Quantum Nernst Effect
Hiroaki Nakamura
It is theoretically predicted that the Nernst coefficient is strongly suppressed and the thermal conductance is quantized in the quantum Hall regime of the two-dimensional electron gas. The Nernst effect is the induction of a thermomagnetic electromotive force in the $y$ direction under a temperature bias in the $x$ direction and a magnetic field in the $z$ direction. The quantum nature of the Nernst effect is analyzed with the use of a circulating edge current and is demonstrated numerically. % The following statement was added by Hatano on 01/13/2005. The present system is a physical realization of the non-equilibrium steady state.
050502Nernst.tex
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% SP 2001/01/05
\documentclass{elsart}

\usepackage{graphicx}% Include figure files
\usepackage{dcolumn}% Align table columns on decimal point
\usepackage{bm}% bold math
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%\nofiles

\begin{document}

\begin{frontmatter}

\title{Quantum Nernst Effect}

\author[nakamura]{Hiroaki Nakamura}
\ead{nakamura@tcsc.nifs.ac.jp}
\author[hatano]{Naomichi Hatano}
\author[shirasaki]{Ry\=oen Shirasaki}

\address[nakamura]{Theory and Computer Simulation Center,
National Institute for Fusion Science,
Oroshi-cho 322-6, Toki, Gifu 509-5292, Japan}
\address[hatano]{Institute of Industrial Science,
University of Tokyo,
Komaba 4-6-1, Meguro, Tokyo 153-8505, Japan}
\address[shirasaki]{Department of Physics, Yokohama National University,
Tokiwadai 79-5, Hodogaya-ku, Yokohama 240-8501, Japan}

\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}
It is theoretically predicted that the Nernst coefficient is strongly suppressed and the thermal conductance is quantized in the quantum Hall regime of the two-dimensional electron gas.
The Nernst effect is the induction of a thermomagnetic electromotive force in the $y$ direction under a temperature bias in the $x$ direction and a magnetic field in the $z$ direction.
The quantum nature of the Nernst effect is analyzed with the use of a circulating edge current and is demonstrated numerically.
% The following statement was added by Hatano on 01/13/2005.
The present system is a physical realization of the non-equilibrium steady state.
\end{abstract}
\begin{keyword}
Nernst effect \sep Nernst coefficient \sep edge current \sep
quantum Hall effect \sep thermoelectric power \sep thermomagnetic effect \sep non-equilibrium steady state

\PACS 73.23.Ad \sep 72.15.Gd \sep 72.20.My
\end{keyword}
\end{frontmatter}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}
The (adiabatic) Nernst effect in a bar of conductor is the generation of a voltage difference in the $y$ direction under a magnetic field in the $z$ direction and a temperature bias in the $x$ direction (Fig.~\ref{Fig.nernst}).
\begin{figure}[b]
\begin{center}
\includegraphics[width=0.35\textwidth,clip]{Nernst}
\end{center}
\caption{A setup for observation of the Nernst effect.
The Nernst voltage $V_\mathrm{N}$ is defined as such that it is positive when the voltage of the upper edge is higher than the voltage of the lower edge.}
\label{Fig.nernst}
\end{figure}
Each of the left and right ends of the conductor is attached to a heat bath with a different temperature, $T_+$ on the left and $T_-$ on the right.
An electric insulator is inserted in between the conductor and each heat bath, so that only the heat transfer takes place at both ends.
%%%% The following was added by N. Hatano on 02/09/05.
(There is no contact on the upper and lower edges.)
%%%%%
A constant magnetic field $B$ is applied in the $z$ direction.
Then the Nernst voltage $V_\mathrm{N}$ is generated in the $y$ direction. 
(In what follows, we always put $\Delta T\equiv T_+-T_->0$ and $B>0$.)

A classical-mechanical consideration on this thermomagnetic effect yields the following:
a heat current flows from the left end to the right end because of the temperature bias;
the electrons that carry the heat current receive the Lorentz force from the magnetic field and deviate to the upper edge;
then we have $V_\mathrm{N}<0$.
The Nernst coefficient is defined as
\begin{equation}\label{eq1}
N\equiv \frac{V_\mathrm{N}/W}{B \nabla_x T},
\end{equation}
where the temperature gradient is given by $\nabla_x T=-\Delta T/L$ with $W$ and $L$ being the width and the length of the conductor bar.
The above naive consideration gives a positive Nernst coefficient.
In reality, the Nernst coefficient can be positive or negative, depending on the scattering process of electrons.

The Nernst effect was extensively investigated in the 1960's\cite{67Harman}
because of a possible application to conversion of heat to electric energy.
The investigation on the energy conversion was eventually abandoned,
since induction of the magnetic field cost lots of energy in those days. 
The Nernst effect, however, has recently seen renewed interest\cite{94Yam2,99Nakamura,04Hasegawa};
improvement of the superconducting magnet has led to more efficient induction of a strong magnetic field.
This is a background of recent studies on the Nernst effect at temperatures higher than the room temperature.

In the present Letter, we direct our attention to the Nernst effect in the regime of the ballistic conduction, that is, the Nernst effect of the two-dimensional electron gas in semiconductor heterojunctions at low temperatures, low enough for the mean free path to be greater than the system size.
Using a simple argument on the basis of edge currents\cite{82Halperin}, we predict that, when the chemical potential is located between a pair of Landau levels,
(i) the Nernst coefficient is strongly suppressed and (ii) the thermal conductance in the $x$ direction is quantized.

