Axion Detection with Quantum Hall Effect

Finding axion dark matter is one of most significant issues in particle physics. It is the important step toward a new physics beyond the standard model of particle 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 axion mass is severely restricted such as $m_{a}=10^{-6}\mbox{eV}\sim 10^{-3}$ eV Wil ; Wil1 ; Wil2 . In the present paper we only consider QCD axion and use physical units, $c=1$, $k_{B}=1$ and $\hbar=1$. Our result is also applicable to dark photon with frequencies of microwaves discussed below.

The QCD axion produces electromagnetic radiations under strong magnetic field $B$. The property is used for the exploration of the axion dark matter admx ; carrack ; haystac ; abracadabra ; organ ; madmax ; brass ; cast ; sumico ; iwazaki01 . Some of them have been proposed and many of them are undergoing at presentnew . Obviously, the radiations produced by the axion are also present in the experiments of quantum Hall effect. It is natural to expect that their effects may be observed in some of properties of quantum Hall effect, although they are quite weak. In particular, as we have shown in previous paperiwa , they may appear in much low temperature less than $100$mK.

Quantum Hall effectvon ; girvin is realized in two dimensional electron system under strong magnetic field. The system shows various intriguing phenomena 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 ; wanli ; sat3 ; sat4 ; sat5 ; sat6 of the width $\Delta B$ in plateau-plateau transition of integer quantum Hall effect. The width defines the range of the magnetic field within which plateau-plateau transition takes place.

Plateau-plateau transition in integer quantum Hall effect is a phase transition between metal and insulator. It has been extensively explored and the width $\Delta B$ in the transition has been shown to follow a scaling law such as $\Delta B\propto T^{\kappa}$ with $\kappa\sim 0.42$deltaB as $T\to 0$. It has also been shown experimentally that the width saturates in low temperature, that is, it never decrease more with the decrease of temperature; $\Delta B=\mbox{const.}$ below a critical temperature. In general, the saturation is considered to arise owing to finite size effect of two dimensional electrons, because the scaling is expected in infinitely large system. But recent experimentssat4 ; sat5 suggest that the saturation is caused by not finite size effect, but intrinsic decoherence, although the mechanism of the decoherence is still unclear.

Furthermore, similar scaling law holds when we imposed microwaves on Hall barengel ; balaban ; hohls ; saeed ; doo . That is, $\Delta B\propto f^{\kappa}$ with $\kappa=0.4\sim 0.7$ as the frequency $f\to 0$ of the microwaves. It has been also observed that the width saturates at a critical frequency $f_{s}$ when we decrease the frequency of the microwaves just as in the case of the temperature. ( In actual experiments we use AC voltage for the measurement of the width. )

In our previous paperiwa , we have shown a possibility that the saturation arises owing to the effect of the axion dark matter. Especially, electromagnetic radiations ( actually, microwaves ) generated by the axion cause the decoherence of electrons in low temperature. Using the analysis, we have proposed a way of axion detection using microwaves imposed on Hall bar. In this paper, we discuss in detail how critical temperature and frequency of the microwaves at which the saturation arises, depends on the physical parameters like size of Hall bar, temperature and axion mass. According to the analysis, we propose more detailed way of the axion detection in the present paper than previous one. In particular, we present two conditions which critical frequency $f_{s}$ must satisfy in order to give the axion mass, i.e. $f_{s}=m_{a}/2\pi$. One is that $f_{s}$ does not depend on temperature and the other one is that $f_{s}$ does not depend on the size of Hall bar. Here we would like to point out that the previous experimentdoo ; doo1 using microwaves strongly suggests axion mass $m_{a}\simeq 0.95\times 10^{-5}$eV, because it seems that the critical frequency observed in the experiment satisfies two conditions presented in this paper.

Here we naively explain how the axion dark matter affects on plateau-plateau transition. We consider infinitely large two dimensional electrons at zero temperature under strong magnetic field. Owing to disorder potential, almost all electrons in each Landau levels occupy localized states except for electrons occupying extended state with energy $E_{c}$ located in the center of each Landau level. As magnetic field $B$ decreases, Fermi energy $E_{f}$ increases. As long as the Fermi energy is less than $E_{c}$, Hall conductivity stays in a plateau. When Fermi energy passes the energy $E_{c}$ of extended states, Hall conductivity goes to next plateau so that the plateau-plateau transition looks like a step function. It is very sharp, i.e. $\Delta B=0$. On the other hand, the axion dark matter generates radiations under the strong magnetic field so that the radiations are absorbed by electrons. Even if Fermi energy $E_{f}$ is below the energy $E_{c}$, some of localized electrons transit to the states with energies larger than $E_{c}$ by absorbing the radiations. These electrons loose their energies by emitting phonons and may occupy the extended states with the energy $E_{c}$. Thus, there is non zero probability of the extended state occupied even if $E_{f}<E_{c}$. Therefore, the plateau-plateau transition becomes smooth function of magnetic field, i.e. $\Delta B\neq 0$. The smooth transition $\Delta B\neq 0$ is not finite size effect. This is a naive explanation how the axion dark matter causes visible effect on the plateau-plateau transition.

We should mention that although thermal effect at $T\neq 0$ contributes to the transition, the axion effect dominates over the thermal effect as long as temperature $T$ is very low. Indeed, we will show later that the axion effect is dominant approximately for $T<100$mK.

