\documentstyle[12pt]{article}
\input bbb.ins
\newcommand{\tr}{\mathop{\rm tr}\nolimits}
\newcommand{\th}{\mathop{\rm th}\nolimits}
\newcommand{\ch}{\mathop{\rm ch}\nolimits}
\newcommand{\sh}{\mathop{\rm sh}\nolimits}
\begin{document}
\title{SURFACE ORBITAL MAGNETISM}
\author{Herv\'e KUNZ \thanks{ Work supported by the Fonds National
Suisse de la Recherche Scientifique No.~${20-33980.92}$}\\
Institut de Physique Th\'eorique\\
Ecole Polytechnique F\'ed\'erale de Lausanne\\
PHB-Ecublens -- CH-l015 Lausanne, Switzerland\\
Email address: KUNZ@eldp.epfl.ch}
\maketitle
\begin{abstract}
We compute the surface correction to the density of states of a particle in a convex
box, submitted to a magnetic field. Applying these results to orbital magnetism,
we find that at high temperatures or weak magnetic fields the surface
magnetisation is always paramagnetic, but oscillations appear at low
temperatures. In two dimensions they can give very large paramagnetic
contributions near integer values of the filling factor. Explicit formulas are given
for the zero field susceptibility and for samples with a cylindrical shape in
arbitrary magnetic field.
\bigbreak
\noindent{\bf Keywords:} Density of states, surface effects, diamagnetism, susceptibility
\end{abstract}
\newpage
\section{INTRODUCTION}
There is presently a renewed interest in size effects on the physical properties
of small metallic or semiconductor systems. In the ballistic regime, it is a
reasonable approximation to ignore the interaction of particles between
themselves or with impurities and to consider collisions with the walls as the
dominant effect. If the particles are submitted to a magnetic field, their motion in
the box leads to the formation of a magnetic moment, resulting in the so called
orbital magnetism. In the bulk limit, the systems shows a diamagnetic behaviour
at low magnetic fields, as was first discovered by Landau in his classic work. An
interesting question is to investigate the possible change in this behaviour when
the size of the system is reduced. The first correction expected is a surface one,
which one can reasonably expect to be able to compute analytically. At still smaller
sizes, in the mesoscopic range, we enter into a regime where the system is a
quantum billiard and the system could only be analysed numerically, at least
presently and properties characteristic of quantum "chaotic" systems should
appear.
In this article, we will be concerned with the problem of computing the
surface correction to the bulk behaviour. Since the 1930 paper of Landau, up to the
present day, many authors have made an attempt at this tricky problem. A brief
summary of the history of the problem, and many references can be found in a
1975 paper of N. Angelescu, G. Nenciu and R. Bundaru [1] and in a very recent
review by J. V. Ruitenbeeck and D. A. van Leeuwen [2]. The Roumanian authors
were the first to compute the surface susceptibility in zero field, of a parallelepiped
in a magnetic field perpendicular to some faces. The other problem on which
exact results are available is that of a thin plate parallel to the field [2]. More
recently, various approximate treatments have indicated a paramagnetic
contribution of the surface term in more general cases [3, 4].
In this work, we first derive a general expression for the surface density of
states of a convex body. Following Kac's strategy in his famous article [5] on the
same problem without a magnetic field, we analyse the partition function : This
allows us to compute the surface magnetisation. In the two-dimensional case, it
can be expressed in terms of zeroes of the Weber cylinder function. In this case
also, we can find a relationship between both the bulk and surface magnetisation
and the surface current of a semi-infinite system.Such relations should hold even
in the presence of interactions. At a very low temperature, when the filling factor
is near an integer, the surface magnetisation is paramagnetic and grows with the
square root of the logarithm of the temperature.
In the usual three dimensional case we give an explicit expression for the surface
magnetisation, for samples with a cylindric shape, in a magnetic field directed
along the axis of the cylinder or perpendicular. The zero field surface susceptibility
is computed for arbitrary convex samples. It is paramagnetic. We show that at
high temperatures (Maxwell-Boltzmann statistics) the surface magnetisation is
paramagnetic, for any value of the magnetic field, and any convex shape.
However at low temperatures, at least for cylindrical shapes, the surface
magnetisation shows de Haas - van Alphen type oscillations.