%%%% The following part was added by Hatano on 01/13/2005.
%%%% Edited by Hatano on 03/15/2005.
Incidentally, the physical state of the present system is a realization of the non-equilibrium steady state (NESS), a new concept much discussed recently in the field of non-equilibrium statistical physics\cite{Ruelle,Ogata}.
The non-equilibrium steady state is almost the first quantum statistical state far from equilibrium that can be analyzed rigorously;
it consists of a couple of independent currents with different temperatures.
Since the study of the non-equilibrium steady state has been almost purely mathematical, we consider it valuable to give it a physical realization.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Predictions}
Let us first briefly explain our basic idea (Fig.~\ref{fig.idea}).
\begin{figure}[b]
\begin{center}
\includegraphics[width=0.4\textwidth,clip]{convection}
\end{center}
\caption{A schematic view of the dynamics of electrons in a Hall bar under the setup for the Nernst effect.}
\label{fig.idea}
\end{figure}
Since there is no input or output electric current, an edge current circulates around the Hall bar when the chemical potential is in between neighboring Landau levels.
The edge current along the left end of the bar is in contact with the heat bath with the temperature $T_+$ and equilibrated to the Fermi distribution $f(T_+,\mu_+)$ with the temperature $T_+$ and a chemical potential $\mu_+$ while running from the corner C$_4$ to the corner C$_1$.
Since the upper edge is not in contact with anything, the edge current there runs ballistically, maintaining the Fermi distribution $f(T_+, \mu_+)$ all the way from the corner C$_1$ to the corner C$_2$.
%%%%%%%%%%%%%%%%%%%%% ���̒��O��Ogata�����p���܂����B(2005-Jan-12 Nakamura) 
%%%%%% This was moved to a few lines below by Hatano on 1/13/2005.
It then encounters the other heat bath with the temperature $T_-$ and equilibrated to the Fermi distribution $f(T_-, \mu_-)$ while running from the corner C$_2$ to the corner C$_3$.
The edge current along the lower edge runs ballistically likewise, maintaining the Fermi distribution $f(T_-, \mu_-)$ all the way from the corner C$_3$ to the corner C$_4$.
%%%%%% The following sentense was added by Hatano on 1/13/2005.
(The circulating edge current constitutes a physical realization of the non-equilibrium steady state\cite{Ogata}.)
The Nernst voltage $V_\mathrm{N}=\Delta\mu/\mathrm{e}\equiv(\mu_+-\mu_-)/\mathrm{e}$ is thus generated, where $\mathrm{e}(<0)$ denotes the charge of the electron.

First, the difference in the chemical potential, $\Delta\mu$, is of a higher order of the temperature bias $\Delta T$, because the number of the conduction electrons is conserved.
The Nernst coefficient (\ref{eq1}), or
\begin{equation}\label{eq10}
N=\frac{1}{|\mathrm{e}|B}\frac{L}{W}\frac{\Delta\mu}{\Delta T},
\end{equation}
hence vanishes as a linear response.
Second, the heat current $I_Q$ in the $x$ direction is carried ballistically by the edge current along the upper and lower edges.
The edge current does not change much when we vary the magnetic field $B$ as long as the chemical potential stays between a pair of neighboring Landau levels.
The heat current hence has quantized steps as a function of $B$.



%%%%% The following part was added by Hatano on 01/15/2005.
%We comment here on the study of the \lq\lq thermoelectric power" in the 1980's\cite{82Zelenin,82Girvin,83Streda,84Jonson}.
%Although the mathematics looks similar to ours, the physical situation is much different.
%The theory of the thermoelectric power considered only the left and right edge currents;
%the upper and lower edge currents were absent.
%The theory related the difference in the chemical potential between the left and right edges to their temperature difference as a \textit{diagonal} response.
%The voltage gradient in the $y$ direction was completely missing in the pure sample;
%it appeared only after taking account of impurity scattering.
%However, the chemical potentials of the right and left edge currents are not well-defined, particularly in the present regime of ballistic transport;
%the currents there are still in the process of equilibration under complicated many-body interactions with the heat baths.
%It is hence quite uncertain whether the experimental \lq\lq confirmation"\cite{thermo.exp} in fact observed what the theory predicted;
%some inconsistency was attributed to impurity scattering, but might have been due to the interactions with the heat baths.

%An essential difference of the present theory is to measure the well-defined voltage on the upper and lower edges and observe the Nernst coefficient, an \textit{off-diagonal} response.
%In fact, Hasegawa and Machida\cite{nernst.experiment} are planning an experiment in the present topology.
%The experimental technique has been developed remarkably since the 1980's, when it was almost impossible to achieve the ballistic transport.
%It would have been in vain then to consider the ballistic currents on the upper and lower edges.
%The present predictions have become first worthwhile in the light of the new technology.
%%%%%%%%%%%%%%

