risk of
terminating the conformal symmetry in the realistic experiments. It is
considered that the conformal symmetry does not survive in some 1+1
dimensional acoustic systems described by minimally-coupled massless scalar
field [29]. Also the conformal symmetry can be broken by inhomogeneity induced
by impurities and/or time-dependence of the realistic QH systems. About the
effective field theory for expanding edge systems, it should be stressed that
answers to the following fundamental questions are unknown at present:

  1. 1.

Does the edge current remain gapless in the expansion? Specifically, is the
conformal symmetry preserved at the boundary?

  2. 2.

Beyond the free field theory, does nontrivial interaction among edge
excitations emerge in the expansion?

  3. 3.

During the expansion, do excitations in the edge interact with gapped
excitations in the bulk?

The expanding edge experiments will provide crucial results for the answers to
the above questions. Note that the experiments are able to achieve this aim
even in classical regimes of the systems. Thus the experiments are quite
promising to reveal new physics of the systems under such dynamical
backgrounds.

In quantum regimes with extremely low noise, QH edges may be applied to future
development in quantum information science. The edge excitations transport
quantum states, i.e., quantum information along an edge connected between
distant points inside a quantum device. This means that the QH edges play a
role of quantum channels. In such situations, the expanding region provides a
useful quantum gate referred to as quantum squeezing for the quantum states
[27]. In addition, measurement of zero-point fluctuation at the downstream of
the edge and its feedback to the upstream are capable of extracting local
zero-point energy at the upstream accompanied by generation of a negative
energy density region in the edge [25]. The protocols are referred to as
quantum energy teleportation (QET) [30, 31, 32]. The inevitable energy cost
$E_{\text{in}}$ of the measurement at the downstream is regarded as the input
energy of QET, and the extracted positive energy $E_{\text{out}}$ at the
upstream is regarded as the output energy of QET satisfying
$0<E_{\text{out}}<E_{\text{in}}$. It is expected that long-distance QET may be
attained utilizing quantum entanglement of the expanding edge systems [27].

The proposed experiment of expanding edges in [27] will also have large
implications to cosmology. The expanding edge can be regarded as a simulator
for the expanding universes in 1+1 spacetime dimensions. In particular, the
charge density of the edge current is capable of playing the analog role of a
quantum field in curved spacetimes. In curved spacetime field theories, many
intriguing phenomena are predicted [33]. One of these phenomena is the Hawking
radiation emission out of black hole horizons [34]. The black hole evaporation
induced by Hawking radiation has not yet been confirmed in astrophysical
observations, but many believe that black holes are evaporating. If a black
hole completely evaporated and its Hawking radiation remained in space, the
thermal radiation would be in a mixed state with a single parameter, that is,
its temperature. Thus, at a glance, the radiation does not appear to carry the
huge amount of information originally stored inside black holes. Where has the
information gone? Suppose that the initial state of the black hole is a pure
state and that the evolution to the mixed state is realized resulting in a
loss of quantum coherence without any environmental interaction. Then
unitarity would be broken, which is one of fundamental laws of quantum
mechanics. This unresolved problem is referred to as the information loss
problem.

In the exploration of information loss, crucial missing links of Hawking’s
original analysis are known [36]. For example, his calculation is only semi-
classical. The matter is quantized, but the spacetime remains classical in his
analysis. Thus, his theory does not cover the final burst of black holes. In
the theory of general relativity for classical spacetime, a curvature
singularity appears and loses predictability by the theory after the burst. To
avoid this flaw, quantum theory is required, which can treat quantum
spacetimes with no singularities. Although several candidates such as string
theory exist, quantum gravity theory remains elusive owing to the lack of
strong guiding principles, actual measurements, and convincing observations to
determine the theory uniquely. To resolve the information loss, any findings
and inspiration, which are delivered by analog black hole experiments in
condensed matter physics including QH systems, are considered useful [37].

Such analog experiments also shed light on the trans-Planckian problem [38].
The typical wavelength of a thermal quantum particle emitted from a black hole
with mass $M$ is computed as $\lambda=O\left(GM/c^{2}\right)$, using the
Hawking temperature $T=\hbar c^{3}/(8\pi k_{B}GM)$ [34]. Here, $c$ is the
velocity of light, $G$ is the gravitational constant, and $k_{B}\,$ is the
Boltzmann constant. The particle experiences severe red shift by gravitational
potential when it propagates to spatial infinity. At the previous point near
the horizon where the particle is created, the wavelength of the particle mode
is much shorter than the Planck length $l_{P}=\sqrt{G\hbar/c^{3}}$. Thus, a
precise description of the mode in the past regime is required for fundamental
microscopic theories such as string theory. It is possible that Hawking’s
approximation is incorrect and has serious discrepancy. However, his results
appear to be correct. In reality, the formula for Hawking temperature $T$ and
derived entropy $S=k_{B}c^{3}\mathscr{A}/(4G\hbar)~{}$for black holes with
horizon area $\mathscr{A}$ are consistent with the other theoretical results
of generalized thermodynamics [39, 40] and statistical mechanics with state
counting in string theory [41]. Thus, why Hawking’s analysis works so well
despite being semi-classical is a mystery. This is called the trans-Planckian
problem.

We can reconsider the trans-Planckian problem in condensed matter physics. If
Hawking’s prediction is correct, analog black holes also emit Hawking
radiation, no matter what the analog black holes are made of [42]. Analog
systems corresponding to black holes possess natural cutoff length, just like
the magnetic length $l_{B}=\sqrt{\hbar/(eB)}$ of electron in QH systems. This
phenomenon poses an analog version of the trans-Planckian problem by regarding
$l_{B}$ as $l_{P}$. Thus, when the original problem is considered, it is
significant to analyze the analog problem. Notably, the trans-Planckian
problem also appears in an inflationary expanding universe owing to the
existence of the cosmological horizon in the universe. Hawking radiation is
emitted out of the horizon and provides nondecaying thermal fluctuation even
in the extreme expansion of the universe. Therefore, expanding edge
experiments using QH systems are also appropriate for exploration of the
corresponding trans-Planckian problems associated with expanding universe.

In the inflationary universe, there exists another conceptual issue related to
quantum field theory in the expanding universe. The accelerated expansion in
the inflationary universe provides large-scale quantum fluctuations of the
inflaton field over the Hubble horizon scale. This primordial fluctuation
leads to gravitational instabilities that ultimately form large-scale
structures in our universe [43]. However, it is not known how the quantum-
classical transition of field fluctuation occurs in a long wavelength region
beyond the Hubble horizon scale. Because the details and mechanisms of the
transition process are not known, it remains far from profound understanding.
In the analysis, quantum entanglement is known to be capable of capturing the
quantum-classical transition [44, 45, 46, 47]. The bipartite entanglement
between two spatial regions will be lost in the course of evolution [44, 47],
and entanglement harvesting from spatially separated regions becomes
impossible [45, 46]. Additionally a study of entanglement structure in quantum
states in de Sitter space [48] may be useful to resolve the issue. It is
possible to simulate the generation of analog primordial fluctuations
experimentally, using expanding analog universes of QH systems. We are able to
detect and study the entanglement of long-wavelength quantum fluctuations in
the expanding universe. This direction of analog experiments in QH systems
facilitates us to capture a detailed understanding of the quantum-classical
transition process in the early universe.

Figure 1: (Color online) Schematic of setup. (a) Top view of the QH system
consisting of the bulk and the edge. Region I and Region III are flat and
Region II is the expanding region. (b) The QH edge in 2D spacetime using the
coordinates $x^{0}=vt$ and $x^{1}=x$.

In this paper, we formulate a general theory of the expanding edge of QH
systems in a cosmology language, i.e., quantum field theory in a curved
spacetime. Taking account of the great success of effective field theory of
static edge systems, we adopt the similar description of effective field
theory for the time-dependent edge systems. It is assumed that such an
effective theory description is valid as long as we consider a long wavelength
regime. In fact, the QH edge experiments usually observe edge excitation
wavelengths about 100 times larger than the magnetic length $l_{B}$ [10] [24].
Hence the effective theory description may be suitable for the realistic
experiments. The derivation of the effective equations obeyed by edge modes
from a microscopic Hamiltonian of the dynamical QH system is out of our aim in
this paper, and will be reported in future. The theory will be directly
applied to analyses of future experiments of the edge expansion, and stimulate
an interdisciplinary interest connecting condensed matter physics and
cosmology. First, we divide the entire edge into three regions, as depicted in
Fig. 1: input flat Region I, expanding Region II, and output flat Region III.
Region I is defined as $x>+L/2$. Region II is defined as $-L/2\leq x\leq+L/2$.
Region III is defined as $x<-L/2$. Here, $L$ denotes the width of Region II
when the edge is static. Edge excitation starts from Region I and runs through
the expanding Region II, and finally enters Region III.