In this paper we also propose a way of confirmation that the axion dark matter really causes the saturation in temperature or frequency of microwave imposed. Using parallel conducting slabs which sandwiches Hall bar, we shield radiations by the axion so that we expect the absence of the saturation of the width $\Delta B$. ( Sometimes in literatures, the derivative $d\rho_{xy}/dB$ at the center in the plateau transition instead of $\Delta B$ is used to see how it behaves with temperature or frequency of radiation. The saturation of $\Delta B$ corresponds to the saturation of $d\rho_{xy}/dB$. The decrease of $\Delta B$ corresponds to the increase of $d\rho_{xy}/dB$. )

In the next section (II), we briefly explain energy scale in integer quantum Hall effect and our notation used in the paper. We proceed to explain localization of electrons in quantum Hall effect in section (III). Most of electrons occupy localized states but a small fraction of electrons occupy extended states, which may carry electric current. The localization leads to plateau of Hall conductivity or resistivity in quantum Hall effect. In this section, using energy distribution of electrons at zero temperature, we discuss how Hall conductivity behaves and forms plateau according to the variation of magnetic field $B$. We define width $\Delta B$ in plateau-plateau transition used in the paper. The dependence of the width on temperature or axion mass is discussed in later sections. In the section (IV), we briefly explain the axion dark matter. We consider QCD axion as dark matter candidate. We show how the axion generates electromagnetic radiations with energy $m_{a}$ under external magnetic field, which is used in experiment of quantum Hall effect. In the next section (V), we discuss in detail how the width $\Delta B$ depends on temperature and axion mass. We find that the saturation of $\Delta B$ in temperature only arises in the presence of the axion dark matter. Without the axion effect, the saturation does not appear even in the system with finite size. In the next section (VI), we discuss the effect of external microwaves on the plateau-plateau transition. We show how the width $\Delta B$ depends on frequency $f$, temperature and axion mass. Especially, we show the presence of critical frequency $f_{s}$ below which the width $\Delta B$ does not depend on the frequency $f$ ( $<f_{s}$ ). We present two conditions which the saturation frequency $f_{s}=m_{a}/2\pi$ must satisfy to give the axion mass. In the section ( VII ), we estimate energy power generated by axion dark matter in two dimensional electrons. We compare it with thermal noise and find that the axion effect can be observable in low temperature less than $100$mK, at least for the surface area of two dimensional electrons being large such as $10^{-3}\rm cm^{2}$ or larger. The axion effect becomes larger as the size of Hall bar becomes larger, because the energy of the axion received by electrons is bigger as the number of electrons becomes larger. In the final section ( VIII ), we propose a way how we confirm the presence of the axion effect in quantum Hall effect. The point is that we shield the radiations generated by the axion dark matter by using conducting slabs put around Hall bar.

We briefly explain energy scales relevant to quantum Hall effectgirvin . The quantum Hall effect is realized in two dimensional electrons of semiconductors under magnetic field $B$. The states of the two dimensional free electrons are specified by integer $n\geq 0$, so called Landau levels. There are a number of degenerate states in each Landau level with the degeneracy $eB/2\pi$ ( i.e. number density of degenerate states in a Landau level ). The typical scale of the magnetic field is of the order of $10$T. Each electrons oscillate with cyclotron frequency $\omega_{c}=eB/m^{\ast}$ where mass $m^{\ast}$ denotes effective one of electron in semiconductors. Generally $m^{\ast}$ is much smaller than real mass $m_{e}\simeq 0.51$MeV of electron, e.g. $m^{\ast}=0.067m_{e}$ in GaAs. Then, cyclotron frequency ( energy ) $\omega_{c}$ is of the order of $\sim 10^{-2}(B/10T)$eV. Furthermore, their energies are specified with integer $n\geq 0$ such that $E_{n}=\omega_{c}(n+1/2)$. The wave functions are extended with typical length scale, so called magnetic length $l_{B}=\sqrt{1/eB}$. $l_{B}\simeq 8.2\times 10^{-7}\rm cm\sqrt{(10T/B)}$. It is cyclotron radius of electron under the magnetic field $B$.

Electron possesses spin components with up and down so that 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. Here we note that $g\simeq 0.44$ and Bohr magneton $\mu_{B}=e/2m_{e}$. Zeeman energy is of the order of $10^{-3}(B/10T)$eV. It is smaller than the cyclotron energy $\omega_{c}$.

We mainly consider axion mass $m_{a}$ in a region $10^{-5}\mbox{eV}\geq m_{a}\geq 10^{-6}\mbox{eV}$ so that the mass is smaller than Zeeman energy and cyclotron energy $\omega_{c}$. The mass is of the order of or less than the width ( extension ) $\Delta E$ in Fig.1 of the energy distribution of electrons in a Landau level, as we will explain in next section. Therefore, because the energies of radiations produced by the axion are almost identical to $m_{a}$, the effect of the radiations can be observable because electrons may absorb the radiations within a Landau level.

We use an index so called filling factor $\nu\equiv\rho_{e}/(eB/2\pi)=2\pi\rho_{e}/eB$ to specify which Landau levels are occupied; $\rho_{e}$ denotes number density of electrons ( typically, $\rho_{e}\sim 10^{11}/\rm cm^{2}$ ). For instance, it implies Landau level with energy $E_{0-}$ is fully occupied but Landau level with $E_{0+}$ is partially occupied when $2>\nu>1$.

