eprint Nisho-1-2023

# A Way of Determination of Axion Mass with Quantum Hall Effect

Aiichi Iwazaki  International Economics and Politics, Nishogakusha University,
6-16 3-bantyo Chiyoda Tokyo 102-8336, Japan

(Jun. 26, 2023)

###### Abstract

Axion dark matter is converted to electromagnetic radiations in the presence
of strong magnetic field. The radiations possibly give rise to non trivial
phenomena in condensed matter physics. Especially, we discuss that saturation
of plateau-plateau transition width observed at low temperature in integer
quantum Hall effect is caused by the axion. The radiations from axions are
inevitably present in the experiment. Although the radiations generated by
axion is extremely weak, Hall conductivity jumps up to next plateau even if
only a single electron occupies an extended state; a localized electron is
transited to the extended state by absorbing the radiation. According to our
analysis, previous experimentsat6  of the saturation in detail suggests that
the axion mass is in the range $10^{-5}\mbox{eV}\sim 10^{-6}$eV. We propose a
way of the determination of the axion mass by imposing microwaves on Hall bar
and also a way of the confirmation that the axion really causes the saturation
of the width.

##  I introduction

One of most significant issues in particle physics is to find an evidence
beyond the standard model. Especially, finding axion is the important step
exploring a new physics. It also gives rise to a solution of dark matter in
the Universe. Axion is the Goldstone boson of Peccei Quinn symmetryaxion1 ;
axion2 ; axion3 , which naturally solves strong CP problem. Such an axion is
called as QCD axion. The allowed mass rangeWil ; Wil1 ; Wil2  of the QCD axion
is severely restricted such as $m_{a}=10^{-6}\mbox{eV}\sim 10^{-3}$ eV.
Hereafter we only consider QCD axion, although our argument may be applicable
to non QCD axion, dark photon e.c.t. In this paper we use the units, $c=1$,
$k_{B}=1$ and $\hbar=1$.

Many projectsadmx ; carrack ; haystac ; abracadabra ; organ ; madmax ; brass ;
iwazaki01  for the detection of the dark matter axion have been proposed and
are undergoing at present. Most of them use a mechanism that the axion is
converted to electromagnetic radiation under strong magnetic field. Namely,
the axion dark matter produces electromagnetic fields when it is imposed by
strong magnetic field. The radiations are detected with the use of resonant
cavity, Rydberg atom, LC circuit, e.t.c.

The axion dark matter may cause some of astrophysical phenomena including
gravitational lens, or phenomena not yet well understood such as fast radio
burstsfastradioburst1 ; fastradioburst2 . The axion dark matter may also give
rise to significant effects in condensed matter physics at low temperature.
Coherence of electrons in conductor, semiconductor, e.t.c. is in general
disturbed by thermal effects. When they are disconnected from thermal bath,
their coherence is restored completely. But, even with the disconnection, the
electrons are in the bath of axion dark matter. Although the coupling of the
axion with individual electron is extremely small, the axion effect may be
observedmagnon ; center  because a large number of electrons are involved in
condensed matter.

In this paper we consider axion effect in quantum Hall phenomenagirvin , which
are realized in two dimensional electrons under strong magnetic field $B$
perpendicular to them. We focus on integer quantum Hall effects. It is well
known that Hall resistance $\rho_{xy}$ ( conductance $\sigma_{xy}$ ) is
quantized such as $\rho_{xy}=(2\pi/e^{2})\times 1/n$ (
$\sigma_{xy}=-\rho_{xy}^{-1}$ ) with positive integer $n$ and electric charge
$-e$ of electron. The diagram of $\rho_{xy}$ in variable $B$ shows plateausvon
; aokiando ; halperin  of $\rho_{xy}$ for each $n$. They look like steps,
while $\rho_{xx}$ ( $\sigma_{xx}$ ) vanishes on the plateaus. But $\rho_{xx}$
( $\sigma_{xx}$ ) takes non zero values in small width $\Delta B$ between two
plateaus. That is, $\rho_{xy}$ increases larger than the value
$(2\pi/e^{2})\times 1/n$ on a plateau and forms next plateau
$\rho_{xy}=(2\pi/e^{2})\times 1/(n-1)$ as magnetic field $B$ increases. There
is small width $\Delta B$ between the two plateaus. The width shrinks as
temperature $T$ decreases. In other wards, the plateau-plateau transition is
not sharp. The plateau $(2\pi/e^{2})\times 1/n$ smoothly connects the plateaus
$(2\pi/e^{2})\times 1/(n-1)$ when temperature $T\neq 0$. But, it is expected
theoretically that at $T=0$ the plateau-plateau transition is sharp because
$\Delta B(T)$ is expected to vanish at $T=0$. But actual experiments show that
the width $\Delta B$ saturates a finite value as $T$ goes to zero. Plateau-
plateau transition is not sharp. It is generally believed that the saturation
is intrinsic although plausible explanation for the intrinsic is still lack.
In this paper, we discuss that axion dark matter causes the saturation. That
is, axions are converted to electromagnetic radiations under the strong
magnetic field $B$. The radiations disturb coherence of electrons, which
result non vanishing $\Delta B$ at $T=0$. The radiations converted from axions
play a role of external noise against the measurement of $\Delta B$ just like
thermal noise. ( Sometimes, the derivative $d\rho_{xy}/dB$ is discussed to see
the sharpness of the plateau-plateau transition. The saturation of $\Delta
B\neq 0$ corresponds to the saturation of $d\rho_{xy}/dB\neq\infty$ as $T\to
0$. )

Quantum Hall effectvon  is realized in two dimensional electron system. The
system is fabricated in MOSFET or heterostructures of GaAs. The system shows
various intriguing phenomena under strong magnetic field such as not only
quantization of Hall resistance but also Josephson-like effectjoseph1 ;
joseph2 ; joseph3  e.t.c.. Quantum Hall system has been extensively
investigated since the discovery, but some of phenomena are not still fully
understood. One of the phenomena is the saturationsat1 ; sat2 ; sat3 ; sat4 ;
sat5 ; sat6  of the width $\Delta B$ in plateau-plateau transition of integer
quantum Hall effect. The width arises at $T\neq 0$ because thermal effects
cause localized electrons below Fermi energy $E_{f}$ to transit to extended
states above $E_{f}$. They occupying the extended states carry electric
current so that the value of electric Hall resistance decreases in the
transition region. We discuss that similar effectsat1 ; micro  to the thermal
one may arise owing to electromagnetic radiations produced by axion dark
matter. The radiations are produced under the strong magnetic field, which is
used in the experiment of quantum Hall effect. The radiations make localized
electrons transit to the extended states so that the non vanishing width
$\Delta B$ arises even at $T=0$. With imposing microwaves on Hall bar, we can
determine the axion mass by observing the saturation of the width. Decreasing
their frequencies $\nu$ of the microwaves, the width $\Delta B(\nu)$
decreasesmicro . But when the frequency is below a critical frequency, we
observe the saturation. The critical frequency $\nu_{c}$ is given by the axion
mass $m_{a}$; $\nu_{c}=m_{a}/2\pi$.

The detection of electron’s absorption of radiations generated by axions has
been performed with the use of Rydberg atoms. In the experiment,
Carrackcarrack , ionized atoms are detected so that a number of the atoms,
which absorb radiations, are needed for the detection. In our case, there are
many electrons $\sim 10^{7}$ in Hall bar with typical size
$200\mu\mbox{m}\times 50\mu$m$=10^{-4}\rm cm^{2}$. Furthermore, only a single
electron is sufficient to make Hall conductivity jump up to next plateau.

In next section(II) we explain integer quantum Hall effect. In section(III) in
order to explain the presence of Hall plateau, we give a brief review of
localization of electrons in integer quantum Hall effect. In section(IV) we
discuss how axion dark matter affects the width $\Delta B$ in plateau-plateau
transition. In section(V) we give a brief review of the features of QCD axion
dark matter. We estimate transition probability of localized electron to
extended states by absorbing radiations converted from axion in section(VI).
In section(VII) we show the number of electrons steady present in extended
states, taking account of the inverse process that electrons occupying
extended states transit back to localized states by spontaneously emitting
radiations. In section(VIII), we calculate the relation between $\Delta B$ and
the width $\Delta E_{f}$ of Fermi energy, within which the transition from
plateau $(2\pi/e^{2})\times 1/(n-1)$ to plateau $(2\pi/e^{2})\times 1/n$ is
complete. We find that the dependence agrees well to experimental results.
Finally, we propose a way of the determination of axion mass by imposing
external microwaves to Hall bar. Furthermore, we propose a way of the
confirmation that axion really produces the saturation of the width $\Delta B$
in section(IX). Finally, conclusions follow in section(X).