The structure of this paper is as follows. In Sec. II, conformally flat
coordinates $x^{\pm}$ of general relativity are introduced for the expanding
Region II. Additionally, we define another conformally flat set of coordinates
$x_{\text{out}}^{\pm}$ of the expanding Region II, which are smoothly
connected with the conformally flat coordinates of the output Region III.
Similarly, we define different conformally flat coordinates
$x_{\text{in}}^{\pm}$ of the expanding Region II, which are smoothly connected
with the conformally flat coordinates of the input Region II. The coordinate
transformation is derived between $x_{\text{out}}^{\pm}$ and
$x_{\text{in}}^{\pm}$. We introduce Bogoliubov coefficients between in-mode
functions for Region I and out-mode functions for Region III. The implication
of the Bogoliubov coefficients measurement in realistic experiments is
discussed for future experiments. In Sec. III, we show that the expanding
edges can be regarded as simulators of homogeneous and isotropic universes
referred to as Friedmann–Lemaître–Robertson–Walker (FLRW) universes in 1+1
dimensions. The dynamics are described by 2D gravity models with a scalar
field $\Phi$, which is called dilaton field. The models include the famous
Jackiw–Teitelboim (JT) model with a negative constant curvature
$R=-4\lambda^{2}$ [49, 50]. The JT model has recently attracted considerable
attention because the gravity model is closely connected with a many-body
model called the Sachdev-Ye-Kitaev (SYK) model [51, 52, 53], and it suggests a
nontrivial example of an anti-de Sitter space (AdS)/CFT correspondence [54].
To connect the edge system with the AdS/CFT correspondence, the three-
dimensional AdS chiral gravity may be interesting, which is dual to 2D CFT
[55]. Kerr/CFT correspondence may be also related to the edge dynamics [56,
57, 58]. The JT model with positive cosmological constant is referred to as
the de Sitter JT (dSJT) model in this paper. The dSJT model achieves an
exponentially fast expansion of the analog universe. Thus, the rapidly
expanding edge experiments are quite fascinating for the study of the trans-
Planckian problem and the quantum-classical transition problem in the dSJT
model. In Sec. IV, as an application of our experimental setup, we present the
analogue Hawking radiation from Region II. With the de Sitter type expanding
edge region, a future horizon is formed and the power of signals measured in
Region III shows the Planckian distribution with Hawking temperature
$\lambda/(\sqrt{2}\pi)$. We summarize our results in Sec. V. In the Appendix,
useful formulas are attached for 2D gravity.

##  II Conformally Flat Coordinates for Expanding region

Let us first consider static edge systems. First, we define the edge
excitation as the left mover with respect to the spatial coordinate $x$ at the
edge. Specifically, excitation runs from the right region with $x>0$ to the
left region with $x<0$. Subsequently, $v$ denotes the edge current velocity of
the system. Introducing relativity notations, as in cosmology, is useful. The
time $t$ is replaced by $x^{0}=vt$. The spatial coordinate $x$ is denoted by
$x^{1}$, and light cone coordinates are then defined as $x^{\pm}=x^{0}\pm
x^{1}$, even though $v$ is not the actual velocity of the light. The edge
excitation is represented by a free real field $\varphi(x^{+})$. The field
$\varphi(x^{+})$ is proportional to the voltage fluctuation on the edge and
can be directly measured in experiments [59]. In this situation, the spacetime
for the edge is just a flat spacetime called the Minkowski spacetime. The
square of the spacetime-invariant distance, which is called metric form, is
given by

$ds^{2}=-v^{2}dt^{2}+dx^{2}=-\left(dx^{0}\right)^{2}+\left(dx^{1}\right)^{2}=-dx^{+}dx^{-}.$
(1)

The aforementioned equation fixes the metric matrix for each coordinate system
as

$\begin{bmatrix}g_{00}&g_{01}\\\
g_{10}&g_{11}\end{bmatrix}=\begin{bmatrix}-1&0\\\
0&1\end{bmatrix},\quad\begin{bmatrix}g_{++}&g_{+-}\\\
g_{-+}&g_{--}\end{bmatrix}=\begin{bmatrix}0&-1/2\\\ -1/2&0\end{bmatrix},$

where $ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$. Note that $\varphi(x^{+})$
satisfies the massless Klein-Gordon equation,

$\left[-\left(\partial_{0}\right)^{2}+\left(\partial_{1}\right)^{2}\right]\varphi=-4\partial_{+}\partial_{-}\varphi=0,$
(2)

and the chirality condition $\partial_{-}\varphi=0$ implying that the edge
current moves only in the left direction.

Let Region II start expanding uniformly at $t=0$, with the remaining Regions I
and III being unchanged. The spacetime metric in Region II is described by the
following equation:

$ds^{2}=-v^{2}d\tau^{2}+a^{2}(\tau)dx^{2},$ (3)

where $\tau$ is the proper time for an observer at $x=const.$, and it is equal
to $t$ when $t<0$. The positive function $a(\tau)$ of $\tau$ satisfies
$a(\tau)=1$ when $\tau<0$. At a fixed time $\tau$, the physical distance $l$
between two points at $x=x_{1}$ and $x=x_{2}~{}(>x_{1})$ is computed as
$l=a(\tau)(x_{2}-x_{1})$. Thus, $a(\tau)$ determines the real size of the
spatial region using the coordinate values, and is called the scale factor of
the expanding universe. Let us rewrite Eq. (3) as

$ds^{2}=a^{2}(\tau)\left[-v^{2}\left(\frac{d\tau}{a(\tau)}\right)^{2}+dx^{2}\right].$
(4)

By defining the relativistic coordinates as

$x^{0}=v\,t=v\int_{0}^{\tau}\frac{d\tau^{\prime}}{a(\tau^{\prime})},\quad
x^{1}=x,$ (5)

and introducing a real function $\Theta$ of $x^{0}$, which obeys
$\Theta(x^{0})=\ln a(\tau)$, we obtain the following metric form in the light
cone coordinates:

$ds^{2}=-\exp\left[2\Theta\\!\left(\frac{x^{+}+x^{-}}{2}\right)\right]dx^{+}dx^{-},$
(6)

where $x^{0}=(x^{+}+x^{-})/2$. The form of Eq. (6) is defined as conformally
flat.

Note that, at least in a local spacetime region, any metric form in 1+1
dimensions can be rewritten using a general coordinate transformation to the
conformally flat form
$ds^{2}=-\exp\left(2\Theta(x^{+},x^{-})\right)dx^{+}dx^{-}$ where
$\Theta(x^{+},x^{-})$ is a real function of the light cone coordinates
$x^{\pm}$. This is because we have two degrees of freedom as a general
coordinate transformation: $x^{\prime+}=f_{+}(x^{+},x^{-})$ and
$x^{\prime-}=f_{-}(x^{+},x^{-})$. By considering the two functions $f_{\pm}$
in the appropriate forms, the two conditions $g^{\prime}_{++}=0$ and
$g^{\prime}_{--}=0$ are satisfied in the new coordinates ${x^{\prime}}^{\pm}$.
Hence, we always obtain conformally flat metric forms. The factor
$\exp\left(2\Theta\left(x^{+},x^{-}\right)\right)$ is called conformal factor.