It is remarkable featurevon ; aokiando ; halperin of quantum Hall effect that Hall resistance ( conductance ) is quantized such that $\rho_{xy}=(2\pi/e^{2})\times 1/n$ ( $\sigma_{xy}=n\times e^{2}/2\pi$ ) with positive integer $n$ specifying Landau level. It is constant within a range $n+1>\nu>n$ when we vary the magnetic field $B$; $\nu=2\pi\rho_{e}/eB$. In the range, 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 $\rho_{xy}$ ( $\sigma_{xy}$ ) in $B$. The plateaus arise owing to the localization of two dimensional electrons discussed soon below. The localization arises owing to disorder potential for electrons. That is, almost of electrons are trapped in the potential.

We explain localization of two dimensional electrons under strong magnetic field $B$. In the case of free electrons we have the density of state $\rho(E)\propto\sum_{n=0,1,,,}\delta(E-E_{n\pm})$. But, there are impurities, defects e.t.c. in actual materials. They lift up the degeneracy in Landau level. Electrons are severely affected by a disorder potential $V$. Most of electrons are localized and they cannot carry electric currents. But, a small fraction of them are not localized so that they can carry electric current. It means that Hall resistance only receives the effect of non-localized electrons. Localized electrons do not contribute to Hall resistance. The essence in integer quantum Hall effect is the presence of non-localized ( extended )aokiando ; ono states of electrons under strong magnetic field $B$. Although the disorder potential $V$ localizes almost of all electrons, there exist a non-localized state with energy $E=E_{n\pm}$. According to numerical simulations we find that in the presence of a potential $V$, the density of states $\rho(E)$ has finite width around the energy $E_{n\pm}$ shown schematically in Fig.1. In the figure we show localized states and extended states. Extended localized state is located at $E=E_{n\pm}$.

The density of state $\rho(E)$ has finite width $\Delta E$ owing to the potential $V$. It is generally supposed that the width is less than the cyclotron frequency $\omega_{c}=eB/m^{\ast}$ in strong magnetic field $B$. That is, the potential energy is much smaller than the cyclotron frequency: $\Delta E\ll\omega_{c}$ ( $\omega_{c}\sim 10^{-2}$eV with $B\sim 10$T ). For instance, $\rho(E)\propto\sqrt{1-((E-E_{n\pm})/\Delta E)^{2}}$ with $|E-E_{n\pm}|\geq\Delta E$ andouemura . Furthermore, it is supposed that the potential has 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 Fig.1.

We may define coherent length as the size of localized state. It depends on the energy of the state. Then, the referencesaoki1 ; aoki2 have shown in the system with infinitely large size that a coherent length $\xi(E)$ diverges such as $\xi(E)\to|E-E_{n\pm}|^{-\nu}$ as $E\to E_{n\pm}$ with $\nu\sim 2.4$. It implies that there are extended states with their energy $E_{n\pm}$.

When Fermi energy $E_{f}$ is less than $E_{n+}$ but larger than $E_{n-}$, the energies of electrons occupying extended states 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 $\rho_{xy}=(2\pi/e^{2})\times 1/3$. As long as the Fermi energy $E_{f}$ is in the range $E_{1+}>E_{f}>E_{1-}$, the Hall resistance does not vary with magnetic field $B$. A plateau is formed. On the plateau, we have vanishing longitudinal resistance $\rho_{xx}=0$. Only electrons in extended states with energies such as $E_{1-}$ and $E_{0\pm}$ carry electric currents. When 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. Hereafter, we mainly state Hall conductivity because it is easy to see the effect on the conductivity of electrons carrying electric currents. Generally, electric conductivity is proportional to electron density carrying 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 the next plateau, for instance, $\sigma_{xy}=4\times e^{2}/2\pi$ 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.

The above argument only holds in Hall bar with the infinite large size. Extended states only have the energies $E_{n\pm}$. In actual Hall bar with finite size, localized states are present whose sizes are larger than the size of Hall bar. Because of the divergence of coherent length $\xi(E)$ as $E$ approaches $E_{n\pm}$, we understand the presence of such localized states with their sizes larger than Hall bar. Electrons in such states have energies $E$ in the range $E_{n\pm}-\delta\leq E\leq E_{n\pm}+\delta$ where $\delta=\delta(L_{h})$ 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 $\Delta E$ in $\rho(E)$. We may call such states as extended states because electrons in the states can carry electric current.

In Hall bar with finite size, plateau-plateau transition takes place smoothly. When Fermi energy $E_{f}$ increases, but stay less than $E_{f}=E_{1+}-\delta$, Hall conductivity stays in a plateau. However, when it reaches at the energy $E_{f}=E_{1+}-\delta$, the transition begins and the conductivity increases smoothly as $E_{f}$ increases. $B$ takes the value of $B_{c}+\Delta B$ at $E_{f}=E_{1+}-\delta$. When $E_{f}=E_{1+}$, the magnetic field is given by $B_{c}=2\pi\rho/4e$. Eventually when $E_{f}$ reaches at $E_{1+}+\delta$, the conductivity stops to increase and stay at next plateau. $B$ takes the value of $B_{c}-\Delta B$ at $E_{f}=E_{1+}+\delta$. Thus, we have $\Delta B\neq 0$. The plateau-plateau transition mentioned here is the one in Hall bar with finite size at zero temperature $T=0$. See Fig.2. In the figure, we define the width $\Delta B$ between two plateaus. The Hall resistance begins to decrease at $E_{f}=E_{1+}-\delta$ and the decrease stops at $E_{f}=E_{1+}+\delta$. In our discussion we use the width $\Delta B$ as specified in the figure.