##  II integer quantum hall effect

We make a brief review of integer quantum Hall effectgirvin . The states of
two dimensional free electrons under magnetic field $B$ are specified by
integer $n\geq 0$, so called Landau levels with their energies specified as
$E_{n}$ ). ( The two dimensional space is specified by the coordinates $x$ and
$y$ and $\vec{B}=(0,0,B)$. The two dimensional electrons are formed by being
trapped in a quantum well between for instance, GaAs and AlGaAs. The motion of
the electrons in the direction $z$ is energetically forbidden, while the
motion in the direction $x$ and $y$ is energetically allowed. ) There are a
number of degenerate states in each Landau level with the degeneracy
$eB/2\pi$; the number density of degenerate state in each Landau level. (
Density of states is described as a delta function $\propto\delta(E-E_{n})$. )
Their wave functions are given by

$\Phi(\vec{x},n,k)=C_{n}H_{n}\Big{(}\frac{x-l_{B}^{2}k}{l_{B}}\Big{)}\exp\Big{(}-\frac{(x-l_{B}^{2}k)^{2}}{2l_{B}^{2}}\Big{)}\exp(-iky)$
(1)

with normalization constant $C_{n}=(2^{n}n!\sqrt{\pi}l_{B}L_{y})^{-1/2}$,
where $L_{y}$ is the length of the Hall bar in the direction $y$ and
$H_{n}(x)$ denotes Hermite polynomials. $l_{B}=\sqrt{1/eB}$ is called as
magnetic length ( $l_{B}\simeq 1.8\times 10^{-7}\rm cm(B/10T)$ ). We use a
gauge potential $\vec{A}_{ex}=(0,Bx,0)$. Their energies are
$E_{n}=\omega_{c}(n+1/2)$ with cyclotron frequency $\omega_{c}=eB/m^{\ast}$;
effective mass $m^{\ast}$ of electron in semiconductor. Generally $m^{\ast}$
is much smaller than real mass $m_{e}\simeq 0.5$MeV of electron. ( For
instance, $m^{\ast}=0.067m_{e}$ in GaAs. Hereafter, we use physical parameters
of GaAs. ) Degenerate states in each Landau level are specified by $k$.
Cyclotron frequency is of the order of $\sim 10^{-2}(B/10T)$eV.

Electron possesses spin components with up and down. Thus, each Landau level
is split to two states with energies $E_{n\pm}=\omega_{c}(n+1/2)\pm g\mu_{B}B$
owing to Zeeman effect; $\pm$ correspond to spin parallel and anti-parallel to
the magnetic field $B$, respectively. Namely, Zeeman energy $\pm g\mu_{B}B$
with $g\simeq 0.44$ and Bohr magneton $\mu_{B}=e/2m_{e}$. Typically, Zeeman
energy is of the order of $10^{-3}(B/10T)$eV. Obviously Zeeman energy is
smaller than the cyclotron energy in semiconductor. The degeneracy ( number
density of electrons ) of each Landau level with definite spin is given by
$eB/2\pi$.

Starting from the lowest energy state, the states in higher Landau levels are
occupied. Namely, increasing the number density of electron $\rho_{e}$, higher
Landau levels with larger energies are occupied. Usually, we use an index so
called filling factor $\nu=2\pi\rho_{e}/eB$. It implies for instance, Landau
level with energy $E_{0-}$ is fully occupied but Landau level with $E_{0+}$ is
partially occupied when $2>\nu>1$.

Flowing an electric current in $x$ direction of Hall bar, we measure voltages
in $x$ and $y$ direction, i.e. $V_{xx}$ and $V_{xy}$, respectively. Then we
obtain $\rho_{xy}$ and $\rho_{xx}$. It is remarkable phenomenavon ; aokiando ;
halperin  that Hall resistance $\rho_{xy}=(2\pi/e^{2})\times 1/n$ is constant
even when the filling factor $\nu$ changes within a range $n+1>\nu>n$: Landau
levels up to $n$ are fully occupied, while the level with $n+1$ is partially
occupied. That is, we see plateaus in the diagram of $\rho_{xy}$ in $B$. The
plateaus arise owing to the localization of two dimensional electrons.

##  III localization of two dimensional electrons

We need to explain localization of two dimension electrons in order to
understand integer quantum Hall effect. In actual materials, there are
impurities, defects e.t.c. which lift up the degeneracy in Landau level.
Electrons do not freely move, but randomly move scattering with them. That is,
electrons move in a random potential $V$ caused by disorders such as
impurities, defects, e.t.c. . As a result, most of electrons are localized so
that they cannot carry electric currents. But, a small fraction of them are
not localized so that they can carry electric current. As is well known, all
of two dimensional electrons are localizedlocalization  around impurities,
defects, e.c.t. when there is no external magnetic field ( i.e. there is no
interactions which break time reversal symmetry. ) Thus, even if magnetic
field $B$ is present, it is naturally expected that most of electrons are
localized, although non-localized electrons are present. We stress that
localized electrons cannot carry electric currents, while non-localized
electrons carry electric currents. The point for understanding integer quantum
Hall effect is the presence of non-localized ( extended )aokiando ; ono
electrons when $B\neq 0$. Although the degeneracy is lifted up, there exist a
small fraction of states with energy $E=E_{n\pm}$. Electrons in such a state
are not localized. Actually, numerical simulations show that when there is a
potential $V$ representing disorder, the density of states $\rho(E)$ has
finite width around the energy $E_{n\pm}$ shown schematically in Fig.1

Figure 1: Density of state $\rho(E)$. Dashed line denotes a density of state
of free electrons under magnetic field $\vec{B}$. Dotted line represents Fermi
energy $E_{f}$

The effect of the potential $V$ makes the density of state $\rho(E)$ to have
finite width $\Delta E$ in the distribution. It is assumed that the width is
much less than the cyclotron frequency $\omega_{c}=eB/m^{\ast}$ in strong
magnetic field $B$. It holds for the potential energy which are much smaller
than the cyclotron frequency. That is, $\Delta E\ll\omega_{c}$. (
$\omega_{c}\sim 10^{-2}$eV with $B\sim 10$T. ) For instance,
$\rho(E)\propto\sqrt{1-((E-E_{n\pm})/\Delta E)^{2}}$ with $E-E_{n\pm}>\Delta
E$ andouemura . Furthermore, the potential is supposed to have the same amount
of attractive and repulsive components. So, the form of $\rho(E)$ is symmetric
around the center $E_{n\pm}$ as shown in the figure.

When the variation length of the potential is much larger than the magnetic
length $l_{B}\simeq 8\times 10^{-7}\rm(10\mbox{T}/B)^{1/2}$cm, we may consider
that electrons move along the contour of the potential, circulating with small
radius $l_{B}$. Because almost contours are closed, electrons moving along the
contours are localized. On the other hand, a small fraction of the contours
extends from boundary to boundary. Such electrons moving along the extended
contours can carry electric current from boundary to boundary. It is the
physical picture of localization. Even if the variation length of the
potential is of the same order of $l_{B}$, the numerical simulationsimulation
shows that there exist extended states with their energies $E_{n\pm}$ and
other states with energies $\neq E_{n\pm}$ are localized in the case of
infinitely large Hall bar.

When Fermi energy $E_{f}$ is less than $E_{n+}$ but larger than $E_{n-}$,
electrons which can carry electric currents, are in extended states whose
energies are equal to $E_{n-}$ or $E_{m\pm}$ with $n>m$. For instance, for
$E_{1+}>E_{f}>E_{1-}$, Hall resistance is given such that
$\rho_{xy}=(2\pi/e^{2})\times 1/3$, while Hall conductance $\sigma_{xy}$ is
such that $\sigma_{xy}=-1/\rho_{xy}$ ( Conventionally, we refer Hall
conductance as positive value like $\sigma_{xy}=n\times e^{2}/2\pi$. ) As long
as the Fermi energy $E_{f}$ is in the range $E_{1+}>E_{f}>E_{1-}$, the Hall
resistance does not vary so that a plateau is formed. On the plateau, we have
longitudinal resistance $\rho_{xx}=0$, which implies that the electric current
flowing in $x$ direction is dissipationless. The currents are carried by
electrons in extended states with energies such as $E_{1-}$ and $E_{0\pm}$.
Namely, when we add an electron, it occupies a localized state with energy
$E_{f}<E_{1+}$ which does not carry electric current. Thus, Hall resistance
does not change. But, if the electron occupies the extended states with the
energy $E_{f}=E_{1+}$, it carry electric current. Then, Hall resistance
suddenly down to $\rho_{xy}=(2\pi/e^{2})\times 1/4$, or Hall conductivity
rises up to $\sigma_{xy}=4\times e^{2}/2\pi$ from $\sigma_{xy}=3\times
e^{2}/2\pi$. The transition is sharp like the step function.