\section{Two-dimensional electron gas}
We now describe the above idea explicitly.
In order to fix the notations, we begin with the basics of the two-dimensional electron gas in a magnetic field.
The dynamics of the two-dimensional electron gas is described by the Schr\"{o}dinger equation
%%%%%%%% The following was modified by Hatano on 01/19/2005.
with the vector potential $(-By,0)$ and the confining potential $V(y)$
%\begin{equation}
%\left[\frac{\left( p_{x} + \mathrm{e} B y \right)^2}{2 m}
%+ \frac{p_{y}^2}{2 m }+V(y) \right]\Psi (x,y)= E \Psi (x,y),
%\label{sh.eq} 
%\end{equation}
%where the energy is measured from the subband of the confining potential in the $z$ direction.
%The potential $V(y)$ is the confining potential in the $y$ direction, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
shown schematically in Fig.~\ref{Fig.vy}.
\begin{figure}
\begin{center}
\includegraphics[width=0.35\textwidth,clip]{potential.eps}
\end{center}
\caption{A schematic view of the transverse confining potential $V(y)$, on which the structure of the Landau levels is superimposed.
The central part of the potential, $|y|<w/2$, is flat, whereas the potential edges $w/2<|y|<W/2$ may have some curvatures.}
\label{Fig.vy}
\end{figure}
We can express the eigenfunction 
%of eq.~(\ref{sh.eq}) 
in the form of variable separation:
%\begin{equation}\label{psi.eq}
$\Psi(x,y) = \mathrm{e}^{\mathrm{i}kx} \chi_k(y)/\sqrt{L}$,
%\end{equation}
where
%\begin{equation}
$k = 2\pi j/L$
%\end{equation}
with an integer $j$.
The transverse part $\chi_k(y)$ is an eigenfunction of the equation
%\begin{equation}\label{chi.eq}
${\mathcal H}_k\chi_k(y)=E\chi_k(y)$,
%\end{equation}
where
% the Hamiltonian is given by
\begin{equation}\label{ham.eq}
{\mathcal H}_k\equiv\frac{p_{y}^2}{2 m }+ \frac{m \omega_\mathrm{c}^2 }{2} \left( y - y_k \right)^2 +V(y)
\end{equation}
with
%\begin{equation}\label{yk.eq}
$\omega_\mathrm{c}\equiv|\mathrm{e}|B/m$
%\quad\mbox{and}\quad
and
$y_k\equiv \hbar k/(|\mathrm{e}|B)$. 
%\end{equation}
We label the discrete eigenfunctions with an integer $n$.
The whole solution is then given by
%\begin{equation}\label{eq100}
$\Psi_{n,k}(x,y)=\mathrm{e}^{\mathrm{i}kx}\chi_{n,k}(y)/\sqrt{L}$
%\end{equation}
with
% an eigenvalue
%\begin{equation}
$E=E(n,k)$.
%\end{equation}
As is schematically shown in Fig.~\ref{Fig.vy}, the eigenvalue $E(n,k)$ in fact scarcely depends on $k$ in the bulk, where the confining potential $V(y)$ is flat\cite{82Halperin,84MacDonald}.

The Hamiltonian~(\ref{ham.eq}) shows that an eigenfunction with the $x$ component of the momentum, $\hbar k$, is centered around $y=y_k\propto k$.
In other words, the state in the upper half of the Hall bar has a current in the positive $x$ direction, while the one in the lower half has a current in the negative $x$ direction.
The velocity of the electron in the state $\Psi_{n,k}$ is
\begin{equation}\label{vdef.eq}
v\left( n,k \right) 
 = \frac{1}{\hbar} \frac{\partial E(n,k)}{\partial k },
\end{equation} 
which remains finite only near the upper and lower edges.
These are the edge currents.

\section{Electric and heat currents}
Now we write down the electric current $I_\mathrm{e}$ and the heat current $I_Q$ in the $x$ direction carried by electrons.
(Note that we will put $I_\mathrm{e}=0$ in the bottom line, observing the boundary conditions in Fig.~\ref{fig.idea}.)
The currents are given by
%%%%%% The following was modified by Hatano on 01/15/2005.
\begin{equation}\label{I.eq}
I_\mathrm{e}=\langle \mathrm{e}v\rangle
\quad\mbox{and}\quad
I_Q=\langle(E-\mu)v\rangle
\end{equation}
with the thermal average
\begin{equation}
\langle\cdots\rangle=\frac{1}{\pi} \sum_{n=0}^{\infty}
  \int_{-k_\mathrm{m}}^{k_\mathrm{m}}     
 \cdots
 f_{n,k}\left(T(y_k),\mu(y_k) \right)\mathrm{d}k,
\end{equation}
%%%%%%%
%\begin{eqnarray}\label{I.eq}
%I_\mathrm{e}
%% This part was deleted by Hatano on 01/13/2005.
%%&=& \frac{1}{L}\sum_{n=0}^{\infty}\sum_{k,\uparrow\downarrow}\mathrm{e}v\left(n,k \right) f_{n,k}\left(T(y_k),\mu(y_k) \right)
%%\nonumber\\
%&=&\frac{\mathrm{e}}{\pi} \sum_{n=0}^{\infty}
%  \int_{-k_\mathrm{m}}^{k_\mathrm{m}}     
% v\left(n,k \right) f_{n,k}\left(T(y_k),\mu(y_k) \right)\mathrm{d}k,
%\\ \label{Q.eq}
%I_Q &=&  \frac{1}{\pi}\sum_{n=0}^{\infty}
%  \int_{-k_\mathrm{m}}^{k_\mathrm{m}}     
%\left( E(n,k)-\mu  \right) \times
%\nonumber\\
%&&\phantom{\frac{1}{\pi}\sum_{n=0}^{\infty}  \int_{-k_\mathrm{m}}^{k_\mathrm{m}}}
%v\left(n,k \right) f_{n,k}\left(T(y_k),\mu(y_k) \right)\mathrm{d}k ,
%\end{eqnarray}
where we made the summation over $k$ to the momentum integration.
The integration limits $\pm k_\mathrm{m}$ are the maximum and minimum possible momenta.
%are defined by
%\begin{equation}\label{ymax.eq}
%$y_{k_\mathrm{m}}= W/2$.
%\end{equation}
The function $f_{n,k}$ denotes the Fermi distribution $f(T,\mu)=\{1+\exp[(E-\mu)/(k_\mathrm{B}T)]\}^{-1}$ with $E=E(n,k)$.
%\begin{equation}
%f_{n,k}(T,\mu)\equiv\frac{1}{1+\exp[(E(n,k)-\mu)/(k_\mathrm{B}T)]}.
%\end{equation}
The layout in Fig.~\ref{fig.idea} yields $(T(y_k),\mu(y_k))=(T_+,\mu_+)$ for 
%$y_k\lesssim W/2$ 
the upper edge states and $(T_-,\mu_-)$ for 
%$y_k\gtrsim -W/2$.
the lower edge states.