The axion is a Nambu Goldstone boson associated with Peccei Quinn symmetry introduced for solving strong CP problem. The symmetry is imposed in a model beyond the standard model, but it is spontaneously broken. As a result, the axion appears. The axion as the boson $a(t,\vec{x})$axion1 ; axion2 ; axion3 acquire its mass $m_{a}$ due to instanton effects in QCD. Because the axion couples with electromagnetic fields with coupling $g_{a\gamma\gamma}$, Maxwell equations are modifiediwazaki01 in the following,

with electric $\vec{E}$ and magnetic $\vec{B}$ fields, where we take the axion field $a(t,\vec{x})$ as the one representing the axion dark matter.

From the equations, we obtain the electric field $\vec{E}_{a}$ generated by the axion $a(t,\vec{x})$ under static external magnetic field $\vec{B}$. Namely, when magnetic field $\vec{B}$ is present, the axion dark matter generates oscillating electric field $\vec{E}_{a}$. Because the parameter $g_{a\gamma\gamma}a(t,\vec{x})$ is extremely small as shown soon below, we obtain

with $g_{a\gamma\gamma}=g_{\gamma}\alpha/f_{a}\pi$, where $\alpha\simeq 1/137$ denotes fine structure constant and $f_{a}$ is 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 mass of the QCD axion is severely restricted such as $m_{a}=10^{-6}\mbox{eV}\sim 10^{-3}$ eV, Wil ; Wil1 ; Wil2 . In the present paper we mainly consider the mass $10^{-5}\mbox{eV}\geq m_{a}\geq 10^{-6}\mbox{eV}$. We should mention that the parameter $g_{a\gamma\gamma}$ is automatically determined in QCD axion when we take the value of the axion mass, i.e. $f_{a}\simeq(6\times 10^{-6}\mbox{eV}/m_{a})\times 10^{12}$GeV.

The amplitude of the axion dark matter $a(t,\vec{x})\simeq a_{0}\cos(m_{a}t)$ is extremely small. ( The momentum of the axion dark matter is of the order of $10^{-3}m_{a}$ so that we may neglect the momentum. ) Supposing that the dark matter in the Universe is composed of the axion, we find that the local energy density $\rho_{d}$ of the dark matter is given as $\rho_{d}=m_{a}^{2}\overline{a(t,\vec{x})^{2}}=m_{a}^{2}a_{0}^{2}/2\sim 0.3\rm GeV/cm^{3}$; $\overline{Q}$ denotes time average of the quantity $Q$. Then we find that $g_{a\gamma\gamma}a(t,\vec{x})\sim 10^{-21}$. Although the electric field $\vec{E}_{a}=-g_{a\gamma\gamma}a(t,\vec{x})\vec{B}$ sikivie ; iwazaki01 is extremely small, it is inevitably produced in the experiment of quantum Hall effect because of the presence of magnetic field $B\sim 10$T. This oscillating electric field $\vec{E}_{a}$ generates electromagnetic radiationsiwazaki01 from conductors. It makes electrons in metals oscillate so that the oscillating electrons emit electromagnetic radiations. Indeed, Hall bar is surrounded by metals composing mixing chamber for cooling the Hall bar, superconducting magnet e.t.c. The frequency $f$ ( wave length ) of the radiations is given by the axion mass, $f=m_{a}/2\pi$ ( $m_{a}^{-1}$ ). Such radiations are absorbed by electrons in Hall bar and they affect on the width $\Delta B$ in plateau-plateau transition, which is our main concerns in the paper.

We mainly focus on the mass $m_{a}$ such as $10^{-5}\mbox{eV}\geq m_{a}>4\times 10^{-6}$eV ( the corresponding frequency $m_{a}/2\pi$ of radiation is about $1\mbox{GHz}\sim 2.4\mbox{GHz}$. The wave length $12\mbox{cm}\sim 30$cm is much larger than typical size of Hall bar. ) We will find later that axion mass $m_{a}=(0.95\sim 0.99)\times 10^{-5}$eV, by analyzing a previous experimentdoo ; doo1 using microwaves imposed in quantum Hall effect.

We explain in detail how the width $\Delta B$ depends on axion mass and temperature. First we suppose the system at zero temperature. Then, the energy distribution of electrons has sharp boundary at Fermi energy $E_{f}$ when axion effect is neglected. That is, the states with energies less than $E_{f}$ are fully occupied and the states with energies larger than $E_{f}$ are empity. Thus, when Fermi energy is less than $E_{1+}-\delta$, electric current does not flow and the conductivity $\sigma_{xy}$ stays in the plateau, $\sigma_{xy}=3e^{2}/2\pi$. Hall resistivity takes the value $\rho_{xy}=1/\sigma_{xy}=2\pi/3e^{2}$. When external magnetic field $B$ decreases, Fermi energy increases and reaches at the value $E_{1+}-\delta$. Then, electric current begins to flow because extended states with energies larger than $E_{1+}-\delta$ begin to be occupied. The conductivity $\sigma_{xy}$ begins to increase or Hall resistivity decreases. As $B$ decreases more, Fermi energy increases. When the Fermi energy $E_{f}$ goes beyond $E_{1+}+\delta$, the conductivity takes the value $4e^{2}/2\pi$ and stay in the next plateau, or Hall resistivity takes $2\pi/4e^{2}$. Thus, the width $\Delta B$ is determined by the magnetic field $B_{c}\pm\Delta B$ at which $E_{f}=E_{1+}\mp\delta$. The width $\Delta E_{f}$ between $E_{f}=E_{1+}-\delta$ and $E_{f}=E_{1+}+\delta$, leading to $\Delta B$ is

We see the plateau-plateau transition of Hall resistance schematically shown in Fig.2.