We should note that when we increase magnetic field $B$ without changing the
number density of electrons, the degeneracy $eB/2\pi$ increase but the Fermi
energy decreases. The Fermi energy changes with magnetic field $B$. Then, we
see plateau in diagram of $\rho_{xy}$ vs $B$, as long as $E_{f}$ does not
cross the energy $E_{n\pm}$. Inversely when we decrease $B$, $E_{f}$ moves
upward ( increases ). If $E_{f}$ passes the energy, for instance, $E_{1+}$,
Hall resistance steps down from $\rho_{xy}=(2\pi/e^{2})\times 1/3$ to
$\rho_{xy}=(2\pi/e^{2})\times 1/4$. In other wards, $\sigma_{xy}$ steps up
from $\sigma_{xy}=3\times e^{2}/2\pi$ to $\sigma_{xy}=4\times e^{2}/2\pi$.
This is because the electrons occupying the states with the energy $E_{1+}$ (
$<E_{f}$ ) carry electric current. It should be noticed that even if a single
electron occupies the extended states with the energy $E_{1+}$, the Hall
conductance jumps to $\sigma_{xy}=4\times e^{2}/2\pi$ from
$\sigma_{xy}=3\times e^{2}/2\pi$. This is a striking feature of quantum Hall
effect. The feature is understood in topological argumenttopology1 ; topology2
.

Hereafter, we examine in detail the case of $E_{1-}<E_{f}<E_{1+}$ for
concreteness.

Up to now, we have discussed localization in the infinite large size of Hall
bar, in which extended states have the energies $E_{n\pm}$. But in the case of
finite size, there are localized states with their sizes larger than the size
of Hall bar. Such states carry electric current and have energies in the range
$E_{n\pm}-\delta\leq E\leq E_{n\pm}+\delta$ where $\delta(L_{h})$ depends on
the size $L_{h}$ of Hall bar such as $\delta(L_{h}\to\infty)=0$. In general
$\delta$ is smaller than the width $\Delta E$ of $\rho(E)$. We call such
states as effective extended states because they can carry electric current.
Hence, when the Fermi energy $E_{f}$ in the range $E_{n\pm}-\delta\leq
E_{f}\leq E_{n\pm}+\delta$, for instance $E_{1+}-\delta\leq E_{f}\leq
E_{1+}+\delta$, Hall resistance $\rho_{xy}$ takes the value
$(2\pi/e^{2})\times 1/4$.

##  IV axion effect on plateau-plateau transition

In the above explanation, we do not include thermal effect on the electrons.
When temperature $T\neq 0$, probability that electrons occupy the states with
energies larger than Fermi energy does not vanish. It is proportional to
$\exp(-|E-\mu(T)|/T)$ with chemical potential $\mu(T)$; $\mu(T)\to E_{f}$ as
$T\to 0$. It implies that even if $\mu(T)$ is less than $E_{1+}$, there are
non vanishing probability of electrons occupying the extended states with the
energy $E_{1+}$. So the Hall conductivity $\sigma_{xy}$ does not keep the
value $\sigma_{xy}=3\times e^{2}/2\pi$, becoming larger than this one. It
gradually increases toward the value $\sigma_{xy}=4\times e^{2}/2\pi$ when
chemical potential $\mu$ approaches the energy $E_{1+}$. ( Inversely, Hall
resistance $\rho_{xy}=2\pi/e^{2}\times 1/3$ goes down to
$\rho_{xy}=2\pi/e^{2}\times 1/4$ as chemical potential increases. On the other
hand, Hall resistance goes up as magnetic field $B$ increases. ) Therefore,
when $T\neq 0$, the transition between plateaus in the variable $B$ is not
sharp as shown in Fig.2.

Figure 2: plateau-plateau transition with width $\Delta B$

In the figure, we show the width $\Delta B$ between two plateaus. We expect
from the above argument that the width $\Delta B\to 0$, as $T\to 0$.
ExperimentsdeltaB  have shown that the width $\Delta B\propto T^{\kappa}\to 0$
as $T\to 0$ with $\kappa\sim 0.4$ as far as $T$ is not small so much. However,
experimentssat1 ; sat2 ; sat3 ; sat4 ; sat5  in detail indicate that $\Delta
B$ approaches non zero constant as $T\to 0$. That is, the width saturates at
low temperature. There is not yet convincing reason why the width saturates at
finite value without vanishing. The similar phenomena can be seen as a
decoherencedecoherence  in states of electrons in topological insulators,
e.c.t.. It is considered that the saturation arises due to intrinsic origin,
not extrinsic one: Effect of temperature induced in measurement, finite size
effect of Hall bar, e.c.t. have been excluded from the cause of the
saturation. But, there is no plausible explanation of the intrinsic origin.
The presence of the finite width $\Delta B$ even at $T=0$ shows the presence
of a decoherence mechanism. Especially, it indicates non vanishing probability
of a transition from an electron in localized state to an electron in extended
state. Such a transition may be caused by axion dark matter.

The plateau-plateau transition is a kind of phase transition. Actually, it has
been shownaoki1 ; aoki2  that a coherent length $\xi(E)$ diverges such as
$\xi(E)\to|E-E_{n\pm}|^{-\nu}$ as $E\to E_{n\pm}$ with $\nu\sim 2.4$. The
coherent length may be considered to be typical length of localized state with
energy $E$. The transition is characterized as the one between insulator and
metal. The state with $E_{f}<E_{1+}$ is insulator, while the state with
$E_{f}\geq E_{1+}$ in which electrons occupy the states with $E=E_{1+}$, is
metal. The divergence of the coherent length at $E=E_{n\pm}$ implies that only
a single state with energy $E_{n\pm}$ is the extended state, in which
electrons can carries electric current in Hall bar with infinite length.

The coherent length is a typical size of localized state with energy $E$.
Thus, the size of localized state becomes larger than the finite size of Hall
bar as $E$ approaches critical energies $E_{n\pm}$. It implies that localized
states with their energies $E$ in the range $E_{n\pm}-\delta\leq E\leq
E_{n\pm}+\delta$ can be effective extended states, which carry electric
currents from boundary to boundary in Hall bar with finite size. Then,
insulator metal transition arises when Fermi energy $E_{f}$ approaches the
energy $E_{n\pm}-\delta$ from below or above. Without thermal effect, the
transition is expected to be sharp. But it is not true in actual Hall bars.
The width $\Delta B$ saturates at finite value as $T\to 0$. Finite size of
Hall barsat3  causes the saturation, but the finite size effect is not only
the causesat4 ; sat5 . In this paper we discuss that the axion dark matter is
the cause of the saturation.

Axion dark matter is converted to radiations in the presence of magnetic
field. In particular, Such radiations are inevitably present in the experiment
of quantum Hall effect, because strong magnetic field generated by
superconducting magnet is used. The magnet plays a role of cavity, with use of
which axion dark matter has been explored. In this paper we consider QCD
axion, whose mass $m_{a}$ is supposed to be in the range $10^{-4}\rm eV\sim
10^{-6}eV$. Among them, we specially focus on the mass range $10^{-5}\rm
eV\sim 4\times 10^{-6}eV$ because previous experiments about the width
suggests the presence of the axion mass in the range, see below. ( The
frequencies of corresponding radiations are in about $1\mbox{GHz}\sim
2.4\mbox{GHz}$. They are microwaves. ) Radiation energies converted from
axions are almost the same as $m_{a}$, because the momenta of the dark matter
axions are of the order of $10^{-3}m_{a}$. Thus, electrons with energy $E$
acquires the energy $E+m_{a}$ when they absorb the radiations.

We make a comment. Two dimensional electrons are formed in a quantum well
fabricated by semiconductor. The width of the well is less than of the order
of $10^{-6}$cm. The semiconductor is transparent for microwaves in low
temperature.

We consider the case in which Fermi energy is slightly smaller than
$E_{1+}-\delta$. By absorbing the radiations from axions, electrons with
energies $E$ ( $<E_{f}-\delta$ ) transit to the states with the energies
$E+m_{a}$ larger than $E_{1+}-\delta$. The process can arise only for
$m_{a}+E_{f}>E_{1+}-\delta$. The transited electrons with energies in the
range $E_{n\pm}-\delta\leq E\leq E_{n\pm}+\delta$ carry electric current.
There are non vanishing probability that the effective extended states are
occupied, even if the Fermi energy $E_{f}$ less than $E_{1+}-\delta$. Thus,
the axions may change the value of Hall resistance. The effect is similar to
thermal one with temperature $T\sim m_{a}$.

Here we should stress that even if only a single electron occupies an
effective extended state, Hall resistance or conductance changes dramatically
to show plateau-plateau transitiontopology1 ; topology2 . Although the
probability of axion conversion to photon is extremely small, the possibility
of a single electron making transition to an effective extended states is not
small because there are a number of localized electrons which may transit to
the effective extended states with energies $E+m_{a}$ (
$E_{1+}-\delta<E+m_{a}<E_{1+}+\delta$ ) in Hall bar.