\section{DENSITY OF STATES}
We want to consider the motion of quantum particle of charge $e$, mass $m$, in
a box $\Lambda_{L}$, submitted to a constant magnetic field $B$, in the $z$ axis. We
choose as unit of energy $\in=\frac{\hbar e B}{m c}$ and of length
$l=\sqrt{\frac{\hbar c}{e B}}$. In these
units the hamiltonian is given by
$$
H_{\Lambda_{L}} = -\frac{1}{2} \partial^{2}_{x} + \frac{1}{2} (\frac{1}{i}\partial_{y}
-x)^{2}-\frac{1}{2}\partial^{2}_{z}
\eqno(2.1)
$$
with Dirichlet boundary conditions on the surface $\partial\Lambda_{L}$. The boxes
$\Lambda_{L}$ are all obtained from the box $\Lambda$, by a dilating factor $L$, i.e.
$$
\Lambda_{L}=\Big\{x\Big| \frac{x}{L}\in\Lambda\Big\}
\eqno(2.2)
$$
When we will discuss the two-dimensional problem, the term
$-\frac{1}{2}\partial^{2}_{z}$ will be absent from the hamiltonian and the motion will
be restricted to the $x$-$y$ plane. We are interested by the integrated density of
states $N_{\Lambda_{L}}(\lambda)$ of this system
$$
N_{\Lambda_{L}}(\lambda)=\sum_{n\colon\varepsilon_{n}\le\lambda}
\eqno(2.3)
$$
where $\varepsilon_{n}$ are the eigenvalues of the hamiltonian. Our purpose is
to compute the asymptotic behaviour of this quantity for large boxes $\Lambda_{L}$,
i.e. when $L\to\infty$. On physical grounds one expects a contribution proportional
to the volume $|\Lambda_{L}|=L^{d}|\Lambda|$ of the box and one of the order of the
surface $|\partial\Lambda_{L}|$ of this box. We will show that, when $d = 2, 3$
$$
N_{\Lambda_{L}}(\lambda)\sim L^{d}|\Lambda| n(\lambda) +
L^{d-1} s_{\partial_{\Lambda}}(\lambda)
\eqno(2.4)
$$
It turns out however that whereas in two dimensions the surface contribution is
indeed proportional to the perimeter of the box, i.e. $ s_{\partial_{\Lambda}}(\lambda)
=|\partial\Lambda|s(\lambda)$, in three dimensions
$ s_{\partial_{\Lambda}}(\lambda)$ depends on the specific shape and of the
orientation of the box with respect to the magnetic field.
Instead of considering directly $N_{\Lambda_{L}}$, we will analyse the behaviour of its
Laplace transform or in more physical terms of the partition function
$$
Z_{\Lambda_{L}} = \tr \exp - t H_{\Lambda_{L}}=
\int^{\infty}_{0} e^{-t\lambda}dN_{\Lambda_{L}}(\lambda)
\eqno(2.5)
$$
We will then essentially follow Kac's strategy, when he analysed the same
problem in the absence of a magnetic field.The partition function can be expressed
by means of the fundamental solution $P_{\Lambda_{L}}(x|y;t)$ of the heat equation
$$
\frac{\partial}{\partial t} P_{\Lambda_{L}}= - H_{\Lambda_{L}} P_{\Lambda_{L}}
\eqno(2.6)
$$
The partition function is given by
$$
Z_{\Lambda_{L}} = \int_{\Lambda _{L}}dx P_{\Lambda_{L}}(x|x;t)
\eqno(2.7)
$$
>From now on, we will assume that $\Lambda$ is compact and {\em convex}. If
$x\in\Lambda$, let $q(x)$ be the point of the boundary $\partial\Lambda$ closest to
$x$ (for Lebesgue almost all $x$ it is unique). If we denote by $\Lambda (x)$ the
half-space bounded by the plane $l(x)$ tangent at $\partial\Lambda$ in $q(x)$, then
$\Lambda\subset\Lambda(x)$.
We can then introduce the fundamental solution of the heat equation
$$
\frac{\partial}{\partial t} P_{\Lambda_{L}(x)}= - H_{\Lambda_{L}(x)}
P_{\Lambda_{L}(x)}
\eqno(2.8)
$$
where $ H_{\Lambda_{L}(x)}$ is the same hamiltonian as the one defined in equation
$(2.1)$, except that now it is defined on the half-space $\Lambda_{L}(x)$ and Dirichlet
boundary conditions are imposed on the plane $l(x)$.