We transform $I_\mathrm{e}$
%eq.~(\ref{I.eq}) 
with the use of eq.~(\ref{vdef.eq}) as
\begin{equation}\label{I2.eq}
I_\mathrm{e}
% This part was deleted by Hatano on 01/13/2005.
%&=& \frac{\mathrm{e}}{\pi} \sum_{n=0}^{\infty} \bigg[
%     \int_\mathrm{upper}  v (n,k)  f_{n,k}\left( T_+,\mu_+ \right)\mathrm{d} k
%\nonumber\\
%&&\phantom{\frac{\mathrm{e}}{\pi} \sum_{n=0}^{\infty}\bigg[}
%+ \int_\mathrm{lower}  v (n,k)  f_{n,k}\left( T_-,\mu_- \right)\mathrm{d} k \bigg], 
%           \nonumber \\
 = \frac{\mathrm{e}}{\pi\hbar} \sum_{n=0}^{\infty} 
      \int_{E_0(n)}^{E_1(n)} 
	  \left[ f\left(T_+,\mu_+\right)-f\left(T_-,\mu_-\right) \right]
	  \mathrm{d}E.
\end{equation}
% This part was deleted by Hatano on 01/13/2005.
%Here $\int_\mathrm{upper}$ and $\int_\mathrm{lower}$ denote the integrations with respect to the upper and lower edge states, respectively.
The integration limits are now $E_0(n)=E(n,0)$ and $E_1(n)=E(n,k_\mathrm{m})$ (see Fig.~\ref{Fig.vy}).
In order to compute the linear response, we here put
$T_\pm=T\pm\Delta T/2$ and $\mu_\pm=\mu\pm\Delta \mu/2$
with $\Delta T \ll T$ and $\Delta \mu \ll \mu$.
%%%%%%%%%%%%%%%%%%%%% (2) �ȉ����C���E�⑫���܂����B(2005-Jan-12 Nakamura)
%�O�̕��ŏq�ׂ��̂ŁA���̕����͍폜���܂����Bby Hatano on 01/13/2005.
%In expantion of eq.~(\ref{I2.eq}) with respect to $\Delta T$ and $\Delta\mu$,
%equations become similar to the previous studies\cite{82Zelenin,82Girvin,83Streda,84Jonson},
%where thermoelectric power was calculated for the edge currents.
%Then we have
By expanding eq.~(\ref{I2.eq}) with respect to $\Delta T$ and $\Delta\mu$, we have
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation} \label{I3.eq}
I_\mathrm{e} 
% This part was deleted by Hatano on 01/13/2005.
%&\simeq& \frac{\mathrm{e}}{\pi\hbar} \sum_{n=0}^{\infty} 
%             \int_{E_0(n)}^{E_1(n)} 
%        \left( 
%         \frac{\partial f}{\partial \mu} \Delta \mu
%      +  \frac{\partial f}{\partial T} \Delta T  
%        \right) 	  \mathrm{d}E \nonumber  \\ 
  \simeq \frac{\mathrm{e}}{\pi\hbar} \left[ \Delta\mu \sum_{n=0}^{\infty} A_0(n)
      +k_\mathrm{B}\Delta T \sum_{n=0}^{\infty} A_1(n)  \right],
\end{equation}
and similarly
\begin{equation}\label{Q3.eq}
I_Q \simeq \frac{k_\mathrm{B}T}{\pi\hbar} \left[  \Delta \mu \sum_{n=0}^{\infty} A_1(n)
      +k_\mathrm{B}\Delta T \sum_{n=0}^{\infty} A_2(n)  \right],
\end{equation}
where
\begin{equation}\label{An.eq}
A_\nu(n)\equiv\int_{x_0(n)}^{x_1(n)}
       \frac{x^\nu\mathrm{d}x}{4 \cosh^2 \left( x/2 \right)}
\end{equation}
with
%\begin{eqnarray}
$x_i (n) \equiv \left(E_i(n)-\mu\right)/(k_\mathrm{B}T)$. 
%       \label{x0.eq} \\ 
%x_1 (n) &\equiv& \left(E_\mathrm{max}(n)-\mu\right)/(k_\mathrm{B}T).
%            \label{x1.eq}
%\end{eqnarray}
The integral~(\ref{An.eq}) can be carried out explicitly for $\nu=0,1,2$.
% This part was deleted by Hatano on 01/13/2005.
% as
%\begin{eqnarray}
%A_0 (n)   &=& \left[\frac{1}{2} \tanh\frac{x}{2} \right]_{x=x_0(n)}^{x_1(n)},
%           \label{A0.eq} \\
%A_1 (n)         &=& \left[
%                \frac{x}{2} \tanh \frac{x}{2} 
%              - \log \cosh \frac{x}{2} 
%            \right]_{x=x_0(n)}^{x_1(n)},
%          \label{A1.eq}   \\
%A_2 (n)         &=& \bigg[2 \mathrm{Li}_2 \left(-\mathrm{e}^{-x} \right)
%- 2 x \log \left( 1+\mathrm{e}^{-x} \right) 
%\nonumber\\
%&&\phantom{\bigg[}
%-\frac{x^2}{2}\left(1-\tanh \frac{x}{2}\right) 
%            \bigg]_{x=x_0(n)}^{x_1(n)}
%         , \label{A2.eq}  
%\end{eqnarray}
%where $  \mathrm{Li}_n (z) = \sum_{m=1}^{\infty} z^m/m^n$ is the polylogarithmic function.