###  II.1 Conformally Flat Coordinates Connecting to the Expanding region and
Output region

The conformally flat coordinates $x^{\pm}$ in Eq. (6) as defined in Region II
cannot extend to Region III as the conformal factor jumps at the boundary
$x=x^{1}=-L/2$. However, another set of conformally flat coordinates
$x_{\text{out}}^{\pm}=x_{\text{out}}^{0}\pm x_{\text{out}}^{1}$ can be
introduced, which smoothly connect Region II and Region III at the boundary
$x=x_{1}=-L/2$. To ensure that $x_{\text{out}}^{\pm}$ are conformally flat
coordinate systems, the original coordinates $x^{\pm}$ must be
$x_{\text{out}}^{\mp}$-independent functions, i.e.,
$x^{\pm}=x^{\pm}\\!\left(x_{\text{out}}^{\pm}\right)$. Then, the metric form
in Eq. (6) is given by the new coordinates $x_{\text{out}}^{\pm}$ as

$ds^{2}=-\exp\left[2\Theta\\!\left(\frac{x^{+}\\!\left(x_{\text{out}}^{+}\right)+x^{-}\\!\left(x^{-}_{\text{out}}\right)}{2}\right)-\ln\left(\frac{dx_{\text{out}}^{+}}{dx^{+}}\,\frac{dx^{-}_{\text{out}}}{dx^{-}}\right)\right]dx_{\text{out}}^{+}\,dx_{\text{out}}^{-}.$
(7)

The above metric form remains the flat metric form in Region III as
$ds^{2}=-dx_{\text{out}}^{+}\,dx_{\text{out}}^{-}$. The coordinate
transformation, $x^{\pm}=x^{\pm}\\!\left(x_{\text{out}}^{\pm}\right)$, is
uniquely determined by the following two conditions:

  * (i)

The spatial position of the boundary in the new coordinates is given by
$x^{1}=x_{\text{out}}^{1}=-L/2$;

  * (ii)

The time coordinates at the boundary coincide with each other:
$x^{0}=x_{\text{out}}^{0}$.

Here, conditions (i) and (ii) can be replaced by (i) and a conformal factor
continuity condition, i.e., the conformal factor in $x_{\text{out}}^{\pm}$ is
continuous at the boundary between Region II and Region III such that

  * (ii)’

$1=\exp\left(2\Theta\\!\left(\frac{x^{+}\\!\left(x_{\text{out}}^{+}\right)+x^{-}\\!\left(x_{\text{out}}^{-}\right)}{2}\right)-\ln\left(\frac{dx_{\text{out}}^{+}}{dx^{+}}\,\frac{dx_{\text{out}}^{-}}{dx^{-}}\right)\right)\quad\text{at~{}}x^{1}=-L/2.$
(8)

The condition (i) yields the following condition:

$x_{\text{out}}^{+}\\!\left(x^{0}-\frac{L}{2}\right)-x_{\text{out}}^{-}\\!\left(x^{0}+\frac{L}{2}\right)=-L.$
(9)

By taking the derivative with respect to $x^{0}$, we get the following
relation:

$\frac{dx_{\text{out}}^{+}}{dx^{+}}\\!\left(x^{0}-\frac{L}{2}\right)=\frac{dx_{\text{out}}^{-}}{dx^{-}}\\!\left(x^{0}+\frac{L}{2}\right).$
(10)

Using the condition (ii)’, the following relation is derived:

$\frac{dx_{\text{out}}^{+}}{dx^{+}}\\!\left(x^{0}-\frac{L}{2}\right)\frac{dx_{\text{out}}^{-}}{dx^{-}}\\!\left(x^{0}+\frac{L}{2}\right)=\exp\left(2\Theta(x^{0})\right).$
(11)

From the above equation and Eq. (10), $x_{\text{out}}^{+}$ obeys the following
relation:

$\frac{dx_{\text{out}}^{+}}{dx^{+}}\\!\left(x^{0}-\frac{L}{2}\right)=\exp\left(\Theta(x^{0})\right).$
(12)

By changing the free parameter $x^{0}$ to $x^{+}=x^{0}-L/2$, this equation can
be replaced by

$\frac{dx_{\text{out}}^{+}}{dx^{+}}\\!\left(x^{+}\right)=\exp\left(\Theta\\!\left(x^{+}+\frac{L}{2}\right)\right).$
(13)

Integration of this equation yields

$x_{\text{out}}^{+}(x^{+})=\int_{0}^{x^{+}}dy\exp\left(\Theta\\!\left(y+\frac{L}{2}\right)\right),$
(14)

where the integration constant is fixed, such that $x_{\text{out}}^{+}=0$ when
$x^{+}=0$. As the coordinate function $x_{\text{out}}^{+}(x^{+})$ is a
monotonically increasing function of $x^{+}$, it has the inverse function
$x^{+}=F_{\text{out}}\left(x_{\text{out}}^{+}\right)$, which satisfies
$x=F_{\text{out}}\left(x_{\text{out}}^{+}(x)\right)$, i.e.,

$x^{+}=F_{\text{out}}\left(x_{\text{out}}^{+}\right)\quad\Leftrightarrow\quad
x_{\text{out}}^{+}(x^{+})=\int_{0}^{x^{+}}dy\exp\left(\Theta\\!\left(y+\frac{L}{2}\right)\right).$
(15)

Similarly, the coordinate function $x_{\text{out}}^{-}(x^{-})$ can be derived
from Eq. (9) to be

$x_{\text{out}}^{-}(x^{-})=x_{\text{out}}^{+}(x^{-}-L)+L=\int_{0}^{x^{-}-L}dy\exp\left(\Theta\\!\left(y+\frac{L}{2}\right)\right)+L,$
(16)

where Eq. (14) is used within the right hand side. In the same way as the
above discussion, it is possible to introduce continuous conformally flat
coordinates $x_{\text{in}}^{\pm}$, which connect Region I and Region II. The
coordinate functions $x_{\text{in}}^{+}(x^{+})$ and $x_{\text{in}}^{-}(x^{-})$
obey

$x_{\text{in}}^{+}\\!\left(x^{0}+\frac{L}{2}\right)-x_{\text{in}}^{-}\\!\left(x^{0}-\frac{L}{2}\right)=L,$
(17)

at the boundary $x^{1}=x^{1}_{\text{in}}=L/2$. As Region I remains flat, the
metric form is given by $ds^{2}=-dx_{\text{in}}^{+}\,dx_{\text{in}}^{-}$ for
$x^{1}_{\text{in}}>L/2$. By changing $L\rightarrow-L$ in Eq. (14), we obtain
the following relation:

$x_{\text{in}}^{+}(x^{+})=\int_{0}^{x^{+}}dy\exp\left(\Theta\\!\left(y-\frac{L}{2}\right)\right).$
(18)

Here, we introduce the inverse function
$x^{+}=F_{\text{in}}\\!\left(x_{\text{in}}^{+}\right)$, which satisfies
$x=F_{\text{in}}\\!\left(x_{\text{in}}^{+}(x)\right)$ such that

$x^{+}=F_{\text{in}}\\!\left(x_{\text{in}}^{+}\right)\quad\Leftrightarrow\quad
x_{\text{in}}^{+}(x^{+})=\int_{0}^{x^{+}}dy\exp\left(\Theta\\!\left(y-\frac{L}{2}\right)\right).$
(19)

In Eq. (16), changing $L\rightarrow-L$ yields

$x_{\text{in}}^{-}\\!\left(x^{-}\right)=x_{\text{in}}^{+}\\!\left(x^{-}+L\right)-L=\int_{0}^{x^{-}+L}dy\exp\left(\Theta\\!\left(y-\frac{L}{2}\right)\right)-L.$
(20)

###  II.2 Conformal Symmetry of Expanding Edge Excitations

In this section, we introduce the coordinate transformation between
$x_{\text{out}}^{+}$ and $x_{\text{in}}^{+}$. Let $F_{\text{in}}^{-1}(x)$
denote the inverse function of $F_{\text{in}}(x)$, i.e.,
$F_{\text{in}}^{-1}(F_{\text{in}}(x))=x$. It should be noted that
$F_{\text{out}}^{-1}(F_{\text{out}}(x))=x$ holds true. We then define the
composite function $F(x)=F_{\text{in}}^{-1}\left(F_{\text{out}}(x)\right)$,
which satisfies $x_{\text{in}}^{+}=F(x_{\text{out}}^{+})$. From Eq. (14) and
Eq. (18), the following relation holds:

$X=\int_{L/2}^{F_{\text{out}}\left(X\right)+L/2}dy\exp\left(\Theta\\!\left(y\right)\right),\quad
F(X)=\int_{-L/2}^{F_{\text{out}}\left(X\right)-L/2}dy\exp\left(\Theta\\!\left(y\right)\right).$
(21)

By combining two equations in Eq. (21), the following relation is derived:

$F(X)=X-\int_{F_{\text{out}}\left(X\right)-L/2}^{F_{\text{out}}\left(X\right)+L/2}dy\exp\left(\Theta\\!\left(y\right)\right)+\int_{-L/2}^{L/2}dy\exp\left(\Theta\\!\left(y\right)\right).$
(22)

If the function $\Theta\\!\left(v\,t\right)$ is a constant function
independent of $t$, the above equations reduce to $F(X)=X$. However, if
$\Theta\\!\left(v\,t\right)$ depends on $t$, then $F(X)$ is a nontrivial
function, which describes the dynamics of the expanding edge region.