When the temperature $T\neq 0$, the energy distribution of electrons has no sharp boundary at chemical potential $\mu$. The boundary is smeared out by thermal effect around chemical potential $\mu$. Because we only consider low temperature $<1$K, we approximately set $\mu=E_{f}$. Although the distribution has no sharp boundary, we may approximately define the effective temperature $T_{e}$ such that the distribution has sharp boundary at $E_{f}+2T_{e}$. Electrons occupy the states below the energy $E_{f}+2T_{e}$ and the states with energies larger than $E_{f}+2T_{e}$ are empty. Especially, the distribution is smeared out over the width $2T_{e}$ schematically shown in Fig.4. Such a simplification arises from the fact that the real energy distribution of electrons decreases exponentially $\propto\exp(-(E-E_{f})/T)$ for $E>E_{f}$. Owing to the simplification, we can easily understand how the width $\Delta B$ depends on temperature. We should note that the temperature $T_{e}$ is not real temperature $T$ and it depends on the density of state $\rho(E)$. Thus, when we set $T_{e}=gT$, the constant $g$ depends on real density of state $\rho(E)$. We expect that $g$ is of the order of $1$.

Then, the width $\Delta B$ is determined in the following. That is, as $B$ decreases, $E_{f}$ increases. When $E_{f}+2T_{e}$ is equal to $E_{1+}-\delta$, electric currents begin to flow. That is the point at which the conductivity begins to increase, i.e. $B=B_{c}+\Delta B$. As $B$ decreases more, Fermi energy $E_{f}$ passes the energy $E_{1+}$ and eventually, $E_{f}$ reaches at $E_{1+}+\delta+2T_{e}$, see Fig.4. At the point, the conductivity stops to increase and stay in next plateau. It is the point $B=B_{c}-\Delta B$. In other words, Fermi energy moves from $E_{f}=E_{1+}-\delta-2T_{e}$ to $E_{f}=E_{1+}+\delta+2T_{e}$ in the plateau-plateau transition. The width is given by

Obviously, the width $\Delta B$ corresponding to $\Delta E_{f}$ decreases as temperature $T_{e}$ ( or $T$ ) decreases. It takes the value $2\delta$ when $T_{e}=0$. It never saturate at non zero temperature. We should make a comment that the dependence of $\Delta B$ on $\Delta E_{f}$ or temperature $T_{s}$ is expected such that $\Delta B\propto|T_{s}|^{\kappa}$ with $\kappa\sim 0.43$ deltaB as $T_{s}\to 0$ according to the scaling analysis, at least in the case of infinitely large Hall bar, i.e. $\delta=0$. Thus, $\Delta B$ is such as $\Delta B\propto(\Delta E_{f})^{\kappa}$ as $T_{e}\to 0$ in the infinitely large Hall bar.

In addition to the decrease of the width $\Delta B$ with temperature, the width $\Delta B$ decreases as size of Hall bar increases. That is, $\Delta B$ decreases with $\delta$, which decreases as the size of Hall bar increases. The phenomena have been well known.

Here we make a comment of the effect of our simplification of sharp boundary at $E_{f}+2T_{e}$ in energy distribution of electrons. Even if we do not take such a simplification, there is no critical temperature $T_{s}$ of saturation. That is, the width $\Delta E_{f}$ defining the difference between a starting point and its ending point of the increase of $\sigma_{xy}$, smoothly decreases as real temperature decreases. This is because the effect of the temperature $T$ on the energy distribution smoothly decreases as $T$ decreases. It never arise that the width does not decrease below a non zero temperature. Therefore, the presence of the saturation cannot be explained simply by finite size effect.

Here we explain thermal effect on $\delta$. Although $\delta$ depends on length scale $L_{h}$ of Hall bar, effectively extended states in which electrons carry electric current are not restricted to the energy region such as $E_{f}+\delta>E>E_{f}-\delta$ at non zero temperature. For sufficiently low temperature, phase coherent length is much larger than the scale $L_{h}$ of Hall bar. The phase coherence lengthdoo1 is the length scale within which quantum states hold keeping quantum coherence. As temperature increases, the phase coherent length decreases and eventually reaches at the physical size $L_{h}$ of Hall bar. There is a critical temperature $T_{c}$ such that at the temperature $T_{c}$, the phase coherent length is equal to $L_{h}$. Beyond the temperature, the coherent length becomes smaller than the physical size of Hall bar. It implies that actual extension of localized states diminishes up to the phase coherent length. It apparently seems that electric current does not flow because all of localized states have extension less than the size of Hall bar. But, electrons in the localized states with spatial extension less than $L_{h}$ may make hopping to to nearby localized states and carry electric current. The hopping arises owing to scattering with phonon or impurities or tunneling under external electric field. It does not arise between localized states with large energy difference among them. In this way, electrons in the localized states even with their energies less than $E_{1+}-\delta$ may carry electric currents at temperature $T_{e}>T_{c}$. As temperature increases more, the phase coherent length decreases more and the hopping of electrons gets more actively. Thus, the energies of effectively extended states becomes smaller than $E_{1+}-\delta$. It implies that we have effective width $\delta_{e}(T_{e})$ instead of real $\delta$ such that $\delta_{e}(T_{e})\geq\delta$ for $T_{e}\geq T_{c}$ and $\delta_{e}(T_{e})=\delta$ for $T_{e}\leq T_{c}$. $\delta_{e}(T_{e})$ increases as $T_{e}$ increases. Thus, the effective extended states at $T_{e}>T_{c}$ are defined as those with energies $E$

with $T_{c}$ given such as $\delta_{e}(T_{c})=\delta$.