##  V axion dark matter

In order to discuss the electron transition by absorbing radiations from axion
dark matter, we explain electromagnetic coupling with axionaxion1 ; axion2 ;
axion3  and smallness of the coupling. The coupling $g_{a\gamma\gamma}$
between axion $a(t,\vec{x})$ and electromagnetic field is given by

$L_{a\gamma\gamma}=g_{a\gamma\gamma}a(t,\vec{x})\vec{E}\cdot\vec{B},$ (2)

with electric $\vec{E}$ and magnetic $\vec{B}$ fields. We note that
$g_{a\gamma\gamma}=g_{\gamma}\alpha/f_{a}\pi$, where $\alpha\simeq 1/137$
denotes fine structure constant and $f_{a}$ does axion decay constant
satisfying the relation $m_{a}f_{a}\simeq 6\times 10^{-6}\rm eV\times
10^{12}$GeV in the QCD axion. The parameter $g_{\gamma}$ depends on the axion
model, i.e. $g_{\gamma}\simeq 0.37$ for DFSZ modeldfsz ; dfsz1  and
$g_{\gamma}\simeq-0.96$ for KSVZ modelksvz ; ksvz1 .

The interaction term is much small in the case of the axion dark matter
$a(t,\vec{x})\simeq a_{0}\cos(m_{a}t)$. ( The momentum $p$ of the axion dark
matter is of the order of $10^{-3}m_{a}$ so that we may neglect the momentum.
) We estimate it supposing that the dark matter in the Universe is composed of
the axion. The local energy density $\rho_{d}$ of the dark matter is given
such that
$\rho_{d}=m_{a}^{2}\overline{a(t,\vec{x})^{2}}=m_{a}^{2}a_{0}^{2}/2\sim 0.3\rm
GeV/cm^{3}$ ( $\overline{Q}$ denotes time average of the quantity $Q$. ) Thus,
the coupling of the CP violating term is found to be much small;
$g_{a\gamma\gamma}a(t,\vec{x})\sim 10^{-21}$ independent of the QCD axion
mass. Thus, the electric field $E_{a}$ of the radiation producedsikivie ;
iwazaki01  in vacuum by the axion under external magnetic field $B$ is much
weak such as $E_{a}\sim g_{a\gamma\gamma}aB$. Such radiations inevitably arise
in experiment of quantum Hall effect because strong magnetic field $\sim 10$T
is used. Furthermore, superconducting magnet plays a role of cavitysikivie ;
iwazaki01  for the radiations.

As we have mentioned, we focus on the mass $m_{a}$ such as
$10^{-5}\mbox{eV}>m_{a}>4\times 10^{-6}$eV ( the corresponding frequency
$m_{a}/2\pi$ of radiation is about $1\mbox{GHz}\sim 2.4\mbox{GHz}$. )

In the next section, we estimate transition probability of electron in a state
with energy $E$ ( $<E_{f}$ ) to a state with energy larger than
$E_{1+}-\delta$ by absorbing the radiation,
$\vec{A}_{a}=\vec{A}_{0}\exp(-i\omega
t+i\vec{p}\cdot\vec{x})\simeq\vec{A}_{0}\exp(-im_{a}t)$ whose amplitude
$|\vec{A}_{0}|\sim g_{a\gamma\gamma}a_{0}B/m_{a}$. We take average over the
direction of the gauge field $\vec{A}_{0}$ because the polarization of
radiations produced in the experiment are not well controlled contrary to the
case of the detection of axions in resonant cavity. The point is that the
magnitudes $|\vec{A}_{0}|$ of the radiations are extremely small.

##  VI transition by absorption of electromagnetic wave caused by axion

We discuss a transition from an electron in a state with energy $E_{\alpha}$
to a state with energy $E_{\alpha}+m_{a}$ ( $>E_{1+}-\delta$ ). A state in a
Landau level $n$ is described by the wave function in the following,

$\Phi_{\alpha}=\exp(-iE_{\alpha}t)\int
dkD_{n}f_{\alpha}(k)H_{n}\Big{(}\frac{x-l_{B}^{2}k}{l_{B}}\Big{)}\exp\Big{(}-\frac{(x-l_{B}^{2}k)^{2}}{2l_{B}^{2}}\Big{)}\exp(-iky),$
(3)

with $D_{n}=(2^{n+1}n!\pi^{3/2}l_{B})^{-1/2}$. The function $f_{\alpha}(k)$ is
taken such that the wave functions are eigenstates of electron’s Hamiltonian
with potential $V$ representing disorder. Normalization is such that that
$1=\int dxdy\overline{\Phi_{\alpha}(x,y)}\Phi_{\alpha}(x,y)=\int
dk\overline{f_{\alpha}(k)}f_{\alpha}(k)$. The state is characterized by index
$\alpha$. Here we assume no mixing between different Landau levels and between
different spin states. Electron in the state $\alpha$ makes a transition to a
state with energy $E_{\alpha}+m_{a}$ by absorbing radiation. Here we only
consider the Landau level $n=1$.

The absorption is described by the following interaction Hamiltonian,

$H_{a}=\frac{-ie\vec{A}_{a}\cdot\vec{P}}{m^{\ast}}$ (4)

with $\vec{P}=-i\vec{\partial}+e\vec{A}_{ex}$ and
$\vec{B}=\vec{\partial}\times\vec{A}_{ex}$, where the gauge potential
$\vec{A}_{a}$ is taken such that $\vec{A}_{a}\sim\vec{A}_{0}\exp(-im_{a}t)$
with $|\vec{A}_{0}|\simeq g_{a\gamma\gamma}a_{0}B$.

The transition amplitude from the state $\alpha$ to the state $\beta$ is
proportional to

$<\beta|H_{a}|\alpha>=\int
dxdy\overline{\Phi}_{\beta}(x,y)\frac{-ie\vec{A}_{a}\cdot\vec{P}}{m}\Phi_{\alpha}(x,y)=i(E_{\beta}-E_{\alpha})e\vec{A}_{a}\cdot<\beta|\vec{x}|\alpha>\equiv
i(E_{\beta}-E_{\alpha})e\vec{A}_{a}\cdot\vec{L}_{\alpha\beta}$ (5)

where $<\beta|z|\alpha>=0$ because the two dimensional states $\alpha$ and
$\beta$ are orthogonal to each other. Electrons trapped in quantum well have
wave function such as $D(z)\Phi_{\alpha}(x,y)$; $D(z)$ is the common factor
for all $\alpha$.

$\vec{L}_{\alpha\beta}$( $\equiv<\beta|\vec{x}|\alpha>$ ) denotes a length
scale of overlapping region of two states $\Phi_{\alpha}$ and $\Phi_{\beta}$,
which is defined in the following,

$\vec{L}_{\alpha\beta}\equiv\int
dxdy\overline{\Phi}_{\beta}(x,y)\vec{x}\Phi_{\alpha}(x,y).$ (6)

Consequently, the number of electrons $N$ making transitions per unit time by
absorbing the radiation $\vec{A}_{a}$ from states with energies $E_{\alpha}$
lower than Fermi energy $E_{f}$ ( $<E_{1+}-\delta$ ) to states with energies
$E_{\beta}$ larger than $E_{1+}-\delta$ is given by

$\displaystyle\dot{N}$ $\displaystyle=$ $\displaystyle 2\pi S^{2}\int
dE_{\beta}dE_{\alpha}\rho({E_{\beta}})\rho(E_{\alpha})|<\beta|H_{a}|\alpha>|^{2}\delta(E_{\alpha}+m_{a}-E_{\beta})$
(7) $\displaystyle=$ $\displaystyle 2\pi S^{2}\int
dE_{\alpha}\rho({E_{\alpha}+m_{a}})\rho(E_{\alpha})m_{a}^{2}\Big{(}e\vec{A}_{a}\cdot\vec{L}_{\alpha\beta}\Big{)}^{2}$

with surface area $S$ of the Hall bar. We take $S=200\mu\rm m\times 50\mu
m=10^{-4}cm^{2}$ as typical value.

As we have explained, even localized states can carry electric current when
the length scale $\xi(E_{\beta})$ of the state $\beta$ is larger than the
length $L_{h}$ of Hall bar. The transition of electron from the state $\alpha$
to the state $\beta$ increases Hall conductivity. It has been discussed that
the scale $\xi(E)$ behaves such as $\xi(E)\propto|E-E_{1+}|^{-\nu}$ as $E\to
E_{1+}$ with $\nu\simeq 2.4$. For instance, it has been numerically shownaoki1
; aoki2  using a finite size scaling analysis that $\xi(E)>100\mu$m for
$B=10$T when $|E-E_{1+}|<0.1\Delta E$ where $\Delta E$ denotes the width of
number density of states $\rho(E)$. For instance
$\rho(E)\propto\sqrt{1-(E-E_{1+})^{2}/\Delta E^{2}}$. Hence, even if the Fermi
energy is less than $E_{1+}-\delta$, Hall conductivity $\sigma_{xy}=3\times
e^{2}/2\pi$ jumps to $\sigma_{xy}=4\times e^{2}/2\pi$, if there are electrons
occupying in the states $\beta$.

We consider the transition of electrons $\alpha$ with the energy $E_{\alpha}$
( $<E_{f}-\delta$ ) to the states $\beta$ with the energy $E_{\beta}$ in the
range $E_{1+}-\delta<E_{\beta}<E_{1+}+\delta$ with, for instance,
$\delta\equiv\Delta E/10$, in the analysis of the Hall bar with $L_{h}\sim
200\mu$m. We call the range as E range. Effective extended states with
energies in the E range can carry electric current. We note that the E range
depends on the size of Hall bar.