If the conformal symmetry (Virasoro symmetry) survives even in the expansion,
the time evolution of the edge excitation is given in Region I and Region II
simply by $\varphi_{\text{in}}\\!\left(x_{\text{in}}+v\,t_{\text{in}}\right)$,
where $\varphi_{\text{in}}\\!\left(x_{\text{in}}^{+}\right)$ is the initial
configuration in Region I. In Region II, the configuration can be expressed
with respect to $x_{\text{out}}^{+}$ as
$\varphi_{\text{in}}\\!\left(F\\!\left(x_{\text{out}}^{+}\right)\right)$.
Assuming the conformal symmetry, the configuration in Region III takes the
same form as that within Region II:

$\varphi_{\text{out}}(x_{\text{out}}+v\,t_{\text{out}})=\varphi_{\text{out}}(x_{\text{out}}^{+})=\varphi_{\text{in}}\\!\left(F\\!\left(x_{\text{out}}^{+}\right)\right).$
(23)

In future experiments of the expanding edge systems, confirmation of the above
equation directly implies verification of the conformal symmetry preservation.

###  II.3 Measurement of Bogoliubov Coefficients and its Implications

In this section, we consider the dynamical analysis of excitations in the
expanding edge using the Bogoliubov coefficients in the curved spacetime field
theory. Suppose plane wave mode functions are defined by

$u_{k}(x)=\frac{e^{-ikx}}{\sqrt{4\pi k}},$ (24)

where the wave number $k$ has continuous positive values. Then the input wave
$\varphi_{\text{in}}(x_{\text{in}}^{\text{+}})$ in Region I can be expanded as

$\varphi_{\text{in}}(x_{\text{in}}^{+})=\int_{0}^{\infty}dk\left(a_{k}^{(\text{in})}\,u_{k}(x_{\text{in}}^{+})+a_{k}^{(\text{in})\ast}\,u_{k}^{*}(x_{\text{in}}^{+})\right),$
(25)

where $a_{k}^{(\text{in})}$ are complex coefficients in this expansion. From
Eq. (23), the output wave is then calculated as

$\varphi_{\text{out}}(x_{\text{out}}^{+})=\int_{0}^{\infty}dk^{\prime}\left(a_{k^{\prime}}^{(\text{in})}\,u_{k^{\prime}}(F\\!\left(x_{\text{out}}^{+}\right))+a_{k^{\prime}}^{(\text{in})\ast}\,u_{k^{\prime}}^{*}(F\\!\left(x_{\text{out}}^{+}\right))\right).$
(26)

Note that the output wave form is given by
$\varphi_{\text{out},k^{\prime}}\left(x_{\text{out}}^{+}\right)=u_{k^{\prime}}(F\\!\left(x_{\text{out}}^{+}\right))$
if the input wave is the plane wave as in Eq. (24) with $k=k^{\prime}$. To
extract information about the dynamics in Region II induced by the edge
expansion, let us expand $u_{k^{\prime}}(F\\!\left(x_{\text{out}}^{+}\right))$
with respect to the plane wave mode functions as

$u_{k^{\prime}}(F\\!\left(x_{\text{out}}^{+}\right))=\int_{0}^{\infty}dk\left(\alpha(k,k^{\prime})\,u_{k}\\!\left(x_{\text{out}}^{+}\right)+\beta(k,k^{\prime})\,u_{k}^{*}\\!\left(x_{\text{out}}^{+}\right)\right).$
(27)

The coefficients $\alpha(k,k^{\prime})$ and $\beta(k,k^{\prime})$ are referred
to as Bogoliubov coefficients and satisfy the unitarity conditions [33] as
follows:

$\displaystyle\int_{0}^{\infty}dq\left(\alpha(k,q)\,\alpha^{*}(k^{\prime},q)-\beta(k,q)\,\beta^{*}(k^{\prime},q^{\prime})\right)$
$\displaystyle=\delta\left(k-k^{\prime}\right),$ (28a)
$\displaystyle\int_{0}^{\infty}dq\left(\alpha(k,q)\,\beta(k^{\prime},q)-\beta(k,q)\,\alpha(k^{\prime},q)\right)$
$\displaystyle=0.$ (28b)

The output wave $\varphi_{\text{out}}(x_{\text{out}}^{+})$ can be expanded
using the plane wave mode functions in Eq. (24) as

$\varphi_{\text{out}}(x_{\text{out}}^{+})=\int_{0}^{\infty}dk\left(b_{k}^{(\text{out})}\,u_{k}(x_{\text{out}}^{+})+b_{k}^{(\text{out})\ast}\,u_{k}^{*}(x_{\text{out}}^{+})\right),$
(29)

where $b_{k}^{(\text{out})}$ are the expansion coefficients and computed as
follows:

$b_{k}^{(\text{out})}=\int_{0}^{\infty}dk^{\prime}\left(a_{k^{\prime}}^{(\text{in})}\,\alpha(k,k^{\prime})+a_{k^{\prime}}^{(\text{in})\ast}\,\beta^{*}(k,k^{\prime})\right).$
(30)

The coefficients $\alpha(k,k^{\prime})$ and $\beta(k,k^{\prime})$ play a
crucial role in the quantum regime [33]. By introducing creation operators
$\hat{a}_{k}^{(\text{in}){\dagger}}\,,\hat{b}_{k}^{(\text{out}){\dagger}}$ and
annihilation operators $\hat{a}_{k}^{(\text{in})},\hat{b}_{k}^{(\text{out})}$,
satisfying

$\left[\hat{a}_{k}^{(\text{in})},~{}\hat{a}_{k^{\prime}}^{(\text{in}){\dagger}}\right]=\hbar\,\delta(k-k^{\prime}),\quad\left[\hat{b}_{k}^{(\text{out})},~{}\hat{b}_{k^{\prime}}^{(\text{out}){\dagger}}\right]=\hbar\,\delta(k-k^{\prime}),$
(31)

the field operators
$\hat{\varphi}_{\text{in}}(x_{\text{in}}^{+}),\hat{\varphi}_{\text{out}}(x_{\text{out}}^{+})$
are given by

$\displaystyle\hat{\varphi}_{\text{in}}(x_{\text{in}}^{+})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dk\left(\hat{a}_{k}^{(\text{in})}\,u_{k}\left(x_{\text{in}}^{+}\right)+\hat{a}_{k}^{(\text{in}){\dagger}}\,u_{k}^{*}\left(x_{\text{in}}^{+}\right)\right),$
(32a) $\displaystyle\hat{\varphi}_{\text{out}}(x_{\text{out}}^{+})$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}dk\left(\hat{b}_{k}^{(\text{out})}\,u_{k}(x_{\text{out}}^{+})+\hat{b}_{k}^{(\text{out})\dagger}\,u_{k}^{*}(x_{\text{out}}^{+})\right).$
(32b)

Using $\alpha(k,k^{\prime})$ and $\beta(k,k^{\prime})$, we obtain proof of the
following relations:

$\displaystyle\hat{b}_{k}^{(\text{out})}$
$\displaystyle=\int_{0}^{\infty}dk^{\prime}\left(\alpha(k,k^{\prime})\,\hat{a}_{k^{\prime}}^{(\text{in})}+\beta^{*}(k,k^{\prime})\,\hat{a}_{k^{\prime}}^{(\text{in}){\dagger}}\right),$
(33a) $\displaystyle\hat{b}_{k}^{(\text{out}){\dagger}}$
$\displaystyle=\int_{0}^{\infty}dk^{\prime}\left(\alpha^{*}(k,k^{\prime})\,\hat{a}_{k^{\prime}}^{(\text{in}){\dagger}}+\beta(k,k^{\prime})\,\hat{a}_{k^{\prime}}^{(\text{in})}\right).$
(33b)

Note that the input vacuum state $|0_{\text{in}}\rangle$ is defined by
$\hat{a}_{k}^{(\text{in})}|0_{\text{in}}\rangle=0$. Thus,
$\beta(k,k^{\prime})$ describes particle creation in Region III from the input
vacuum in Region I. The expectation values of particle number density
$\hat{n}_{k}^{(\text{out})}$, with wave number $k$, are computed as

$\left\langle\hat{n}_{k}^{(\text{out})}\right\rangle=\frac{1}{\hbar}\,\langle
0_{\text{in}}|\,\hat{b}_{k}^{(\text{out}){\dagger}}\,\hat{b}_{k}^{(\text{out})}\,|0_{\text{in}}\rangle=\int_{0}^{\infty}dk^{\prime}\left|\beta(k,k^{\prime})\right|^{2}.$
(34)

The total number of particles is evaluated as

$\left\langle\hat{N}^{(\text{out})}\right\rangle=\int_{0}^{\infty}dk\left\langle\hat{n}_{k}^{(\text{out})}\right\rangle.$

This implies that quantum particle creation in the expanding analog universes
can be predicted only from $\beta(k,k^{\prime})$, which can be measured in the
classical regime of the system. Thus, experimentally determining
$\beta(k,k^{\prime})$ is crucial.