Since there is no input or output current in the setup in Fig.~\ref{Fig.nernst}, we put $I_\mathrm{e}\equiv0$ in eq.~(\ref{I3.eq}), relating $\Delta\mu$ with $\Delta T$.
%%%%%%%%% The following part was removed by Hatano on 01/19/2005.
as
\begin{equation}\label{muT.eq}
\Delta \mu = -\frac{\sum_n A_1(n)}{\sum_n A_0(n)}k_\mathrm{B}\Delta T.
\end{equation}
%%%%%%%%%%%%%%%%%%%
We thereby arrive at the Nernst coefficient~(\ref{eq10}) as
\begin{equation}\label{Nernstresult}
N  = -\frac{k_\mathrm{B}}{|\mathrm{e}|B} \frac{L}{W} \frac{\sum_n
                  A_1(n)}{\sum_n A_0(n)}.
\end{equation}
The heat current~(\ref{Q3.eq}) 
%and eq.~(\ref{muT.eq}) 
yields the thermal conductance as
\begin{equation}\label{Conductanceresult}
G_Q\equiv \frac{I_Q}{\Delta T}
=\frac{{k_\mathrm{B}}^2T}{\pi\hbar} \left[  \sum_n A_2(n)- \frac{\left(\sum_n A_1(n)\right)^2 }{ \sum_n A_0(n) }
                         \right]. 
\end{equation}

\textit{Low-temperature limit}:
In the low-temperature limit, the upper and lower limits of the integration in eq.~(\ref{An.eq}) goes to $\pm\infty$, depending on their signs.
First, in the usual experimental situation, the confining potential at its edges (of the order of eV) is considerably higher than the chemical potential (of the order of meV);
hence we assume $E_1(n)>\mu$ for all $n$.
The upper integration limit $x_1(n)$ thus always goes to $+\infty$ as $T\to 0$.
Next, suppose that the chemical potential is located in between the bottom of the $(M-1)$th Landau level and the bottom of the $M$th one.
The lower integration limit $x_0(n)$ goes to $+\infty$ for $n\geq M$ and the integral vanishes as $T\to0$.
The integral can survive only for $n\leq M-1$, for which the lower integration limit $x_0(n)$ goes to $-\infty$ as $T\to0$, yielding
%\begin{eqnarray}
$A_0(n)=1$,
%\\
$A_1(n)=0$,
%\\
and $A_2(n)=\pi^2/3$.
%\end{eqnarray}
%%%%%%%%% The following part was removed by Hatano on 01/19/2005.
%(The integration $A_\nu$ for odd $\nu$ vanishes as the integrand is an odd function.)
%%%%%%%%%%%%%%%%%%%%%%%%%%
Thus we arrive at the predictions
\begin{equation}\label{eq.lowT}
N=0\qquad\mbox{and}\qquad
\frac{G_Q}{T}=\frac{\pi{k_\mathrm{B}}^2}{3\hbar}M
\end{equation}
when the chemical potential is located in between the bottoms of a pair of the neighboring Landau levels.


\section{Numerical demonstration}
Let us demonstrate the above by adopting the following confining potential\cite{92Komiyama}:
\begin{equation}
 V(y) = 
  \left\{
    \begin{array}{ll}
      0           & \mbox{for}\quad |y| \leq \frac{w}{2} , \\
      \frac{m \omega_0^2}{2} \left( |y| - \frac{w}{2} \right)^2
             & \mbox{for}\quad \frac{w}{2}< |y| <\frac{W}{2}.
    \end{array}
  \right.
  \label{v.eq}
\end{equation}
%Hamiltonian can be expressed as 
%\begin{equation}
%{\mathcal H} %   \left\{
%    \begin{array}{ll}
%         \frac{p_{y}^2}{2 m }
%          + \frac{m \omega_\mathrm{c}^2 }{2} \left( y - y_k \right)^2   
%%            \ \ \ \equiv {\mathcal H}_\mathrm{I}   
%         & \mbox{in I } , \\
%         \frac{p_{y}^2}{2 m }
%         + \frac{m \omega_\mathrm{c0}^2 }{2} \left( y - Y_k \right)^2
%         + \left( \frac{ \omega_\mathrm{c0} }{ \omega_\mathrm{c} } \right)^2
%              \frac{m \omega_\mathrm{0}^2 \left( |Y_k| - w \right)^2}{2}   
%%           \ \equiv {\mathcal H}_\mathrm{II}          
%         & \mbox{in II and II$^{\prime}$ } .
%    \end{array}
%  \right.
%  \label{hami2.eq}
%\end{equation}
%%%%%%%% This part was deleted by Hatano on 01/27/2005.
%The parabolic parts near the edges cause a shift of the center of $\chi_k(y)$ from $y_k$ to $Y_k \equiv (\omega_\mathrm{c}^2 y_k +  
%\omega_0^2 w/2 )/\omega_\mathrm{c0}^2$.
%The upper limit of the momentum integration is then given by $Y_{k_\mathrm{m}}= W/2$.
%%%%%%%%%%%%%
The eigenvalues are well approximated in each region of eq.~(\ref{v.eq}) by tentatively regarding that the potential there continues for all $y$\cite{82Halperin}.
%%%%%%%%%%%% This part was deleted by Hatano on 01/13/2005.
%This gives\cite{90Martin,86Smrcka,86Berggren}
%\begin{equation}\label{energy.eq}
%E(n,k)\simeq
%  \left\{
%    \begin{array}{ll}
%      \left( n+ \frac{1}{2} \right) \hbar \omega_\mathrm{c}	  & \mbox{for}\quad |y_k| \leq \frac{w}{2} , \\
%      \multicolumn{2}{l}{\left( n+ \frac{1}{2} \right) \hbar \omega_\mathrm{c0} 
%           + \left( \frac{\omega_\mathrm{c0} }{\omega_\mathrm{c} }\right)^2  
%             \frac{ m \omega_0^2 \left( |Y_k| - w/2  \right)^2}{2}} \\
%       \phantom{\left( n+ \frac{1}{2} \right) \hbar \omega_\mathrm{c}}  & \mbox{for}\quad \frac{w}{2}< |y_k| <\frac{W}{2},
%    \end{array}
%  \right.
%\end{equation}
%where 
%%\begin{eqnarray}
%$\omega_\mathrm{c0} \equiv \sqrt{ \omega_\mathrm{c}^2+ \omega_0^2 }$.
%%%%%%%%%%%%%%
%               \label{omega.eq} \\
%and 
%\label{yk1.eq}
%\end{eqnarray}
This approximation is valid because, in the parameter range that we use below,
% This part was deleted by Hatano on 01/13/2005.
%the width of the eigenfunction $\chi_{n,k}(y)$ for low $n$ is much less than $w$, $W$ and $W-w$;
%that is, 
each eigenfunction is well localized in the $y$ direction and insensitive to the potential elsewhere.
The mismatch of the approximated eigenvalue
% This part was deleted by Hatano on 01/13/2005.
%~(\ref{energy.eq}) 
at $|y|=w/2$ is much smaller than the eigenvalue itself
% This part was deleted by Hatano on 01/13/2005.
%because $\omega_0 \ll \omega_\mathrm{c}$
in the parameter range below.\cite{92Komiyama}
Furthermore, the eigenvalue is shifted
% This part was deleted by Hatano on 01/13/2005.
% to
%\begin{equation}
%\textstyle{
%E(n,k_\mathrm{m})\simeq
%\left[ (2n+1)+ \frac{1}{2} \right] \hbar \omega_\mathrm{c0} 
%           + \left( \frac{\omega_\mathrm{c0} }{\omega_\mathrm{c} }\right)^2  
%             \frac{ m \omega_0^2 \left( W- w \right)^2}{8}}
%\end{equation}
right on the edges because of the potential walls\cite{84MacDonald}.
This shift contributes only to a shift of $x_1(n)$, which is irrelevant as $x_1(n)$ is virtually infinite anyway.
%\begin{equation}
%v\left( n,k \right) 
% = \frac{1}{\hbar} \frac{\partial E(n,k)}{\partial k } 
% = \left\{
%    \begin{array}{ll}
%      0           & \mbox{in I}, \\
%     -\frac{\omega_0^2}{\omega_\mathrm{c}} \left( Y_k-   w/2  \right) 
%                  & \mbox{in II and II$^{\prime}$}.
%    \end{array}
%  \right.
% \label{v1.eq}
%\end{equation} 
%where the lower and upper limits of energy integral 
%are defined as 
%\begin{eqnarray}
%E_\mathrm{min}(n)&\equiv& \left( n+\frac{1}{2}\right) \hbar \omega_\mathrm{c},  
%                                                 \label{Emin.eq} \\
%E_\mathrm{max}(n)&\equiv& \left( n+\frac{1}{2}\right) \hbar \omega_\mathrm{c0} 
%                     + \frac{m\omega_\mathrm{0}^2
%       \left( \frac{W}{2} - w/2 \right)^2}{2}
%       \left( \frac{ \omega_\mathrm{c0} }{ \omega_\mathrm{c} } \right)^2. 
%  \label{Emax.eq} 
%\end{eqnarray}