We stress that even if only a single electron is present in the effective
extended states, Hall resistance takes a value of $\rho_{xy}=2\pi/e^{2}\times
1/4$. The probability that the states in the E range are occupied is non
vanishing even if the Fermi energy is smaller than $E_{1+}-\delta$. Thermal
fluctuation or electromagnetic noise by axions mentioned above make the
probability non vanishing. It leads to the finite width of $\Delta B$ in
plateau-plateau transition even at $T=0$.

##  VII number of electrons steady present in effective extended states

Now, we estimate the transition probability of localized electrons to
effective extended states. In particular, we estimate the number $\dot{N}$ of
electrons per unit time occupying the states in the E range, $E_{1+}-\delta$
to $E_{1+}+\delta$. In the estimation we need to know the number density of
the states $\rho(E)$. It is shown schematically in Fig.1. The detail of the
distribution depends on each of Hall bars used in experiments. It is
characterized by the width $\Delta E$ of the distribution. Hereafter, we
suppose the distribution

$\rho(E)=\rho_{0}\sqrt{1-\Big{(}\frac{E-E_{1+}}{\Delta
E}\Big{)}^{2}}\quad\mbox{with}\quad|E-E_{1+}|\leq\Delta
E\quad\mbox{otherwise}\quad\rho(E)=0$ (8)

with $\rho_{0}=(eB/2\pi)\times 2/(\pi\Delta E)$, where $\int
dE\rho(E)=eB/2\pi$ represents the number density of electrons in a Landau
level; $\int dE\rho(E)=eB/2\pi\simeq 2.4\times 10^{11}/\rm cm^{2}$$(B/10T)$.
The width $\Delta E$ width  is of the order of $\sim 10^{-4}$eV. Tentatively,
we take $\delta=\Delta E/10\sim 10^{-5}$eV. Thus, the E range is much smaller
than Landau level spacing $eB/m^{\ast}\simeq 1.7\times
10^{-2}\mbox{eV}(B/10\mbox{T})(0.067m_{e}/m^{\ast})$.

The length $\vec{L}_{\alpha\beta}$ denotes length scale of overlapping region
between two localized states $\alpha$ and $\beta$. It depends on both the
states. In general, there is no overlapping when the energies $E_{\alpha}$ and
$E_{\beta}$ are much far from the critical energy $E_{1+}$;
$\vec{L}_{\alpha\beta}\simeq 0$. This is because the sizes of the states are
small. The size of a localized state, however, becomes much larger as its
energy approaches to the critical energy. The size becomes infinity at the
critical energy $E_{1+}$. Because we consider the localized states $\alpha$
with their energies $E_{\alpha}$ ($<E_{f}$ ) such as $E_{\alpha}+m_{a}$ in the
E range ( i.e. $E_{1+}-\delta-m_{a}<E_{\alpha}<E_{1+}+\delta-m_{a}$ ), the
energies $E_{\alpha}$ are very close to the critical energy $E_{1+}$ because
$m_{a}\sim\delta$ which we mainly consider as axion mass $m_{a}$ equal to or
less than $10^{-5}$eV . So, their sizes are large and comparable to the
effective extended states. On the other hand, the states $\beta$ are in the E
range so that their length scales are larger than the scale
$L_{h}=200\mu$m$\sim 10^{4}l_{B}(10\mbox{T}/B)^{1/2}$ of the Hall bar.
Therefore, their overlapping length $L_{\alpha\beta}$ can be also large such
as for instance $L_{\alpha\beta}\sim 10^{3}l_{B}$. For simplicity, we assume
that the length $L_{\alpha\beta}$ is independent of the states $\alpha$ and
$\beta$; $L_{\alpha\beta}=A\l_{B}$. Then, it follows that

$\dot{N}=\int dE_{\alpha}2\pi
S^{2}\rho({E_{\alpha}+m_{a}})\rho(E_{\alpha})m_{a}^{2}\Big{(}e\vec{A}_{a}\cdot\vec{L}_{\alpha\beta}\Big{)}^{2}\simeq
2\pi S^{2}m_{a}^{2}\Big{(}\frac{eB}{2\pi}\frac{2}{\pi\Delta
E}\Big{)}^{2}e^{2}A_{0}^{2}A^{2}l_{B}^{2}\times\Delta,$ (9)

where $\int dE_{\alpha}\equiv\Delta$ and we put
$(e\vec{A}_{a}\cdot\vec{L}_{\alpha\beta})^{2}\equiv
e^{2}A_{0}^{2}A^{2}l_{B}^{2}$ with $A_{0}=g_{a\gamma\gamma}a_{0}B/m_{a}$.

It is obvious that the integration $\int dE_{\alpha}$ is taken over the range
$E_{f}\geq E_{\alpha}\geq E_{1+}-\delta-m_{a}$. Thus,
$\Delta=E_{f}-(E_{1+}-\delta-m_{a})$ for $E_{1+}+\delta-m_{a}\geq E_{f}$,
otherwise $\Delta=2\delta$.

When the Fermi energy is smaller than $E_{1+}-\delta-m_{a}$, i.e.
$E_{f}<E_{1+}-\delta-m_{a}$, there are no electrons transiting to states in
the E range, i.e. $\dot{N}=0$ because $\Delta=0$. Hall conductance
$\sigma_{xy}$ keeps the value $3e^{2}/2\pi$. Once the Fermi energy goes beyond
the value $E_{1+}-\delta-m_{a}$ ( $\Delta=E_{f}-(E_{1+}-\delta-m_{a})>0$ ),
the probability that electrons occupy the effective extended states is non
vanishing so that $\dot{N}>0$. The Hall conductance $\sigma_{xy}$ gradually
increases larger than $3e^{2}/2\pi$ as Fermi energy $E_{f}$ increases. But the
Fermi energy $E_{f}$ goes beyond $E_{1+}+\delta-m_{a}$ ( $E_{f}>E_{1+}+\delta-
m_{a}$ ), $\dot{N}$ takes its maximum and does not increase, keeping the value
with $\Delta=2\delta$ in eq(9). This is because electron in the state with
energy $E_{\beta}=E_{\alpha}+m_{a}$ does not carry electric current.
Furthermore, when $E_{f}$ goes beyond the value $E_{1+}-\delta$, $\sigma_{xy}$
becomes $4e^{2}/2\pi$ because there are electrons always present in the
effective extended states without the transition. Then, the plateau
$\sigma_{xy}=4\times e^{2}/2\pi$ newly arises. The width of Fermi energy
$\Delta E_{f}=E_{1+}-\delta-(E_{1+}-\delta-m_{a})$ is given by $\Delta
E_{f}=m_{a}$. Within the width, plateau-plateau transition takes place. It
corresponds to the width $\Delta B$. Later we present the explicit relation
$\Delta E_{f}\propto\Delta B$ between $\Delta E_{f}$ and $\Delta B$.

Typically, the value of $\dot{N}$ is

$\dot{N}\simeq
3\,\mbox{s${}^{-1}$}\Big{(}\frac{A}{10^{3}}\Big{)}^{2}\Big{(}\frac{S}{\mbox{10${}^{-4}$cm${}^{2}$}}\Big{)}^{2}\Big{(}\frac{10^{-4}\mbox{eV}}{\Delta
E}\Big{)}^{2}\Big{(}\frac{\rho_{d}}{0.3\,\mbox{GeVcm${}^{-3}$}}\Big{)}\Big{(}\frac{B}{10\rm
T}\Big{)}^{3}\Big{(}\frac{\delta}{10^{-5}\rm eV}\Big{)}$ (10)

where we simply assumed $\Delta=2\delta$. The value has been obtained under
the assumption that the size of Hall bar $L_{h}\simeq 200\mu$m and the length
scale, tentatively, $Al_{B}\sim 10^{3}l_{B}$ of the overlapping between
localized states $\alpha$ and effective extended states. The energies
$E_{\alpha}$ of the localized states $\alpha$ is very near the E range (
$E_{1+}-\delta\sim E_{1+}+\delta$ ) because $E_{\alpha}+m_{a}$ must be in the
E range. Being close is necessary for the overlapping $L_{\alpha\beta}=Al_{B}$
to be large such as $A=10^{3}$. As we can see, it is unclear how large the
parameter $A$ is. However, it turns out later that the assumption of large $A$
is not irrelevant for the estimation of the number of electrons steady
occupying the effective extended states. Electrons steady occupying the states
are those defined such that they are determined by the balance between
incoming electrons to the states and outgoing electrons from the states.

( In the above estimation, we have assumed that the radiations $\vec{A}_{a}$
absorbed by electrons are generated by axion dark matter. When the radiations
are externally imposed on Hall bar by ourselves, we find that the formula
$\dot{N}$ is proportional to $\nu^{2}$ with the frequency $\nu$ of the
radiations. )

We have estimated the transition probability of localized electrons to
effective extended states by absorbing radiations from axion dark matter at
the temperature $T=0$. But, such transition also arises owing to the black
body radiation when $T\neq 0$. The energy density of the radiations are much
bigger than that of radiations from axions. The effect of black body radiation
may be considered as thermal noise. Thus, we calculate the ratio of the signal
to the noise.