Here it may be useful to comment about a relation between the conformal
symmetry and the coefficient $\beta(k,k^{\prime})$. If a conformally flat
coordinate system covers the entire spacetime region (Region I + Region II +
Region III) in the experiments and remains the flat metric in Region I and
Region III, $x_{\text{out}}^{+}$ is equal to $x_{\text{in}}^{+}$ up to a
factor and a constant. Thus the conformal symmetry always results in
$\beta(k,k^{\prime})=0$. It should be stressed that this is not the case in
our situation since we do not have such a global conformally flat coordinate
system. Due to this fact, the function
$u_{k^{\prime}}(F\left(x_{\text{out}}^{+}\right))$ becomes a nontrivial
function with a transient behavior in a region of $x_{\text{out}}^{+}$, and
includes negative frequency out modes. This implies that
$\beta(k,k^{\prime})\neq 0$ in general even if the conformal symmetry is
preserved.

As Region I and Region III remain flat during the expansion of Region II, the
excitations in the regions are always described by the free fields in Eq.
(32a) and Eq. (32b). The coefficients $\alpha(k,k^{\prime})$ and
$\beta(k,k^{\prime})$ can be estimated using the experimental data in Region
III, even if the conformal symmetry is broken in Region II. Thus, the question
of whether the conformal symmetry survives can be answered by the experimental
data analysis of the wave form $\varphi_{\text{out}}(x)$ in Eq. (23),
$\alpha(k,k^{\prime})$, and $\beta(k,k^{\prime})$ in Eq. (27). Let us consider
a deviation $\delta\varphi_{\text{out},k^{\prime}}(x)$ from the predicted
function form $\varphi_{\text{out},k^{\prime}}(x)$. Then the following
relation holds for the deviations of $\alpha(k,k^{\prime})$ and
$\beta(k,k^{\prime})$:

$\delta\varphi_{\text{out},k^{\prime}}(x_{\text{out}}^{+})=\int_{0}^{\infty}dk\left(\delta\alpha(k,k^{\prime})\,u_{k}\\!\left(x_{\text{out}}^{+}\right)+\delta\beta(k,k^{\prime})\,u_{k}^{*}\\!\left(x_{\text{out}}^{+}\right)\right).$
(35)

From the above equation, the following relations are directly computed via the
Fourier transformation:

$\displaystyle\delta\alpha(k,k^{\prime})=\sqrt{\frac{k}{\pi}}\int^{\infty}_{-\infty}dx^{+}\,\delta\varphi_{\text{out},k^{\prime}}(x^{+})\exp\left(ikx^{+}\right),$
(36)
$\displaystyle\delta\beta(k,k^{\prime})=\sqrt{\frac{k}{\pi}}\int^{\infty}_{-\infty}dx^{+}\,\delta\varphi_{\text{out},k^{\prime}}(x^{+})\exp\left(-ikx^{+}\right).$
(37)

By substituting observed deviation $\delta\varphi_{\text{out},k^{\prime}}$
into Eq. (36) and Eq. (37), we are able to quantify the conformal symmetry
breaking for every wavelength $k^{\prime}$ of the input wave and wavelength
$k$ of the output plane wave. It is possible that the conformal symmetry
breaking may occur only for some regions of $(k,k^{\prime})$. In such a case,
Eq. (36) and Eq. (37) provide the information of detailed structure of the
symmetry breaking.

If the future experiments will confirm breakdown of the conformal symmetry,
then we will be able to consider various kinds of extended effective field
theories in Region II to compute $\alpha(k,k^{\prime})$ and
$\beta(k,k^{\prime})$, and compare the predictions with the experimental
results. To construct simple effective theories describing the possible
breakdown of the conformal symmetry, we propose to assume the general
covariance, i.e., symmetry under general coordinates transformations as a
working hypothesis for the dynamics. Then the equation of motion takes a
general form as follows:

$\left(\nabla^{2}-\left(\frac{m(t)v}{\hbar}\right)^{2}-\xi
R(t)\right)\varphi(t,x)-U^{\prime}(\varphi,t)=J(t,x).$ (38)

Here $\nabla^{2}$ is the covariant Laplacian operator in Region II, $m(t)$ is
the time-dependent effective mass induced by the expansion. The scalar
curvature of the FLRW spacetime in Region II is denoted by $R(t)$, and $\xi$
is a real parameter controlling the curvature interaction. The terms with
$m(t)$ and $\xi R(t)$ induce different velocity of the edge excitation in
Region II for each wave number $k$, and break the conformal symmetry. The term
$U^{\prime}(\varphi,t)$ represents a time-dependent nonlinear interaction of
the field $\varphi(t,x)$, which may also break the conformal symmetry. The
term $J(t,x)$ is a possible source. When the source term $J(t,x)$ exists, an
additional wave $\varphi_{J}(t,x)$ is classical-mechanically generated by the
source, even in the zero input wave case. Note that the source $J(t,x)$ also
breaks the conformal symmetry, but it remains harmless when the nonlinear
interaction does not exist, or when the amplitude of the input wave is so
small that the nonlinear interaction can be neglected. In such cases, it turns
out that the source term can be eliminated by subtracting $\varphi_{J}(t,x)$
from $\varphi(t,x)$. The dynamics information about the edge excitations in
the Region II is imprinted onto $\alpha(k,k^{\prime})$ and
$\beta(k,k^{\prime})$, which enables us to check the validity of Eq. (38).

Increasing the amplitude of the initial wave
$\varphi_{\text{in}}(x_{\text{in}}^{+})$ in Region I, the experiments are able
to determine whether nonlinear interactions like $U^{\prime}(\varphi,t)$ are
generated in Region II. If the amplitude of
$\varphi_{\text{out}}(x_{\text{out}}^{+})$ does not increase linearly as the
amplitude of $\varphi_{\text{in}}(x_{\text{in}}^{+})$, the nonlinear
interaction exists. Furthermore, it should be emphasized that an experimental
check of the edge unitarity conditions of $\alpha(k,k^{\prime})$ and
$\beta(k,k^{\prime})$ in Eq. (28a) and Eq. (28b) may detect the existence of
bulk-edge interactions in Region II. If the edge excitations interact with
bulk excitations like magneto-roton, the edge dynamics are coupled with the
bulk dynamics and cannot be determined only by the edge information.
Specifically, the relations in Eq. (28a) and Eq. (28b) may be extended to the
following relations:

$\displaystyle\int_{0}^{\infty}\\!\\!dq\left(\alpha(k,q)\,\alpha^{*}(k^{\prime},q)-\beta(k,q)\,\beta^{*}(k^{\prime},q)\right)+\int\\!\\!d^{2}q\left(\tilde{\alpha}(k,\vec{q})\,\tilde{\alpha}^{*}(k^{\prime},\vec{q})-\tilde{\beta}(k,\vec{q})\,\tilde{\beta}^{*}(k^{\prime},\vec{q})\right)=\delta\left(k-k^{\prime}\right),$
$\displaystyle\int_{0}^{\infty}\\!\\!dq\left(\alpha(k,q)\,\beta(k^{\prime},q)-\beta(k,q)\,\alpha(k^{\prime},q)\right)+\int\\!\\!d^{2}q\left(\tilde{\alpha}(k,\vec{q})\,\tilde{\beta}(k^{\prime},\vec{q})-\tilde{\beta}(k,\vec{q})\,\tilde{\alpha}(k^{\prime},\vec{q})\right)=0,$

where $\tilde{\alpha}(k,\vec{q})$ and $\tilde{\beta}(k^{\prime},\vec{q})$ are
contributions to the Bogoliubov coefficients induced by the bulk mode
functions. This implies that Eq. (28a) and Eq. (28b) are not satisfied if the
bulk-edge interactions appear in Region II. Consequently, the experiments are
able to provide upper bounds of coupling constants of bulk-edge interactions
in various situations. More detailed analyses in this section will be reported
in forthcoming papers.