In the above argument, we need to replace $\delta$ by the effective width $\delta_{e}(T)$ or $\delta_{e}(T_{e})$,

It decreases smoothly as $T_{e}$ decreases. It takes the value $\Delta E_{f}=2\delta+4T_{e}$ below $T_{e}\leq T_{c}$ and reaches $2\delta$ as $T_{e}\to 0$. Therefore, we find that $\Delta E_{f}$ never saturate for any temperature $T_{e}$ when axion effect is absent. The width $\Delta B$ corresponding to $\Delta E_{f}$ also does not saturate. It behaves such as $\Delta B=\mbox{const.}+c\,T_{e}$ as $T_{e}\to 0$ with a numerical constant $c$.

Now we take into account the axion effect. The dark matter axions produces radiations with energy $m_{a}$ under strong magnetic field. Such radiations with their energy $m_{a}$ are absorbed by electrons, which make electrons transit from the state with energy $E$ to the state with energy $E+m_{a}$. The electrons in the state with energy $E+m_{a}$ may loose their energies by emitting phonon. This is similar process to the one that electron with energy $E$ transits to the state with energy $E+T$ by absorbing thermal energy $T$ and it loose its energy by emitting phonon. ( We note that black body radiation with energy $T$ is absorbed by electron just as radiation generated by axion. ) Consequently, the energy distribution of electrons is smeared out. But, the effect only slightly modifies the energy distribution because the axion effect is quite small. It is not important for later discussion to specify precisely how large the axion effect smears the energy distribution when temperature is low sufficiently for axion effect to be dominant over thermal effect. We only need to know the energy at which the energy distribution has sharp boundary. Thus, we simplify the effect of the axion such that the energy distribution smeared by axion also has a sharp boundary in the following. When $T_{e}>m_{a}$, the sharp boundary caused by thermal effect does not change even if the axion effect is taken into account. There are no occupied states with energies larger than $E_{f}+2T_{e}$ ( $>E_{f}+2m_{a}$ ). On the other hand, when $T_{e}<m_{a}$, sharp boundary moves upward beyond $E_{f}+2T_{e}$. That is, there are occupied states with energies larger than $E_{f}+2T_{e}$, while states with energies larger than $E_{f}+2m_{a}$ are empty. Consequently, the sharp boundary is located at $E_{f}+2m_{a}$ ( $>E_{f}+2T_{e}$ ) when $m_{a}>T_{e}$, while at $E_{f}+2T_{e}$ when $m_{a}<T_{e}$. See Fig.5. Similarly to the case of temperature $T_{e}$, the axion mass $m_{a}$ in the energy distribution of electrons is not necessarily equal to real axion mass. But, when we discuss the effect of microwave imposed externally on Hall bar in later section, the microwave plays the similar role to the radiation by the axion. Energy distribution of electrons modified by the radiation is supposed to have sharp boundary associated with the frequency $f$ of the radiation. Both radiations by axion and external setup are supposed to cause sharp boundary on the energy distribution such as the boundary located at $E_{f}+2m_{a}$ or $E_{f}+4\pi f$ when $m_{a}>T_{e}$ or $2\pi f>T_{e}$. In this case we may directly compare the frequency $f$ with axion mass as real ones.

Later we estimate how the axion effect is large compared with thermal effect. We note that black body radiation present at $T_{e}\neq 0$ gives similar effect to the one of radiation generated by the axion. It will turns out that the axion effect is dominant over the thermal effect when $m_{a}>T$ ( not $T_{e}$ ) as long as real temperature $T$ is less than $100$mK, when axion mass $\sim 10^{-5}$eV. When temperature is much larger than $100$mK, the energy distribution of electrons has no contribution of axion. So even for $m_{a}>T_{e}$, the sharp boundary is located at $E_{f}+2T_{e}$ when real temperature is larger than $100$mK. But we must remember that it is unclear what is the value $T_{e}$ corresponding to real temperature $100$mK. The relation depends on each samples.

Now we explain how the width $\Delta B$ depends on axion mass and temperature. We discuss it in the cases of $T_{c}>m_{a}$ and $T_{c}\leq m_{a}$ separately. First, we start the case of $T_{c}>m_{a}$.

When the temperature $T_{e}$ is larger than the axion mass $m_{a}$, the energy distribution of electrons is almost identical to the previous one with no axion effect. Thus, the width $\Delta B$ is given in the way mentioned above. Namely, the width $\Delta E_{f}$ leading to $\Delta B$ is given such that $\Delta E_{f}=2\delta_{e}(T_{e})+4T_{e}$ for $T_{e}>m_{a}$. The width decreases as $T_{e}$ decreases up to $T_{c}$ when $T_{c}>m_{a}$. At temperature $T_{e}=T_{c}$ ( $>m_{a}$ ), it takes $2\delta+4T_{c}$. It further decreases as $T_{e}$ decreases up to $T_{e}=m_{a}$. When $T_{e}\leq m_{a}$, the width $\Delta E_{f}$ is given such that $\Delta E_{f}=2\delta+4m_{a}$ because the energy distribution of electrons has sharp cutoff at $E_{f}+4m_{a}$. $\Delta E_{f}=2\delta+4m_{a}$ does not depend on $T_{e}$. Thus, the width saturates at $T_{e}=m_{a}$ when $T_{c}>m_{a}$. That is, $\Delta E_{f}$ is given in the following,

in the Hall bar with $T_{c}>m_{a}$. It saturates at $T_{e}=m_{a}$ and takes the value $\Delta E_{f}=2\delta+4m_{a}$. When we decrease the size of Hall bar ( increase $\delta$ ), the saturation temperature does not change, while the width $\Delta B$ corresponding to $\Delta E_{f}$ increases.