The energy power generated by the dark matter axion in the Hall bar is

$P_{a}\equiv\dot{N}m_{a}\sim 4.8\times
10^{-24}\mbox{W}\Big{(}\frac{A}{10^{3}}\Big{)}^{2}\Big{(}\frac{S}{10^{-4}\mbox{cm${}^{2}$}}\Big{)}^{2}\Big{(}\frac{10^{-4}\mbox{eV}}{\Delta
E}\Big{)}^{2}\Big{(}\frac{\rho_{d}}{0.3\,\mbox{GeVcm${}^{-3}$}}\Big{)}\Big{(}\frac{B}{10\rm
T}\Big{)}^{3}\Big{(}\frac{\delta}{10^{-5}\rm
eV}\Big{)}\Big{(}\frac{m_{a}}{10^{-5}\mbox{eV}}\Big{)}$ (11)

We compare it with thermal noise, for instance $P_{T}=T\times 10\rm Hz$. SN
ratio is given by

$\frac{P_{a}\sqrt{10\mbox{Hz}\times 1\mbox{s}}}{P_{T}}\sim 10$ (12)

Thus, the axion effect can be observable at the low temperature. Hall
conductance measured at temperature $\sim 10$mK is determined mainly by the
axion effect. Actually, a measurement in detail of the saturation
performedsat6  has shown that the saturation of the width $\Delta B$ is
observed at low temperature $\sim 30$mK. It has also been shown that the cause
of the saturation is not finite size effect of Hall bar. The experiment is
consistent with our argument that the saturation at the low temperature is
caused by axion dark matter.

We have obtained the number of electrons transited from localized states
$\alpha$ to effective extended states $\beta$. These electrons do not remain
to stay in the effective extended states. They loose their energies and go
back to localized states. It is difficult to find how fast they loose their
energies because there are several mechanisms of the energy loss; energy loss
by emission of electromagnetic radiation, emission of phonon, or by
interaction with other electrons, e.t.c..

Here, we only consider a mechanism of energy loss by emission of
electromagnetic radiations such that the electrons in the states $\beta$ go
back to the original localized states $\alpha$ by spontaneously emitting
radiations. Such a transition is possible because the localized states
$\alpha$ are unoccupied. The mechanism of the energy loss can be explicitly
treated. We would like to find the average number of electrons staying in the
effective extended states. The number is determined by the balance between
incoming electrons and outgoing electrons. Because we have obtained the
incoming electrons $\dot{N}$, we need to know the number of outgoing
electrons.

So, we calculate the probability that an electron in the state $\beta$ with
energy $E_{\beta}$ ( $E_{1+}+\delta\leq E_{\beta}\leq E_{1+}-\delta$ )
transits to the state $\alpha$ with energy $E_{\alpha}=E_{\beta}-m_{a}$ by
emitting radiation with energy $m_{a}$.

That is, we calculate the transition amplitude,

$\langle
E_{\alpha},k,\gamma|\frac{e\vec{A}\cdot\vec{P}}{m^{\ast}}|E_{\beta},vac\rangle=i(E_{\alpha}-E_{\beta})\langle
E_{\alpha}|\vec{x}|E_{\beta}\rangle\cdot\langle k,\gamma|e\vec{A}|vac\rangle$
(13)

where the state $|k,\gamma\rangle$ denotes a photon with momentum
$k=|\vec{k}|$ and polarization $\gamma$. The radiation is described by the
gauge potential $\vec{A}$. We make approximation
$\exp(i\vec{k}\cdot\vec{x})\simeq 1$ because wave length
$k^{-1}=m_{a}^{-1}\simeq 2$cm $(10^{-5}$eV$/m_{a})$ of the radiation is much
larger than the scale of Hall bar.

Then, by summing over all directions of momentum $\vec{k}$ with
$|\vec{k}|=E_{\beta}-E_{\alpha}$ and polarization $\gamma$, we find that the
transition probability per unit time $\dot{T_{p}}$ is

$\dot{T_{p}}=\frac{4\alpha}{3}|\vec{L}_{\alpha\beta}|^{2}m_{a}^{3}=\frac{4\alpha}{3}A^{2}l_{B}^{2}m_{a}^{3}$
(14)

where we have used the length scale
$|\vec{L}_{\alpha\beta}|^{2}=A^{2}l_{B}^{2}$ of the overlapping region between
the state $\beta$ and the state $\alpha$. Thus, we obtain

$\dot{T_{p}}\sim 2.2\times
10\,\mbox{s}^{-1}\Big{(}\frac{m_{a}}{10^{-5}\mbox{eV}}\Big{)}^{3}\Big{(}\frac{10\mbox{T}}{B}\Big{)}\Big{(}\frac{A}{10^{3}}\Big{)}^{2}.$
(15)

By absorbing radiations with the energy $m_{a}$, localized electrons make
transition to effective extended states, while the electrons go back to the
original localized states by spontaneously emitting radiations with the energy
$m_{a}$. Consequently, a stationary state is realized. We have estimated both
numbers of incoming and outgoing electrons in the effective extended states.
Average number $N_{av}$ of electrons stationary occupying the effective
extended states is determined by the valance between incoming electrons
$\dot{N}$ and outgoing electrons $N_{av}\dot{T_{p}}$, i.e.
$\dot{N}=N_{av}\dot{T_{p}}$,

$N_{av}=\frac{\dot{N}}{\dot{T_{p}}}\sim
0.14\Big{(}\frac{S}{\mbox{10${}^{-4}$cm${}^{2}$}}\Big{)}^{2}\Big{(}\frac{10^{-4}\mbox{eV}}{\Delta
E}\Big{)}^{2}\Big{(}\frac{\rho_{d}}{0.3\,\mbox{GeVcm${}^{-3}$}}\Big{)}\Big{(}\frac{B}{10\rm
T}\Big{)}^{4}\Big{(}\frac{\delta}{10^{-5}\rm
eV}\Big{)}\Big{(}\frac{10^{-5}\mbox{eV}}{m_{a}}\Big{)}^{3}.$ (16)

It is important to note that the final result $N_{av}$ does not depend on the
size of overlapping region $Al_{B}$.

Even if Fermi energy is less than $E_{1+}-\delta$, there is the presence of
electrons in the effective extended states. It leads to the finite width
$\Delta B$ in plateau-plateau transition. Although the value $N_{av}$ depends
on magnetic field $B$, size of Hall bar, or disorder potential $V$ ( namely
$\delta$ and $\Delta E$ ), the rough estimation implies that the axion dark
matter can be a cause of the saturation of the width $\Delta B$ in plateau-
plateau transition. This is because the value $N_{av}$ is of the order of $1$
and only if a single electron occupies an effective extended state, the Hall
conductivity changes by large step. Although radiations from axions are
extremely weak, a large number of electrons of the order of $S\times
eB/2\pi\sim 10^{7}(B/10\mbox{T})(S/10^{-4}\rm cm^{2})$ are involved in the
transition. Thus, a small fraction of localized electrons transiting to
effective extended states may change the Hall conductivity by large step.

As we have explained, incoming electrons $\dot{N}$ gradually increases as the
Fermi energy increases from $E_{1+}-\delta-m_{a}$. Accordingly, the number
$N_{av}$ of electrons increases, because the transition probability
$\dot{T_{p}}$ does not depend on the Fermi energy. The Hall conductivity also
increases as $N_{av}$ increases. In this way, the finite width $\Delta
B(E_{f})$ arises in the plateau-plateau transition.

We make a comment that we have not yet clarified the explicit relation between
$\Delta B$ and $N_{av}$. We simply state that when $N_{av}=0$,
$\sigma_{xy}=3\times e^{2}/2\pi$, but as $N_{av}$ increases more than $0$,
$\sigma_{xy}$ increases more than $3\times e^{2}/2\pi$. Finally, when $N_{av}$
reaches at $1$, Hall conductivity takes the value $\sigma_{xy}=4\times
e^{2}/2\pi$. The increase of the Fermi energy $E_{f}$ makes $N_{av}$ increase.
Thus, the width $\Delta E_{f}$ corresponds to the range of the value $N_{av}$
from $0$ to $1$. It also corresponds to the width $\Delta B$. Later we derive
the explicit relation between $\Delta E_{f}$ and $\Delta B$. But we do not
know how $N_{av}$ depends on the Fermi energy $E_{f}$. In other words, when
the value $N_{av}$ changes from $0$ to $1$, what amount of the value $E_{f}$
changes, i.e. $\Delta E_{f}$.