##  III Expanding Edges as Universe Simulators in 2D Dilaton Gravity

In this section, we show that the expanding edges of QH systems are regarded
as universe simulators of 2D dilaton gravity models. Some useful formula for
2D gravity are given in the Appendix. Let us suppose the following action for
a real scalar field $\Phi$, which is called the dilaton field, and the metric
field $g_{\alpha\beta}$:

$S=\int d^{2}x\sqrt{-g}\left(\Phi R-4\lambda^{2}\,V(\Phi)\right),$ (39)

where $\lambda$ is a positive constant and $V(\Phi)$ is the potential term of
$\Phi$. As shown later, the determination of the time schedule of edge
expansion corresponds to the determination of a form of $V(\Phi)$. Thus, the
field $\varphi$ in Eq. (38) can be interpreted as a matter field in a curved
spacetime background of 2D dilation gravity with the potential term $V(\Phi)$.
The metric equation $\delta S/\delta g^{\alpha\beta}(x)=0$ derived from Eq.
(39) is written as

$\left(g_{\alpha\beta}\nabla^{2}-\nabla_{\alpha}\nabla_{\beta}\right)\Phi+2\lambda^{2}\,V(\Phi)\,g_{\alpha\beta}=0.$
(40)

Here we have used a property of the 2D gravity, such that
$R_{\alpha\beta}-R\,g_{\alpha\beta}/2=0$. The field equation $\delta
S/\delta\Phi(x)=0$ reads

$R=4\lambda^{2}\,V^{\prime}(\Phi),$ (41)

where $V^{\prime}(\Phi)$ is the derivative function of $V(\Phi)$ with respect
to $\Phi$. As already mentioned in Sec. II, any 2D spacetime is able to be
described, at least locally, by the conformally flat metric as
$ds^{2}=-\exp\left(2\Theta\left(x^{+},x^{-}\right)\right)dx^{+}dx^{-}$. In
this coordinate system, Eq. (40) is simplified to the following three
equations:

$\displaystyle\partial_{+}\partial_{-}\Phi-\lambda^{2}\,V(\Phi)\,e^{2\Theta}=0,$
(42a)
$\displaystyle\partial_{+}^{2}\Phi-2\partial_{+}\Theta\,\partial_{+}\Phi=0,$
(42b)
$\displaystyle\partial_{-}^{2}\Phi-2\partial_{-}\Theta\,\partial_{-}\Phi=0.$
(42c)

Similarly Eq. (41) is reduced to

$2\partial_{+}\partial_{-}\Theta-\lambda^{2}\,V^{\prime}(\Phi)\,e^{2\Theta}=0.$
(43)

To describe FLRW universes, let us consider metric forms such that $\Theta$
does not have any $x$ dependence:

$ds^{2}=-e^{2\Theta((x^{+}+x^{-})/2)}\,dx^{+}dx^{-}=e^{2\Theta(vt)}\left(-v^{2}dt^{2}+dx^{2}\right).$

The conformal factor $e^{2\Theta(vt)}$ directly corresponds to the conformal
factor in Eq. (6) of the expanding-edge realistic experiments. In this case,
Eq. (42a) and Eq. (43) become

$\displaystyle\frac{d^{2}\Phi}{dt^{2}}-4v^{2}\lambda^{2}\,V(\Phi(t))\,e^{2\Theta(vt)}=0,$
(44a)
$\displaystyle\frac{d^{2}\Theta}{dt^{2}}-2v^{2}\lambda^{2}\,V^{\prime}(\Phi(t))\,e^{2\Theta(vt)}=0.$
(44b)

Equations (42b) and (42c) result in the same equation as

$\frac{d^{2}\Phi}{dt^{2}}-2\frac{d\Theta}{dt}\frac{d\Phi}{dt}=0.$ (45)

Let us divide both sides by $(d\Phi/dt)$ in Eq. (45), yielding
$d/dt\left(\ln(d\Phi/dt)-2\Theta(v\,t)\right)=0$. By integrating the
aforementioned equation with respect to $t$, we obtain
$d\Phi/dt=A\,e^{2\Theta(vt)}$ where $A$ is a positive integration constant.
The constant $A$ can be replaced by $\lambda\,v$ using a transformation such
that $\Theta(vt)\rightarrow\Theta(vt)-1/2\ln A+1/2\ln(\lambda v)$. Thereby,
the equation can be written as

$\frac{d\Phi}{dt}=\lambda v\,e^{2\Theta(vt)}.$ (46)

By integrating Eq. (46), we obtain the following equation:

$\Phi(t)=\lambda v\int_{0}^{t}dt^{\prime}\,e^{2\Theta(vt^{\prime})}+\Phi_{0},$
(47)

where $\Phi_{0}$ is a constant. In the expanding-edge experiment, the factor
$e^{2\Theta(vt)}$ is fixed as a function of $t$. This determines the
monotonically increasing function $\Phi(t)$ in Eq. (47). Subtracting Eq. (44a)
from Eq. (45) yields

$2\frac{d\Theta}{dt}\frac{d\Phi}{dt}=4v^{2}\lambda^{2}\,V(\Phi(t))\,e^{2\Theta(v\,t)}.$

By substituting Eq. (46) into the above equation, the potential as a function
of time $t$ can be determined by

$V(\Phi(t))=\frac{1}{2v\lambda}\frac{d\Theta}{dt}(v\,t).$ (48)

Here we define the inverse function of $\Phi(t)$ as $t\left(\Phi\right)$.
Since $\Phi(t\left(\Phi\right))=\Phi$ holds, the potential term $V(\Phi)$ is
fixed by the function $\Theta(vt)$ as

$V(\Phi)=\frac{1}{2v\lambda}\frac{d\Theta}{dt}(v\,t\left(\Phi\right)).$ (49)

It is worth highlighting that various function forms of $\Theta(vt)$ can be
realized by changing the time schedule of the edge expansion in the
experiments. This implies that the expanding edge experiments provide universe
simulators of the dilaton gravity models.

When we consider $V(\Phi)=-\Phi$, the model becomes the Jackiw–Teitelboim (JT)
model [49, 50]. The equation of motion possesses a solution of anti-de Sitter
(AdS) spacetime with negative constant curvature $R=-4\lambda^{2}$. The edge
expansion experiment is capable of simulating the JT model by taking the
conformal factor as

$e^{\Theta(vt)}=\frac{1}{\cosh\left(Hvt\right)},$ (50)

where we introduced $H=\sqrt{2}\,\lambda$; $H^{-1}$ corresponds to the AdS
curvature radius. The conformal factor yields the following FLRW metric of AdS
spacetime with proper time $\tau$:

$\displaystyle ds^{2}=-v^{2}d\tau^{2}+\cos^{2}\left(Hv\tau\right)dx^{2}.$ (51)

From Eq. (47), the time evolution of the dilaton field is given by

$\Phi(t)=\frac{Hv}{\sqrt{2}}\int_{0}^{t}\frac{dt^{\prime}}{\cosh^{2}\left(Hvt^{\prime}\right)}+\Phi_{0}=\frac{1}{\sqrt{2}}\tanh\left(Hvt\right)+\Phi_{0}.$
(52)

Next, let us consider the dSJT model, which is constructed by $H\rightarrow
iH$ in the JT model with the negative cosmological constant $-\lambda^{2}$.
The corresponding potential term in Eq. (39) is given by $V(\Phi)=\Phi$. This
model reproduces the de Sitter spacetime with a positive constant curvature as
$R=4\lambda^{2}$. The conformal factor is fixed as

$e^{\Theta\left(vt\right)}=\frac{1}{\cos\left(Hvt\right)}.$ (53)

In this case, $H^{-1}$ corresponds to the Hubble radius of de Sitter
spacetime. Since the FLRW metric is computed using proper time $\tau$ as

$ds^{2}=-v^{2}d\tau^{2}+\cosh^{2}\left(Hv\tau\right)dx^{2},$ (54)

the analog universe experiences an inflationary expansion when $\tau>0$. From
Eq. (47), the time evolution of the dilaton field is derived as follows:

$\Phi(t)=\frac{1}{\sqrt{2}}\tan\left(Hvt\right)+\Phi_{0}.$

We demonstrate the effect of the expanding (shrinking) region on the edge
mode’s waveform. The mode function in Region III is related to the mode
function in Region I by Eq. (23), which is explicitly obtained by finding the
function $F(X)$. The function $F(X)$ is explicitly given by Eq. (56) when the
boundary of Region II is the de Sitter spacetime (54), while $F(X)$ for the
anti-de Sitter spacetime is given by the analytic continuation. The derivation
of $F(X)$ for the de Sitter case is presented in the next section. From Eq.
(23), the form of edge excitation $\varphi_{\text{out}}(x_{\text{out}}^{+})$
in Region III is predicted, assuming the conformal symmetry in Region II.