Next, we discuss the case of $T_{c}\leq m_{a}$. The width is given such that $\Delta E_{f}=2\delta_{e}(T_{e})+4T_{e}$ for $T_{e}>m_{a}$, while $\Delta E_{f}=2\delta_{e}(T_{e})+4m_{a}$ for $T_{c}\leq T_{e}\leq m_{a}$. Obviously, it decreases with $T_{e}$, but saturates at $T_{e}=T_{c}$ because $\delta_{e}(T_{c})=\delta$. The width is given such that $\Delta E_{f}=2\delta+4m_{a}$ for $T_{e}\leq T_{c}$. That is, $\Delta E_{f}$ is given in the following,

in the Hall bar with $T_{c}<m_{a}$. It saturates at $T_{e}=T_{c}<m_{a}$ and takes the value $\Delta E_{f}=2\delta+4m_{a}$. See Fig.6. The saturation temperature $T_{e}=T_{c}$ depends on the size of Hall bar.

To be summarized, the saturation temperature $T_{s}$ and the width of $\Delta E_{f}$ at $T_{e}\leq T_{s}$ are given in the following. When $T_{c}>m_{a}$

while, when $m_{a}>T_{c}$

We should notice misunderstanding that from the above formula $T_{s}=m_{a}$, the observation of the saturation temperature give real axion mass. As we have stated, the temperature $T_{e}$ in the formula is not real temperature, although it is not far from real temperature $T$. The formula $T_{a}=m_{a}$ in eq(9) implies that the saturation temperature does not depend on the size of Hall bar. On the other hand, the formula $T_{s}=T_{c}$ implies that the saturation temperature depends on the size of Hall bar. In order to find the saturation temperature $T_{s}=m_{a}$, we must check the independence of $T_{s}$ on the size of Hall bar.

Although the above result shows that the saturation temperature $T_{s}$ is equal to or less than the axion mass $m_{a}$ in any cases, it is possible to have real saturation temperature large such as $1$K$\sim 10^{-4}$eV, which is larger than axion mass $\sim 10^{-5}$eV expected later in the present paper.

In the previous paperwanli , it has been shown that as the size decreases, both of the saturation temperature $T_{s}=T_{c}$ and the width $\Delta B$ corresponding to $\Delta E_{f}$ increases. Indeed, it has been observed that $T_{s}\propto L_{h}^{-1}$. It indicates that $T_{c}\propto L_{h}^{-1}$. It has been understood that the saturation is caused by finite size effect. According to our analysis, such a feature of the saturation suggests that the sample used in the paper has the feature $T_{c}<m_{a}$. ( The real saturation temperatures in the paper are in the range $300\rm mK\sim 10mK$. )

On the other hand, it has been shown in the referencesat5 that the saturation temperatures $T_{s}$ in samples with sizes $50\mu$m$\times 200\mu$m, $200\mu$m$\times 800\mu$m and $800\mu$m$\times 3200\mu$m are almost identical ( real temperature $T\sim 30$mK ). Obviously, finite size effect does not appear. Thus, it has been stated in the referencesat5 that the feature is caused by intrinsic decoherence. On the other hand, according to our analysis, these samples may have the feature of $T_{c}>m_{a}$ and the saturation temperature is given by the axion mass, $T_{s}=m_{a}$. Therefore, the saturation, in other words, the intrinsic decoherence observed in the paper is caused by the axion dark matter.

The papersat5 suggests that the axion mass is given by the real saturation temperature $\simeq 30$mK. Although the temperature is not identical to the axion mass $m_{a}$. it suggests that it is near to the axion mass. Later we will find that the axion mass is strongly suggested to be equal to $(0.95\sim 0.99)\times 10^{-5}\mbox{eV}$ ( $\sim 100$mK ) by experiments using microwaves imposed on Hall bar.

We proceed to discuss the effect of external microwave imposed on Hall bar. We would like to know how the width $\Delta B$ ( or $\Delta E_{f}$ ) depends on the frequency $f$ of the microwave. Contrary to the observation of saturation temperature $T_{s}$, it will turn out that we can determine the axion mass by observing the saturation frequency $f_{s}$ of $\Delta B$.

The effect of the microwave is identical to the one of radiation generated by the axion. The difference is that we can change frequency and power of the microwave. The power is in general much larger than the power generated by the axion. The energy distribution of electrons is modified significantly, but its power must be taken small enough not to increase the temperature of the Hall bar. In actual experiments, the power has been taken sufficiently small for the temperature not to increase.

We also simplify the effect of the radiations similarly to that of radiation generated by the axion. That is, they modify energy distribution of electrons such that there is a sharp boundary in the distribution. For instance it has sharp boundary at the energy $E_{f}+2\pi f$ when $2\pi f>T_{e}$. The states with energies larger than $E_{f}+2\pi f$ are empty. On the other hand, it has sharp boundary at the energy $E_{f}+2T_{e}$ when $2\pi f\leq T_{e}$.

We should make a comment that although the frequency $f$ used to define the sharp boundary is not real frequency, the formula $f=m_{a}/2\pi$ obtained in subsequent discussions gives real axion mass $m_{a}$ by observing the real frequency $f$ in the experiment using external microwave. This is because both effects of radiations generated by axion and experimental apparatus are identically simplified on energy distribution of electrons. Therefore, the formula $f=m_{a}/2\pi$ obtained later implies that the real frequency $f$ is equal to the real axion mass divided by $2\pi$.