Consider now the fundamental solutions of the usual heat equations
$$
\partial_{t}Q_{\Lambda_{L}}= \frac{1}{2}\Delta_{\Lambda_{L}}Q_{\Lambda_{L}}
\eqno(2.9)
$$
$$
\partial_{t}Q_{\Lambda_{L}(x)}= \frac{1}{2}\Delta_{\Lambda_{L}(x)}Q_{\Lambda_{L}(x)}
\eqno(2.10)
$$
where $\Delta_{\Lambda}$ is the Laplacian with Dirichlet boundary conditions on
$\partial\Lambda$. We have the basic inequality, if $x\in\Lambda$.
$$
|P_{\Lambda_{L}}(x|x;t) - P_{\Lambda_{L}(x)} (x|x;t)|\leq Q_{\Lambda_{L}(x)} (x|x;t)
- Q_{\Lambda_{L} (x|x;t)}
\eqno(2.11)
$$
This inequality follows easily from the following functional integral
representation of $P_{\Lambda}$ and $Q_{\Lambda}$ $[6]$.
$$
P_{\Lambda} (x|x;t) = \int d\mu^{\Lambda}_{0x;tx}(\omega) \exp - i \int^{t}_{0}
\omega_{1}(s) d\omega_{2} (s)
\eqno(2.12)
$$
$$
Q_{\Lambda} (x|x;t) = \int d\mu^{\Lambda}_{0x;tx}(\omega)
\eqno(2.13)
$$
$d\mu^{\Lambda}_{0x;tx}$ being the conditional Wiener measure for paths starting from
$x$ at time $0$ and ending at $x$ at time $t$, but remaining in $\Lambda$.
The inequality follows from the fact that the magnetic field contribution in
$P_{\Lambda}$ is of modulus one, and $\Lambda \subset \Lambda (x)$ as a consequence of
the convexity of~$\Lambda$.
>From this inequality we can see that
$$
Z_{\Lambda_{L}}(t) = \int_{\Lambda_{L}}dx P_{\Lambda_{L}(x)} (x|x;t) +
r_{\Lambda_{L}}(t)
\eqno(2.14)
$$
and
$$
0 \leq r_{\Lambda_{L}}(t) \leq \int_{\Lambda} dx [Q_{\Lambda_{L}(x)} (x|x;t)
- Q_{\Lambda_{L}} (x|x;t)]
\eqno(2.15)
$$
But we have the scaling relationship
$$
Q_{\Lambda_{L}} (x|x;t) = L^{-d} Q_{\Lambda}
(\frac{x}{L}|\frac{x}{L};\frac{t}{L^{2}})
\eqno(2.16)
$$
and we can see that it follows from Kac's result that
$$
r_{\Lambda_{L}}(t) = O (L^{d-2})
\eqno(2.17)
$$
In this way we have reduced the computation of the partition function (with an
$L^{d-2}$ accuracy) to the problem of computing $ P_{\Lambda_{L}(x)} (x|x;t)$.
This amounts essentially to replace the boundary locally by its tangent plane.We will
show that
$$
P_{\Lambda_{L}(x)} (x|x;t) = \rho_{t} (u; n^{3} (x))
\eqno(2.18)
$$
Where $u$ denotes the distance of $x$ to the boundary $\partial\Lambda_{L}$
and $n^{3}(x)$ denotes the $z$-component (i.e. along the direction of the magnetic
field) of the inward normal $\overrightarrow{n}(x)$ at the point $x$ of the boundary
closest to $x$. $\rho _{t}(u; n^{3}(x))$ can be interpreted as the density of
particles, constrained to move in the half-space $\Lambda_{L}(x)$, at distance $u$
from the boundary. One expects on physical grounds that
$\lim_{u \to \infty} \rho_{t}(u; n^{3}(x)) = \rho_{t}(\infty)$ where $\rho_{t}(\infty)$
is the density in the infinite system. The approach
of $\rho _{t}(u; n^{3}(x))$ to its limit $\rho_{t}(\infty)$ is rapid enough so that
$$
\int^{\infty}_{0}du u |\frac{\partial}{\partial u}\rho _{t}(u; n^{3}(x))| < \infty
\eqno(2.19)
$$
We can therefore write
$$
Z_{\Lambda_{L}}(t) = L^{d}|\Lambda|\rho_{t}(\infty) -L^{d-1} \int^{\infty}_{0}dz