The above estimation of $\dot{N}_{av}$ is very naive one. Electrons in the
effective extended states $\beta$ loose their energies not only by transition
to the original localized states $\alpha$, but also by transition to other
localized states with energy larger than $E_{\alpha}$ (
$E_{\beta}=E_{\alpha}+m_{a}$ ). That is, they loose their energies by
emissions of radiations or phonons with energies less than $m_{a}$. ( In this
case, $\dot{T}$ does not depend on $m_{a}$ ). Furthermore, electrons loose
their energies by colliding with other electrons, transiting to localized
states. The precise estimation of real transition probability $\dot{T}_{rp}$
is difficult and beyond our scope.

But we need to take into account the following fact in the estimation. The
transitions to localized states with energies less than Fermi energy $E_{f}$
are not allowed. This is because the localized states are occupied. The
transition is only allowed such that electrons in effective extended states
transit to localized states with energies $E_{\alpha}$ such as
$E_{1+}-\delta>E_{\alpha}>E_{f}$. The states are not occupied. Of course, the
transition to the effective extended states with energies larger than
$E_{1+}-\delta$ is also allowed. But the transition does not change the number
of electrons steady staying in the effective extended states. With these
consideration, we need to examine how the allowed real transition probability
$\dot{T}_{rp}$ to localized states depends on the Fermi energy $E_{f}$, in
order to see the dependence of $N_{av}=\dot{N}/\dot{T}_{rp}$ on $E_{f}$.

The dependence of $N_{av}$ on $E_{f}$ arises not only from $\dot{N}$, but also
$\dot{T}_{rp}$. Then, when the Fermi energy $E_{f}$ begins to go beyond the
energy $E_{1+}-\delta-m_{a}$, $\dot{N}$ gradually increases more than $0$, but
$\dot{T}_{rp}$ is still large enough for $N_{rav}=\dot{N}/\dot{T}_{rp}\ll 1$.
This is because the range $E_{1+}-\delta-E_{f}$ is large. $\dot{N}$ increases
more but $\dot{T}_{rp}$ decreases as $E_{f}$ increases more beyond
$E_{1+}-\delta-m_{a}$. The reason of the decrease of $\dot{T}_{rp}$ is that
the number of localized states with energies $E$ within the range
$E_{1+}-\delta>E>E_{f}$, decreases as $E_{f}$ increases. Finally, $N_{rav}$
approaches its maximum when $E_{f}$ approaches $E_{1+}-\delta$. At the point,
$\dot{T}_{rp}=\dot{T}$ because the range $E_{1+}-\delta-E_{f}$ vanishes. The
point corresponds to the end of plateau-plateau transition. Therefore, actual
relation between $\Delta E_{f}$ and $m_{a}$ is more complicated than the naive
one $\Delta E_{f}=m_{a}$ mentioned above.

We may infer a real relation from a microwave experimentmicro , in which
microwaves with frequencies $\nu$ are imposed on Hall bar. It has been
observed that the relation $\Delta B\propto\nu^{\gamma}$ with $\gamma\sim
0.4$. As we will demonstrate soon below, the relation $\Delta B\propto\Delta
E_{f}$ holds in general. Thus, the experiment shows that $\Delta
E_{f}\propto\nu^{\gamma}$ with $\gamma\sim 0.4$. The microwave may be
identified as the one converted from the axion dark matter. Hence, $\Delta
E_{f}\propto m_{a}^{\gamma}$. This is the observed relation between axion mass
$m_{a}$ and the width of Fermi energy $\Delta E_{f}$.

According to the above argument, we speculate that the width $\Delta E_{f}$
does not depend on Landau level $n$. It is determined only by the difference
between $E_{n\pm}-\delta$ and $E_{f}(>E_{n\pm}-\delta-m_{a}$ ). In next
section, we present an observation which shows that it is true.

##  VIII relation between $\Delta E_{f}$ and $\Delta B$ and its dependence on
Landau level

There has been an observationsat6  which shows how the width $\Delta B(n)$
depends on plateau-plateau transition characterized with
$\rho_{xy}=(2\pi/e^{2})\times 1/n$; the dependence of $\Delta B(n)$ on $n$,
which represents the transition width from the plateau
$\rho_{xy}=(2\pi/e^{2})\times 1/n$ to the plateau
$\rho_{xy}=(2\pi/e^{2})\times 1/(n+1)$. First, we show the relation between
$\Delta B(n)$ and Fermi energy $\Delta E_{f}$. According to the observation,
it turns out that $\Delta E_{f}/\Delta E$ is independent on Landau level $n$.
( $\Delta$E is the width of the density of states $\rho(E)$. )

For simplicity, we assume that Zeeman energy is negligibly small. Thus, there
are the number $2eB/2\pi$ of states degenerates in each Landau level with
energy $E_{n}=\omega_{c}(n+1/2)$ with $n\geq 0$. The degeneracy is lifted up
by potential $V$. The density of states $\rho_{n}(E,B)$ is characterized by
the bandwidth $\Delta E$. We assume the density
$\rho_{n}(E,B)=2\rho_{0}\sqrt{1-((E-E_{n})/\Delta E)^{2}}$ as used above; the
factor $2$ comes from spin degree of freedom.

Suppose that Landau levels up to $n$ are occupied and that the half of the
Landau level $n+1$ is also occupied at the magnetic field $B=B_{c}$. That is,
the Fermi energy is located at the center of the number density
$\rho_{n+1}(E,B)$ of Landau level $n+1$ at $B=B_{c}(n)$; $E_{f}=E_{n+1}$. We
assume that $\delta/\Delta E\ll 1$ and we neglect $\delta$. Now, we make
magnetic field $B$ increase such as $B_{c}(n)+\Delta B$. Then, the number of
electrons occupying each Landau level $m(\leq n)$ increases so that the Fermi
energy $E_{f}$ decreases such as $E_{f}=E_{n+1}-\Delta E_{f}$. Then, by noting
that the total number density $\rho_{e}$ of electrons does not change, it
follows that

$\displaystyle\frac{2eB_{c}}{2\pi}(n+\frac{1}{2})$ $\displaystyle=$
$\displaystyle\frac{2e(B_{c}(n)+\Delta B)}{2\pi}n+\int_{E_{n+1}-\Delta
E}^{E_{n+1}-\Delta E_{f}}dE\rho_{n+1}(E,B_{c}(n)+\Delta B)$ (17)
$\displaystyle\simeq$ $\displaystyle\frac{2e(B_{c}(n)+\Delta
B)}{2\pi}n+\frac{1}{2}\frac{2e(B_{c}(n)+\Delta B)}{2\pi}-2\rho_{0}\Delta
E_{f}$ (18)

with $\int_{E_{n+1}-\Delta E_{f}}^{E_{n+1}}dE\rho(E,B_{c}(n)+\Delta B)\simeq
2\rho_{0}\Delta E_{f}$ for $\Delta E_{f}/\Delta E\ll 1$.

It turns out that

$\Delta B(n)=\frac{4B_{c}(n)}{(2n+1)\pi}\frac{\Delta E_{f}}{\Delta E}.$ (19)

It describes how the width $\Delta B(n)$ depends on Landau level $n$ and the
width of Fermi energy $\Delta E_{f}$. The relation is very general. Although
we have used the explicit formula $\rho(E)\propto\sqrt{1-(E-E_{n})^{2}/(\Delta
E)^{2}}$, we can derive similar result i.e. $\Delta B\propto\Delta E_{f}$,
even if we use a different formula such as
$\rho(E)\propto\exp(-(E-E_{n})^{2}/(\Delta E)^{2})$.

As we have mentioned, the width $\Delta E_{f}$ satisfies $\Delta E_{f}\propto
m_{a}^{\gamma}$ with $\gamma\sim 0.4$. Setting $\Delta E_{f}=C\times
m_{a}^{\gamma}$ with a constant $C$, we have

$\Delta B(n)=\frac{4B_{c}(n)}{(2n+1)\pi}\frac{Cm_{a}^{\gamma}}{\Delta E}.$
(20)

We find that the formula in eqs(19) and (20) well agrees with the
observationsat6 . That is, the factor $\Delta E_{f}/\Delta
E=Cm_{a}^{\gamma}/\Delta E$ is independent on $n$. Here, we present the result
$\Delta E_{f}/\Delta E\sim 0.27$ in the observation,

$\displaystyle\frac{(2\times 2+1)\pi}{4}\frac{\Delta B(2)}{B_{c}(2)}$
$\displaystyle\simeq$
$\displaystyle\frac{5\pi}{4}\frac{0.7\mbox{T}}{11\mbox{T}}\simeq
0.25,\quad\frac{(2\times 3+1)\pi}{4}\frac{\Delta
B(3)}{B_{c}(3)}\simeq\frac{7\pi}{4}\frac{0.4\mbox{T}}{8\mbox{T}}\simeq 0.27$
$\displaystyle\frac{(2\times 4+1)\pi}{4}\frac{\Delta B(4)}{B_{c}(4)}$
$\displaystyle\simeq$
$\displaystyle\frac{9\pi}{4}\frac{0.23\mbox{T}}{6\mbox{T}}\simeq
0.27,\quad\frac{(2\times 5+1)\pi}{4}\frac{\Delta
B(5)}{B_{c}(5)}\simeq\frac{11\pi}{4}\frac{0.16\mbox{T}}{5\mbox{T}}\simeq
0.28.$ (21)

Because the width $\Delta E$ andouemura1  in the density of state $\rho(E)$
does not depend on Landau levels $n$, $\Delta E_{f}$ does not depend on $n$ as
we have discussed in the previous section. The observation shows the validity
of our previous discussion.