We remind that the effect of external microwave on the width $\Delta B$ is very similar to the effect of temperature, as shown experimentally. For example, $\Delta B\propto T^{\kappa}$ for $T\to 0$ and $\Delta B\propto f^{\kappa}$ for $f\to 0$ with $\kappa=0.4\sim 0.7$ doo1 . The behavior is expected in scaling theory as a critical behavior. The external microwaves with frequency $f$ diminish phase coherent length just as thermal fluctuations in temperature $T$ diminish it with correspondence $f\sim T$. Thus, we may take the effect of the external microwave on electrons in a similar way to the one of temperature.

The similarity can be understood by comparing the effect of the radiation with thermal effect. Electrons absorb thermal energy $T$ and transit to states with higher energy, but they lose their energies emitting phonons. Similarly electrons absorb radiation energy $f$ and transit to states with higher energy, but they lose their energies emitting phonons. Only difference is that the thermal energy $E$ distributes around $T$, while radiation energy is given by a single frequency $f$. Thus, we expect that both effect of temperature and radiation is almost identical.

Here we should notice that external microwaves makes the coherent length of electrons decrease just as temperature does. Thus, $\delta_{e}$ depends not only on temperature but also frequency of microwave such that $\delta_{e}(T_{e},f)$ increases as $T_{e}$ or $f$ increases with the condition $\delta_{e}(T_{e},f=0)=\delta_{e}(T_{e})$ and $\delta_{e}(T_{e}\leq T_{c})=\delta$. It is naturally supposed that there is a critical frequency $f_{c}(T_{e})$ such that $\delta_{e}(T_{e},f)=\delta_{e}(T_{e},f_{c}(T_{e}))$ for $f\leq f_{c}(T_{e})$. That is, $\delta_{e}(T_{e},f)$ does not decrease even more in $f$ for $f<f_{c}(T_{e})$. It saturates at the critical frequency $f_{c}(T_{e})$ when frequency $f$ decreases. In other words, the energy region $E_{1+}+\delta_{e}(T_{e},f)>E>E_{1+}-\delta_{e}(T_{e},f)$ of the effectively extended states does not change as long as $f<f_{c}(T_{e})$. The critical frequency $f_{c}(T_{e})$ decreases with the decrease of the temperature $T_{e}$. Furthermore, for the temperature $T_{e}\leq T_{c}$, $f_{c}(T_{e})$ does not decrease less than $f_{c}(T_{c})$. Namely, phase coherent length at temperature $T_{e}<T_{c}$ and frequency $f<f_{c}(T_{e}=T_{c})$ is larger than the size of Hall bar. That is, $\delta_{e}(T_{e},f_{c}(T_{e}))=\delta$ for $T_{e}\leq T_{c}$.

Therefore, we speculate the dependence of $\delta_{e}(T_{e},f)$ on temperature $T_{e}$ and frequency $f$ in the following,

Based on the speculation, we examine how the width $\Delta E_{f}$ leading to $\Delta B$ behaves depending on temperature $T_{e}$ and frequency $f$.

As discussed before, we discuss two cases of Hall bar separately. One is Hall bar with $T_{c}>m_{a}$, that is, Hall bar with small size of length. The other one is that with $T_{c}\leq m_{a}$, that is, Hall bar with large size.

First we consider the case $T_{c}>m_{a}$. When high temperature $T_{e}>T_{c}>m_{a}$, the energy distribution of electrons is modified by the microwave with frequency $f$ such that its sharp boundary is located at $E_{f}+4\pi f$ for $2\pi f>T_{e}$ or $E_{f}+2T_{e}$ for $T_{e}\geq 2\pi f$. Thus, the width $\Delta E_{f}$ is given in the following,

We can see that the width $\Delta E_{f}$ decreases with the decrease of the frequency $f$, but it saturates at the frequency $f=T_{e}/2\pi$ when $2\pi f_{c}(T_{e})>T_{e}$, while it does at $f_{s}=f_{c}(T_{e})$ when $2\pi f_{c}(T_{e})\leq T_{e}$. In both cases, $\Delta E_{f}=2\delta_{e}(T_{e},f_{c}(T_{e}))+4T_{e}$. Therefore, in the case $T_{e}>T_{c}>m_{a}$, both of saturation frequency $f=T_{e}/2\pi$ and $f_{s}=f_{c}(T_{e})$ decrease with the decrease of the temperature $T_{e}$.

On the other hand, in the case, low temperature $T_{e}<m_{a}$ ( $<T_{c}$ ), we have

When $f_{c}(T_{c})>m_{a}/2\pi$, we have two cases,

because $\delta_{e}(T_{e},f)=\delta_{e}(T_{e},f_{c}(T_{c}))=\delta$ for $f\leq f_{c}(T_{c})$.

The width $\Delta E_{f}=2\delta_{e}(T_{e},f)+8\pi f$ saturates at $f_{s}=m_{a}/2\pi$ where $\Delta E_{f}=2\delta+4m_{a}$. While, when $f_{c}(T_{c})\leq m_{a}/2\pi$, according to the equation(13), the width $\Delta E_{f}$ saturates at $f_{s}=f_{c}(T_{c})\leq m_{a}/2\pi$ where $\Delta E_{f}=2\delta+4m_{a}$. Therefore, in the case $T_{e}<m_{a}$ ( $<T_{c}$ ), both saturation frequency $f_{s}=m_{a}/2\pi$ and $f_{s}=f_{c}(T_{c})$ does not decrease with temperature $T_{e}$.