We set the parameters as follows:
%\cite{Machida}:
the effective mass is $m=0.067 m_0$ for GaAs, where $m_0$ is the bare mass of the electron;
the sample size is $L=20\mu$m and $W=20\mu$m (less than the mean free path at low temperatures\cite{Tarucha}) with $w=16\mu$m;
the confining potential is given by $V(\pm W/2) = 5.0$eV, the work function of GaAs;
the chemical potential is $\mu=15$meV, which means the carrier density $n_\mathrm{s}=4.24\times 10^{15}\mathrm{m}^{-2}$.
%The temperature bias is $\Delta T=1$K.

Using these values, we obtained the adiabatic Nernst coefficient~(\ref{Nernstresult}) as in Fig.~\ref{Fig.coeff}
and the thermal conductance~(\ref{Conductanceresult}) as in Fig.~\ref{q.fig}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{coeff}
\end{center}
\caption{The magnetic-field dependence of the adiabatic Nernst coefficient at $T=1,5,10$ and $20$K for $1\mathrm{T}\leq B\leq 20\mathrm{T}$;
(a) the raw data and (b) a scaling plot of $N \times B$ against $m\mu/\hbar|\mathrm{e}|B$.}
\label{Fig.coeff}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{quantized}
\end{center}
\caption{The magnetic-field dependence of the thermal conductance at $T=1,5,10$ and $20$K for $1\mathrm{T}\leq B\leq 20\mathrm{T}$;
(a) the raw data and (b) a scaling plot of $G_Q/T\times3\hbar/\pi{k_\mathrm{B}}^2$ against $m\mu/\hbar|\mathrm{e}|B$.}
\label{q.fig}
\end{figure}
We see that our predictions~(\ref{eq.lowT}) are indeed realized at low temperatures.
We also note that the Nernst coefficient is negative in the present case.




%%%%% The following part was further modified by Hatano on 02/06/2005.
%%%%% The following part was modified by Nakamura on 01/19/2005.
\section{Thermopower}
We comment here on the study of \lq\lq the thermopower of the two-dimensional electron gas" in the 1980's.
Although the mathematics resembles ours, the physical situation is much different.

%%%%% The following part was yet further modified by Hatano on 02/24/2005.
The theoretical study of the thermopower\cite{82Zelenin,82Girvin,83Streda,84Jonson} considered edge currents with different temperatures on opposing edges with adiabatic boundary conditions.
This situation is indeed the same as the upper and lower edge currents in Fig.~\ref{fig.idea}.
Therefore, Eq.~(\ref{muT.eq}) is the same as the relation that they derived between the temperature difference and the chemical-potential difference of the two edge currents as a longitudinal response.
The theory, however, did not come up with any experimental setup that could realize the above situation.
The experiments\cite{84Obloch,86Obloch,86Davidson,86Fletcher}, in fact, were carried out in a situation where edge currents were in contact with heat baths with different temperatures;
that is, the experimental study considered non-adiabatic boundary conditions on the upper and lower edges in Fig.~\ref{fig.idea}.
In spite of this essential difference did  they compare the theoretical predictions with the experimental results!