Furthermore, the independence of $\Delta E_{f}/\Delta E$ on $n$ shows that the
width $\Delta B(n)$ becomes smaller such as $\Delta B(n)\propto 1/n^{2}$ as
$n$ increases, because $B_{c}(n)\propto n^{-1}$. Plateaus in higher Landau
levels are realized with smaller magnetic field $B$ when electron number
density is constant. Since the number $N_{av}$ in eq(16) of electron steady
present in extended states is proportional to $B^{4}$, the decrease of the
width $\Delta B(n)\propto 1/n^{2}$ in higher Landau level is consistent with
our idea that the saturation of the width is caused by axion dark matter.

Such a saturation of $\Delta B$ at low temperature have been observed in many
experimentssat1 ; sat2 ; sat3 ; sat4 ; sat5  using various sizes of Hall bar.
It is generally considered that the cause of the saturation is intrinsic.
Similar phenomena, as is called phase decoherence, have been
observeddecoherence  in various condensed matters. But there are no plausible
explanations of the intrinsic cause. In this paper we show that the saturation
$\Delta B$ is caused by axion dark matter. It might be the intrinsic cause for
the phase decoherence observed in various condensed matters. ( Even without
magnetic field, axion directly couples with electrons so that the electrons
are in the bath of the axion dark matter at $T=0$. Hence, the coherence of
electrons at $T=0$ may be disturbed. )

##  IX determination of axion mass

Under strong magnetic field, axion is converted to electromagnetic radiation.
Because experiment of quantum Hall effect uses such a strong magnetic field,
Hall bar is inevitably imposed by the radiation. We have discussed that the
radiation produces finite width $\Delta B\neq 0$ at temperature $T=0$ in the
plateau-plateau transition region.

The saturation of the width ( $\neq 0$ ) has been observed, contrary to the
expectation that the width behaves such as $\Delta B\propto T^{\kappa}\to 0$
for $T\to 0$ with $\kappa\sim 0.4$. How large the width is depends on each
Hall bar. In order to suppress the finite size effect, we should use large
Hall bar such as $L_{h}>500\mu$m.

Similar saturation has been observed in a different situation. Namely, when we
impose microwaves on Hall bar, the width also depends on the frequency of the
microwavemicro . The width increases as the frequency increases, in other
words, it decreases with the decrease of the frequency. It has been shown at
sufficiently low temperature that the width behaves such as $\Delta
B\propto\nu^{\gamma}$ for $\nu=10\mbox{GHz}\sim 2\mbox{GHz}$ with $\gamma\sim
0.4$. But, the width appears to be saturated at lower frequency $\nu<2$GHz.
Obviously, the frequency of the radiation plays a similar role of temperature.
The experiment is very suggestive when we notice that radiation from axion can
be identified as radiation imposed externally on Hall bar. Without radiations
imposed externally, we still have radiations from axion dark matter in
experiments of quantum Hall effect.

( We make a comment that the observational results of $\Delta
B\propto\nu^{\gamma}$ with $\gamma\sim 0.4$ is consistent with our estimation
that $\dot{N}_{av}\propto\nu^{2}$ in the case of the radiations being
externally imposed. The increase of the frequency makes $\dot{N}_{av}$ large
so that $\Delta B$ becomes large. )

Our proposal for the determination of axion mass is in the following. We
impose microwave on Hall bar and measure Hall conductivity at sufficiently low
temperature for the width $\Delta B$ to be saturated without microwaves. The
size of Hall bar should be larger than $500\mu$m. That is, the experiment is
performed when the width $\Delta B$ has already been saturated at extremely
low temperature. The temperature should be below several $10$mK because the
effect of black body radiation is less effective than the effect of the
radiation produced by axion. We note that the microwave micro  plays a role of
temperature. That is, the width behaves such that $\Delta
B\propto\nu^{\gamma}$ with $\gamma\sim 0.4$.

The width $\Delta B$ decreases as the frequency of the microwave decreases as
well as as the temperature decreases. But at sufficient low temperature, it
saturates at a finite value $\Delta B_{c}$ in the absence of the external
microwaves. On the other hand, at such a sufficient low temperature, the
external microwave leads to the width larger than $\Delta B_{c}$. The decrease
of the frequency $\nu$ leads to the decrease of the width $\Delta B$ such that
$\Delta B\propto\nu^{\gamma}$. When the frequency is below a critical
frequency, the width saturates at $\Delta B=\Delta B_{c}$ as shown in Fig.(3).
Such a critical frequency is equal to the frequency $m_{a}/2\pi$ of the
radiation produced by axion. In this way, we can determine the axion mass.
Using the previous studymicro , we can read the frequency is about $1$GHz to
$1.3$GHz although there is large ambiguity in the data. The previous study did
not intended the determination of axion mass. So in order to derive the mass,
the experiment needs to be performed carefully.

The experimentsat6  in detail has shown that the saturation arises at
temperature below $30$mK. When we take into account the correspondence between
temperature and frequency of microwave, we expect that the axion mass is in
the range $10^{-5}\mbox{eV}\sim 4\times 10^{-6}$eV.

Figure 3: saturation of $\Delta B$ in radiation frequency Figure 4: Hall bar
sandwiched by conducting plates

It is favorable to check whether the saturation is really caused by axion. For
the purpose, we make radiations by axion much weaker and examine whether or
not the width diminishes. In order to make the radiation weak, we use two
parallel conducting plates ( slabs ). That is, we put the slabs parallel to
magnetic field imposed on Hall bar as we have discussed in previous our
paperiwazaki01 . Then, the radiation produced by axion changes its strength
depending on the position between the slabs. When we set Hall bar just in the
center Fig.4 between two slabs separated by the distance $l\ll m_{a}^{-1}$,
the strength of the radiation is suppressed by the factor $(m_{a}l)^{2}$ such
that $E_{a}^{\prime}\sim g_{a\gamma\gamma}a_{0}B(m_{a}l)^{2}/2\sim
10^{-2}(m_{a}/10^{-5}\mbox{eV})^{2}(l/0.2\mbox{cm})^{2}g_{a\gamma\gamma}a_{0}B$.
( This suppression factor can be easily obtained using results in the previous
paperiwazaki01 . ) The spacing $l$ should be much smaller than Compton wave
length $1/m_{a}\sim 2\mbox{cm}(10^{-5}\mbox{eV}/m_{a})$. Consequently,
$N_{av}$ is suppressed by the factor
$10^{-4}(m_{a}/10^{-5}\mbox{eV})^{4}(l/0.2\mbox{cm})^{4}$ when we choose the
spacing $l=0.2$cm.

Therefore, we will see that the width $\Delta B$ may diminish when we put Hall
bar in the center between two parallel conducting plates. The decrease of the
width implies that the saturation is caused by axion dark matter.

##  X conclusion

We have discussed the effect of the axion dark matter on integer quantum Hall
effect. The axion produces electromagnetic radiation with energy $m_{a}$ under
strong magnetic field used in the experiments. Electrons in localized states
absorb the radiations and transit to effective extended states in the quantum
Hall state. The electrons in the states carry electric current. Even if a
single electron occupies an extended state, the Hall conductivity or
resistance can change its value and forms a new plateau. The probability of
the single electron occupying the extended state at the energy $E_{n\pm}$ is
zero in the absence of the axion dark matter when Fermi energy $E_{f}$ is less
than the energy $E_{n\pm}$. The plateau-plateau transition becomes sharp like
step function. On the other hand, the probability is non negligible in the
presence of the axion dark matter because the naive estimation shows that the
number of electrons $N_{av}$ in extended states can be of the order of $1$.
$N_{av}$ gradually increases from $0$ to $1$ as Fermi energy $E_{f}$ increases
to approach the energy $E_{n\pm}$. Consequently, the plateau-plateau
transition becomes smooth even at $T=0$.

Furthermore, we discuss how we determine the axion mass with quantum Hall
effect. By observing the saturation of the width $\Delta B$ with changing
frequencies of microwaves imposed on Hall bar, we can find the axion mass.
Additionally, we propose a way of the confirmation that the axion dark matter
really causes the saturation $\Delta B\neq 0$ even as $T\to 0$. Using two
parallel conducting flat plates, we can make radiations generated by axion
much weak. The Hall bar is set in the center between two plates. Then, the
radiation becomes much weak. As a result, we expect that the width $\Delta B$
of the saturation becomes much smaller compared with the case without two
parallel conducting plates.

The author expresses thanks to A. Sawada for useful comments. He also
expresses thanks to members of theory group in KEK for their hospitality. This
work is supported in part by Grant-in-Aid for Scientific Research ( KAKENHI ),
No.19K03832.

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