Figure 2: Behavior of $\varphi_{\text{in}}$ (black curves) and
$\varphi_{\text{out}}$ (red curves). (a) de Sitter ($H=1/\sqrt{2}$) (b) AdS
spacetime ($H=i/\sqrt{2}$) cases. All parameters, i.e. $a$, $b$, $c$, $L$, are
identical except for $H$. Here $x$ denotes $x_{\rm in}^{+}$ for $\varphi_{\rm
in}$ and $x_{\rm out}^{+}$ for $\varphi_{\rm out}$.

Figures 2 (a) and (b) show the behavior of the input wave
$\varphi_{\text{in}}$ (black curves) and output wave $\varphi_{\text{out}}$
(red curves) in de Sitter and AdS spacetime. Here, the input wave is assumed
to be the following Gaussian form:

$\displaystyle\varphi_{\text{{in}}}(x_{\rm in}^{+})=a\exp\left(-\frac{(x_{\rm
in}^{+}-b)^{2}}{2c^{2}}\right),$

where $a,b$, and $c$ are arbitrary parameters. Note that the parameter $H$ in
de Sitter spacetime (54) is replaced by $iH$ in AdS spacetime (51). In our
case, de Sitter spacetime is expanding, so the output wave is also spreading,
but, in AdS spacetime, it is shrinking, so the output wave is narrowing. In
both cases, the longer the input wave stays in Region II, the more it will be
affected by the expansion or shrinkage, and the output wave will behave as if
it is expanding or shrinking.

When Region II is the expanding de Sitter spacetime (54), i.e., inflationary
universe, there appear the wave modes whose wavelength stretched infinitely.
Namely, the cosmological horizon appears. This predicts the Hawking radiation
in the edge modes on the quantum Hall systems, which we will discuss in the
next section.

##  IV Edge mode’s Hawking radiation

In this section, we will discuss the Hawking radiation when Region II is an
inflationary universe. We first evaluate $F_{\text{out}}\left(X\right)$ in
(21) for the inflationary universe with the conformal factor (53), which
covers a whole (1+1)-dimensional de Sitter spacetime. From Eq. (21),
$F_{\text{out}}\left(X\right)$ obeys the following equation:

$X=\frac{1}{2H}\ln\left[\frac{1+\sin\left(H\left(F_{\text{out}}\left(X\right)+L/2\right)\right)}{1-\sin\left(H\left(F_{\text{out}}\left(X\right)+L/2\right)\right)}\times\frac{1-\sin\left(HL/2\right)}{1+\sin\left(HL/2\right)}\right].$

Thus, the functions $F_{\text{out}}(X)$ and $F(X)$ are computed as

$\displaystyle
F_{\text{out}}\left(X\right)=-\frac{L}{2}+\frac{1}{H}\arcsin\left[\frac{\left(1+\sin\left(HL/2\right)\right)e^{2HX}-\left(1-\sin\left(HL/2\right)\right)}{\left(1+\sin\left(HL/2\right)\right)e^{2HX}+\left(1-\sin\left(HL/2\right)\right)}\right],$
(55) $\displaystyle
F(X)=\frac{1}{2H}\ln\left[\frac{1+\sin\left(H\left(F_{\text{out}}\left(X\right)-L/2\right)\right)}{1-\sin\left(H\left(F_{\text{out}}\left(X\right)-L/2\right)\right)}\times\frac{1+\sin\left(HL/2\right)}{1-\sin\left(HL/2\right)}\right].$
(56)

The function $F(X)$ behaves as is shown in Fig. 3. For a negative value of
$X=X_{*}<0$, $F(X_{*})=-\infty$ and for $X\rightarrow+\infty$,
$F\rightarrow\text{const.}$. The domain and the range of $F(X)$ are semi-
infinite, and this behavior indicates existence of horizons in the present
spacetime. About $X=0$, $F(X)\approx X$ which corresponds to behaviour of flat
spacetime. Indeed, in the limit of $L\rightarrow 0$, a whole spacetime region
becomes flat and $F(X)=X$ is recovered.

Figure 3: Behavior of the function $F(X)$ which relates $x_{\text{in}}^{+}$
and $x_{\text{out}}^{+}$ as $x_{\text{in}}^{+}=F(x_{\text{out}}^{+})$.

The domain and the range of $F(X)$ become semi-indefinite, which indicates
existence of horizons in the present spacetime.

Figure 4 shows Penrose diagrams of the analog spacetime for the QH system with
an expanding and contracting edge. Regions I and III are flat spacetimes, and
Region II is a part of the de Sitter spacetime representing the expanding
edge, whose metric is described by (54) with the global chart. This form of de
Sitter metric covers a whole region of the de Sitter spacetime, and represents
a contracting universe for $t<0$ and an expanding universe for $t>0$. Three
spacetime Regions I, II, III are joined along spatial points
$x_{\text{in}}=L/2$ and $x_{\text{out}}=-L/2$. Owing to its global structure,
only a part of data prepared on $\mathscr{I}^{-}$ of Region I can reach
$\mathscr{I}^{+}$ of Region III, which implies existence of the event horizon
in this spacetime ($\mathscr{H}^{+}$ in Fig. 4).

Figure 4: Penrose diagram for the QH system with an expanding ($t>0$) and a
contracting ($t<0$) edge Region II, which is assumed to be a part of the de
Sitter spacetime with the global chart. Owing to its global structure, this
spacetime possesses a future horizon $\mathscr{H}^{+}$ and a past horizon
$\mathscr{H}^{-}$. The shaded region of the right panel shows the region that
the edge modes in $\mathscr{I}^{-}$ in Region I move to reach
$\mathscr{I}^{+}$ in Region III.

Reachability of signals from the in-region I to the out-region III depends on
the size $L$ of the expanding edge Region II (Fig. 5); for $\pi/2<LH$, signals
from Region I cannot reach Region III. For $LH<\pi/4$, signals prepared at
$t=0$ in Region I reach Region III at $t>0$.

Figure 5: $L$ dependence of the global structure of spacetime. Red lines
represent world lines for fixed spatial points in Region I and Region III.
From left panel to right panel, $0<LH<\pi/4,\pi/4<LH<\pi/2,\pi/2<LH$. For
$\pi/2<LH$, signals emitted from the in-region I cannot reach the out-region
III.

Figure 6 shows the wave form
$\varphi_{\text{out}}(X)=\varphi_{\text{in}}(F(X))$ with
$\varphi_{\text{in}}(X)=e^{-ikX}$. Owing to the existence of the future
horizon $\mathscr{H}^{+}$, the wave is stretched and freezes out as
$X\rightarrow\infty$. This behavior of wave is same as that for black hole
formation via gravitational collapse. Thus, we expect the emission of Hawking
radiation from vicinity of the future horizon $\mathscr{H}^{+}$ if the scalar
field is quantized and the vacuum condition for the in-vacuum state is imposed
at $\mathscr{I}^{-}$ in Region I, that corresponds to the Unruh vacuum state
in the standard scenario of Hawking radiation via gravitational collapse [34,
35, 33]. We do not discuss detail of quantum effect in this paper (we will
discuss this aspect in our forthcoming paper [60]), but just investigate the
Bogoliubov coefficient which can be obtained from the relation between
classical wave modes $\varphi_{\text{in}}$ and $\varphi_{\text{out}}$.

Figure 6: The wave form received at a detection point in Region III
($LH=0.5$). $X$ corresponds to $x_{\text{out}}^{+}$. The input signal is a
plane wave $\varphi_{\text{in}}=e^{-ikX}$ with $k=10$. For $X\rightarrow
X_{*}\approx-1.6$, the input wave is infinitely blue-shifted owing to the past
de Sitter horizon $\mathscr{H}^{-}$ in Region II, and for
$X\rightarrow+\infty$, the input wave is infinitely red-shifted owing to the
future de Sitter horizon $\mathscr{H}^{+}$ in Region II.