In comparison with the study in the 1980's, our novel point is to propose a way of realizing the situation where edge currents with different temperatures coexist, by considering the circulating edge current in Fig.~\ref{fig.idea}.
%
%Their theory of the thermopower considered the isothermal condition;
%that is, while there is a temperature gradient in one direction, there is none in the other direction.
%%The theory of the thermopower considered only the left and right edge currents;
%%the upper and lower edge currents were absent in their situation.
%%As a result, the thermopower tensor $S_{ij}$ was diagonal and the thermal-conductivity tensor $K_{ij}$ lacked the diagonal elements in the pure sample at low temperatures.
%
%Most experiments on the transport coefficients are carried out in the adiabatic condition that the heat current in the $y$ direction vanishes.
%If we try to convert their isothermal response functions to the adiabatic ones, however, we end up in a contradiction;
%the temperature gradient in the $x$ direction, where there was originally the temperature gradient, now must vanish!
%%$S_{xy}$ (proportional to the isothermal Nernst coefficient) and $K_{xx}$ appeared only after taking account of impurity scattering.
%The adiabatic condition would not have been satisfied without considering the phonon thermal conductivity, which, however, is negligible in the regime of ballistic transport.
%We therefore claim that the earlier theory of the thermopower is not applicable to the present situation.
%
%In our theory, in contrast, we do not use the temperature and the chemical potential of the right and left edge currents.
%The quantities there are indeed ill-defined, particularly in the ballistic regime;
%the currents there are in the process of equilibration under complicated interactions with the heat baths.
%The point of our theory is that, 
Thanks to the setup of the Nernst effect,  the \textit{external} temperature gradient in the $x$ direction appears in the \textit{internal} temperature gradient in the $y$ direction.
%thus we avoid the above contradiction.
%%In the present study, in contrast, the circulating edge current yields both $S_{xy}$ and $K_{xx}$ in the pure sample.
%%Moreover, the adiabatic condition is satisfied without considering the phonon effect;
%%we can thereby obtain the adiabatic Nernst coefficient directly.
%%Finally, the chemical potential is measured where it is well-defined.
%
%%They calculated  the thermoelectric ($S_{ij}$)  and thermal conductivity  ($K{ij}$) tensors which are defined by 
%%\begin{equation}
%%\partial_i  \phi  = - S_{ij} \partial_j T  , 
%%{J_Q\}_i = K_{ij} \partial_j T + \kappa_p \partial_i T,\label{tensor}
%%\end{equation}
%%where $i$ and $j$ denotes $x$ or $y$ direction, and potential $\phi$  and temperature  $T$.
%%$\{J_Q\}_i$ is $i$-component of the heat current density. 
%%They considered\cite{83Streda}   phonon thermal conductivity $\kappa_p$  
%%in Eq. (\ref{tensor}) to satisfy the adiabatic  condition $(\{J_Q\}_y = 0 )$ to measure
%%the adiabatic Nernst voltage $V_N$ in Fig. \ref{Fig.nernst}.
%
%%After similar mathematical calculations to ours,
%%they concluded that $S_{ij}$ is diagonal, $K_{ij}$ is off-diagonal  
%
%%On the other hand, our result shows that $S_{ij}$ and  $K_{ij}$  have non-zero both diagonal and off-diagonal components. The adiabatic condition  $(\{J_Q\}_y = 0 )$, therefore, is satisfied without the phonon thermal conductivity term in Eq. (\ref{tensor}).
%%This is quite consistent with the ballistic transport situation,
%%because  carrier-phonon interaction is not dominant in the ballistic edge current.
%
%%The differences in the transport tensors ($S_{ij}$, $K_{ij}$) between the present and the previous works come from selection of the edge current pair in
%%Fig. \ref{fig.idea}.
%%They used  the right and left edge currents. 
%%However, 

%It was reported\cite{84Jonson} that experiments\cite{84Obloch,86Obloch,86Davidson,86Fletcher} on the thermopower agreed well with the theory when impurity scattering was considered.
As yet another point, the experiments in the 1980's were 
%carried out on samples of size $\sim7$mm at $T>1.5$K, 
out of the ballistic regime.
%, where the ballistic transport cannot be achieved.
%It is hence quite uncertain whether the experimental \lq\lq confirmation"\cite{84Obloch,86Obloch} in fact observed what the they predicted by the theory using the left and right edge currents;
%some inconsistency was attributed to impurity scattering, but might have been due to the interactions with the heat baths.
%
%An essential difference of the present theory is to measure the well-defined voltage on the upper and lower edges and observe the Nernst coefficient, an \textit{off-diagonal} response.
The experimental technique has been developed remarkably since the 1980's, when it was almost impossible to achieve the ballistic transport.
In fact, Hasegawa and Machida\cite{nernst.experiment} are planning an experiment in the situation of the present theory.
%It would have been in vain to consider the ballistic currents along the upper and lower edges in those days.
The present predictions have become worthwhile in the light of new technology.
%%%%%%%%%%%%%%


\section{Summary}
We predicted a prominent quantum effect of the two-dimensional electron gas, which is closely analogous to the quantum Hall effect.
As long as the chemical potential stays in between the bottoms of the neighboring Landau levels, the quantized nature of the edge currents suppresses the Nernst coefficient and fixes the thermal conductance.
We also noted that the system is a physical realization of the non-equilibrium steady state.

The precise forms of the peaks in Fig.~\ref{Fig.coeff} and the risers of the steps in Fig.~\ref{q.fig} may be different from the reality.
This is because our argument using the edge currents is not applicable when the chemical potential coincides with the bottom of a Landau level, namely when $\mu=(n+\frac{1}{2})\hbar\omega_\mathrm{c}$, or $1/B=(n+\frac{1}{2})\hbar|\mathrm{e}|/m\mu$.
There the heat current is carried by bulk states as well as the edge states.
We then have to take account of impurities and possibly electron interactions\cite{future}.

Comments on other approaches to the quantum Nernst effect are in order.
Kontani derived\cite{02Kontani,03Kontani} by the Fermi liquid theory general expressions of the Nernst coefficient and the thermal conductivity of strongly correlated electron systems such as high-$T_\mathrm{c}$ materials.
%Akera\cite{Akera} analyzed the Ettingshausen effect, the reciprocal of the Nernst effect, by thermohydrodynamics. 
%The quantum behavior predicted in the present paper, however, was not reported in either studies.

\section*{Acknowledgments}
The authors express sincere gratitude to Dr.~Y.~Hasegawa and Dr.~T.~Machida for useful comments on experiments of the Nernst effect and the quantum Hall effect.
This research was partially supported by the Ministry of Education, Culture, Sports, 
Science and Technology, 
Grant-in-Aid for Exploratory Research, 2005, No.17654073.
N.H.~gratefully acknowledges the financial support by Casio Science Promotion Foundation and the Sumitomo Foundation.

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\end{document} 
cond-mat0506044.extracted_bib
\begin{thebibliography}{99} %% The number "99" means that this list has more than nine items.
\bibitem{67Harman}
T. C. Harman and J. M. Honig:
\textit{Thermoelectric and Thermomagnetic Effects and Applications}, McGraw-Hill, New York, 1967, Chap. 7, p. 311.
%\bibitem{94Yam}
%S. Yamaguchi, A. Iiyoshi, O. Motojima, M. Okamoto, S. Sudo, M. Ohnishi, M. Onozuka and C. Uesono:
%in Proc. 7th Int. Conf. Emerging Nuclear Energy Systems, Chiba, 1994
%(World Scientific, Singapore, 1994) p. 495.
\bibitem{94Yam2}
S. Yamaguchi, A. Iiyoshi, O. Motojima, M. Okamoto, S. Sudo, M. Ohnishi, M. Onozuka, and C. Uesono: 
\textit{Proc. 7th Int. Conf. Emerging Nuclear Energy Systems, Chiba, 1994}
(World Scientific, Singapore, 1994) p. 502.
\bibitem{99Nakamura}
H. Nakamura, K. Ikeda, and S. Yamaguchi:
Jpn. J. Appl. Phys. \textbf{38} (1999) 5745. 
\bibitem{04Hasegawa}
Y. Hasegawa, T. Komine, Y. Ishikawa, A. Suzuki, and H. Shirai:
Jpn. J. Appl. Phys. \textbf{43} (2004) 35.
\bibitem{82Halperin}
B. I. Halperin:
Phys. Rev. B \textbf{25} (1982) 2185.
\bibitem{Ruelle}
D. Ruelle:
J. Stat. Phys. \textbf{98} (2000) 57.
\bibitem{Ogata}
Y. Ogata: 
Phys. Rev. E \textbf{66} (2004) 016135.
\bibitem{84MacDonald}
A. H. MacDonald and P. St\v{r}eda:
Phys. Rev. B \textbf{29} (1984) 1616.
\bibitem{92Komiyama}
S. Komiyama, H. Hirai, M. Ohsawa, Y. Matsuda, S. Sasa, and T. Fujii:
Phys. Rev. B \textbf{45} (1992) 11085. 
%%%%%%%%%%%%%%%%%%% This part was deleted by Hatano on 01/13/2005.
%\bibitem{90Martin}
%T. Martin and S. Feng:
%Phys. Rev. Lett. \textbf{64} (1990) 1971.
%\bibitem{86Smrcka}
%L. Smrcka, H. Havlova, and A. Isihara:
%J. Phys. C \textbf{19} (1986) L475.
%\bibitem{86Berggren}
%K.-F. Berggren, T. J. Thornton, D. J. Newson, and M. Pepper:
%Phys. Rev. Lett. \textbf{57} (1986) 1769.
%%%%%%%%%%%%%%%%%
%\bibitem{Machida}
%T. Machida: private communications.
\bibitem{Tarucha}
S. Tarucha, T. Saku, Y. Hirayama, and Y. Horikoshi:
Phys. Rev. B \textbf{45} (1992) 13465.
\bibitem{82Zelenin}
S. P. Zelenin, A. S. Kondrat'ev, and A. E. Kuchma:
Sov. Phys. Semicond. \textbf{16} (1982) 355.
\bibitem{82Girvin} 
S. M. Girvin and M. Jonson:
J. Phys. C \textbf{15} (1982) L1147.
\bibitem{83Streda}
P. St\v{r}eda:
J. Phys. C \textbf{16} (1983) L369.
\bibitem{84Jonson}
M. Jonson and S. M. Girvin:
Phys. Rev. B \textbf{29} (1984) 1939.
\bibitem{84Obloch}
H. Obloh, K. von Klitzing and K. Ploog:
Surf. Sci. \textbf{142} (1984) 236.
\bibitem{86Obloch}
H. Obloh, K. von Klitzing, K. Ploog and G. Weimann:
Surf. Sci. \textbf{170} (1986) 292.
\bibitem{86Davidson}
J.S. Davidson, E.D. Dahlberg, A.J. Valois and G.Y. Robinson:
Phys. Rev. B \textbf{33} (1986) 2941.
\bibitem{86Fletcher}
R. Fletcher, J.C. Maan, K. Ploog and G. Weimann:
Phys. Rev. B \textbf{33} (1986) 7122.
\bibitem{nernst.experiment}
Y. Hasegasa and T. Machida:
private communication.
\bibitem{future}
H. Nakamura, N. Hatano, and R. Shirasaki:
in preparation.
\bibitem{02Kontani}
H. Kontani:
Phys. Rev. Lett. \textbf{89} (2002) 237003.
\bibitem{03Kontani}
H. Kontani:
Phys. Rev. B \textbf{67} (2003) 014408.
%\bibitem{Akera}
%H. Akera and H. Suzuura:
%cond-mat/0409498.
%\bibitem{89Ono}
%Y. Ono, T. Ohtsuki and B. Kramer:
%J. Phys. Soc. Jpn. \textbf{58} (1989) 1705. 
\end{thebibliography}