For the input plane wave $\varphi_{\text{in}}(X)=e^{-ikX}$, from Eq. (27), the
Bogoliubov coefficients are obtained by the following Fourier transformation

$\alpha(k,k^{\prime})=\sqrt{\frac{k}{k^{\prime}}}\int_{-\infty}^{\infty}dX\,e^{-ik^{\prime}F(X)}e^{ikX},\quad\beta(k,k^{\prime})=\sqrt{\frac{k}{k^{\prime}}}\int_{-\infty}^{\infty}dX\,e^{-ik^{\prime}F(X)}e^{-ikX}.$
(57)

To extract information of late time particle creations in Region II, we
consider the asymptotic behavior of the function $F(X)$ for $HX\gg 1$:

$F(X)\approx-c_{0}-c_{1}e^{-HX},$ (58)

with

$c_{0}=\frac{2}{H}\ln(\tan(HL/2)),\quad
c_{1}=\frac{4}{H}\sqrt{\frac{1-\sin(HL/2)}{1+\sin(HL/2)}}.$ (59)

Using this asymptotic form of $F(X)$, we obtain the Bogoliubov coefficient as

$\displaystyle\beta(k,k^{\prime})$
$\displaystyle\approx\sqrt{\frac{k}{k^{\prime}}}\,\frac{e^{ik^{\prime}c_{0}}}{H}\int_{0}^{\infty}dy\,y^{-ik/H-1}\,e^{ik^{\prime}c_{1}y}$
$\displaystyle=\sqrt{\frac{k}{k^{\prime}}}\,\frac{e^{ik^{\prime}c_{0}}}{H}(-ik^{\prime}c_{1})^{-ik/H}\,\Gamma\left(\frac{ik}{H}\right).$
(60)

Therefore

$|\beta(k,k^{\prime})|^{2}\approx\frac{2\pi}{Hk^{\prime}}\,\frac{1}{\exp\left(2\pi
k/H\right)-1},$ (61)

and it shows the Planckian distribution with a temperature

$T_{\text{H}}=\frac{H}{2\pi}.$ (62)

This temperature coincides with the Gibbons-Hawking temperature in the de
Sitter spacetime. Equation (61) is a signal of the classical counterpart of
Hawking radiation from the future horizon in Region II. Although Region II
corresponds to an expanding de Sitter universe, owing to our setup of
experiment, the global structure of spacetime resembles the situation of black
hole formations by gravitational collapse, and it is possible to detect
Hawking radiation from the analog black hole. The temperature of the emitted
Hawking radiation is the same as that of the de Sitter horizon. More detailed
analysis on Hawking radiation from the expanding edge region will be presented
in our forthcoming paper [60].

##  V Summary

In this paper, we formulated a general theory for the analysis of future
experiments of the expanding edges of QH systems. The dynamics of edge
expansion is described by the wave form
$\varphi_{\text{out}}(x_{\text{out}}^{+})$ of edge excitation measured in the
Region III and its Bogoliubov coefficients $\alpha(k,k^{\prime})$ and
$\beta(k,k^{\prime})$. Based on Eq. (23), the experiments determine whether
the conformal symmetry survives in Region II. By increasing the amplitude of
the initial wave $\varphi_{\text{in}}(x_{\text{in}}^{+})$ in Region I, the
experiments determine whether nonlinear interactions of the edge waves are
generated in Region II. If the amplitude of the output wave $\varphi_{\text{
out}}(x_{\text{out}}^{+})$ does not increase linearly with the increase in the
amplitude of $\varphi_{\text{in}}(x_{\text{in}}^{+})$, the nonlinear
interactions exist. If the experiments show a breakdown of the edge unitarity
conditions of $\alpha(k,k^{\prime})$ and $\beta(k,k^{\prime})$ in Eq. (28a)
and Eq. (28b), the edge excitations interact with bulk excitations. More
detailed analyses will be reported in forthcoming papers.

We have also shown that the expanding edges can be regarded as analogs of
expanding universes in 2D dilaton gravity models, including the JT model and
dSJT model. By controlling the time schedule of the edge expansion, the
experiments are capable of simulating the gravity models with the potential
term $V\left(\Phi\right)$ in Eq. (49). Furthermore, we demonstrated the analog
Hawking radiation from the de Sitter horizon formed in the expanding edge
region, which might be detected in our experimental setup. Since the field
$\varphi$ in Eq. (38) for a QH system in a low-noise situation is regarded as
a quantum field in an expanding analog universe, the time-dependent QH systems
enable us to explore the trans-Planckian problem and the quantum-classical
transition problem in 2D dilaton gravity.

In conclusion, future studies of the expanding edge experiments are expected
to reveal new physics of QH systems, yielding numerous milestones that
endeavour to resolve the fundamental problems in early universe.

###### Acknowledgements.

The authors are grateful to K. Nakayama, T. Nishioka, N. Shibata, T.
Shiromizu, T. Takayanagi, M. Tezuka, and K. Yonekura for their fruitful
discussions. This work was supported by a Grant-in-Aid for Scientific Research
(Grant Nos. 21H05188, 21H05182 (M. H., G. Y.), JP19K03838 (M. H.), 17H01037
(G. Y.), 19H05603 (G. Y.), 21F21016 (G. Y.), and 19K03866 (Y. N.)) from the
Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan.

*

##  Appendix A Formula for 2D gravity

  * •

Flat spacetime metric:

$ds^{2}=-c^{2}dt^{2}+dx^{2}.$

  * •

Definition of Christoffel symbols for general metric form
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$:

$\Gamma_{\beta\gamma}^{\alpha}=\frac{1}{2}\,g^{\alpha\mu}\left(\partial_{\beta}g_{\mu\gamma}+\partial_{\gamma}g_{\mu\beta}-\partial_{\mu}g_{\beta\gamma}\right).$

  * •

Definition of Riemann curvature tensor:

$\displaystyle R^{\alpha}{}_{\beta\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Gamma_{\nu\beta}^{\alpha}-\partial_{\nu}\Gamma_{\mu\beta}^{\alpha}+\Gamma_{\mu\gamma}^{\alpha}\Gamma_{\nu\beta}^{\gamma}-\Gamma_{\nu\gamma}^{\alpha}\Gamma_{\mu\beta}^{\gamma},$
$\displaystyle R_{\alpha\beta\mu\nu}$ $\displaystyle=$ $\displaystyle
g_{\alpha\gamma}R^{\gamma}{}_{\beta\mu\nu}$

  * •

Definition of Ricci curvature tensor:

$R_{\alpha\beta}=R^{\mu}{}_{\alpha\mu\beta}=g^{\mu\nu}R_{\mu\alpha\nu\beta}.$

  * •

Definition of scalar curvature:

$R=g^{\alpha\beta}R^{\mu}{}_{\alpha\mu\beta}=g^{\alpha\beta}g^{\mu\nu}R_{\mu\alpha\nu\beta}.$

  * •

In 2D gravity theory, $R_{\alpha\beta\mu\nu}$ and $R_{\alpha\beta}$ are
uniquely determined by $R$ and $g_{\mu\nu}$ as

$\displaystyle R_{\alpha\beta\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}R\left(g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\nu}g_{\beta\mu}\right),$
$\displaystyle R_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,g_{\alpha\beta}R.$

  * •

Light cone coordinate systems:

$x^{\pm}=c\,t\pm x=x^{0}\pm x^{1}.$

  * •

Partial derivatives with respect to $x^{\pm}$:

$\displaystyle\partial_{+}$ $\displaystyle:$
$\displaystyle=\frac{\partial}{\partial x^{+}}=\frac{\partial x^{0}}{\partial
x^{+}}\,\partial_{0}+\frac{\partial x^{1}}{\partial
x^{+}}\,\partial_{1}=\frac{1}{2}(\partial_{0}+\partial_{1}),$
$\displaystyle\partial_{-}$ $\displaystyle:$
$\displaystyle=\frac{\partial}{\partial x^{-}}=\frac{\partial x^{0}}{\partial
x^{-}}\,\partial_{0}+\frac{\partial x^{1}}{\partial
x^{-}}\,\partial_{1}=\frac{1}{2}(\partial_{0}-\partial_{1}).$

  * •

Any metric form can be rearranged into a conformally flat metric form as

$ds^{2}=-\exp\left(2\Theta(x^{+},x^{-})\right)dx^{+}dx^{-},$

at least, in any local region of whole spacetime.

  * •

Nonvanishing $\Gamma_{\beta\gamma}^{\alpha}$ in conformally flat coordinate
system:

$\displaystyle\Gamma_{++}^{+}$ $\displaystyle=$ $\displaystyle
2\,\partial_{+}\Theta,$ $\displaystyle\Gamma_{--}^{-}$ $\displaystyle=$
$\displaystyle 2\,\partial_{-}\Theta.$

  * •

Scalar curvature in conformally flat coordinate system:

$R=8\,e^{-2\Theta}\,\partial_{+}\partial_{-}\Theta.$

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