\int_{ {\Lambda\hfill\atop {\rm dist}(r,\partial\Lambda)\leq\frac{z}{L} }}dr
\frac{\partial}{\partial z}\rho_{t}(z; n^{3}(r)) + r_{\Lambda_{L}}(t)
\eqno(2.20)
$$
so that we get
$$
Z_{\Lambda_{L}}(t) = L ^{d}|\Lambda| \rho_{t} (\infty) - L^{d-1}\int^{\infty}_{0} dz
\int_{\partial \Lambda} d\sigma z \frac{\partial}{\partial z}\rho_{t}
(z; n^{3}(\sigma)) + O (L^{d-2})
\eqno(2.21)
$$
We have thus shown that if we can find the representations
$$
\rho^{t}(\infty) = t \int^{\infty}_{0} e^{-t\lambda} n(\lambda) d\lambda
\eqno(2.22)
$$
and
$$
\int^{\infty}_{0} du[\rho_{t} (u; n^{3} (\sigma)) - \rho_{t} (\infty)] = t
\int^{\infty}_{0} e^{-t\lambda} s(\lambda; n^{3} (\sigma)) d\lambda
\eqno(2.23)
$$
then
$$
\int^{\infty}_{0} e^{-t\lambda} N_{\Lambda_{L}}(\lambda) d\lambda = L^{d}|\Lambda|
\int^{\infty}_{0} e^{-t\lambda} n(\lambda) + L^{d-1}\int^{\infty}_{0} e^{-t\lambda}
s_{\partial\Lambda}(\lambda) d\lambda + O (L^{d-2)}
\eqno(2.24)
$$
with
$$
s_{\partial\Lambda}(\lambda) = \int_{\partial\Lambda} d\sigma s(\lambda;
n^{3}(\sigma))
\eqno(2.25)
$$
>From equation $(2.24)$ should follow that
$$
N_{\Lambda_{L}}(\lambda) \sim L^{d}|\Lambda| n (\lambda) + L^{d-1}s_{\partial\Lambda}
(\lambda)
\eqno(2.26)
$$
This result however is stronger than $(2.24)$, and more information is needed to
derive it. We discuss this point in the appendix. It can be proved that quite
generally
$$
\lim_{L \to \infty} \frac{N_{\Lambda_{L}}}{L^{d}}(\lambda) = n (\lambda)
$$
For the surface term, one can only prove that
$$
\lim_{L \to \infty} \frac{1}{L^{d-1}}[N_{\Lambda_{L}} (\lambda) - |\Lambda| L^{d}
n (\lambda)]= s_{\partial\Lambda}(\lambda)
$$
when $\Lambda$ is a parallelepiped. The convergence is the so called weak convergence
of measures, appropriate for the thermodynamics.
To proceed further, we need to analyse the density $\rho_{t}(u)$ of a half-infinite
system.
In the two-dimensional case, if we choose the origin on the line delimiting the
allowed domain, the hamiltonian becomes
$$
h_{+} = - \frac{1}{2} \partial^{2}_{u} + \frac{1}{2} (\frac{1}{i} \partial_{v}
- u)^{2}
\eqno(2.27)
$$
defined on the half-space $u\geq0$, and with Dirichlet boundary conditions at$u =
0$.
We therefore have
$$
\rho_{t}(u) = \int^{+\infty}_{- \infty}\frac{dk}{2\pi}(u|\exp - \th_{+}
(k)|u)
\eqno(2.28)
$$
where
$$
h_{+}(k) = - \frac{1}{2} \partial^{2}_{u} + \frac{1}{2} (k + u)^{2} \eqno(2.29)
$$
is now defined on the half-line $u\geq0$. .
If we call $h(k)$ the same operator as the one in $(2.29)$, but defined on the full
line, then we claim that
$$
\rho_{t}(\infty) = \int^{+\infty}_{-J\infty}\frac{dk}{2\pi} (u|\exp -
\th(k)|u)
\eqno(2.30)
$$
$\rho_{t}(\infty)$ is easily computed
$$
\rho_{t}(\infty) = \frac{1}{2\pi}\sum^{\infty}_{n = 0}\exp - t (n + \frac{1}{2})
\eqno(2.31)
$$
we have $\rho_{t}(u) \leq \rho_{t}(\infty)$
and if we write
$$
\rho_{t} (u) - \rho_{t}(\infty) = \int^{+\infty}_{-\infty}\frac{dk}{2\pi}
(u|\exp - \th_{+}(k) - \chi \exp - \th(k)\chi|u)
\eqno(2.32)
$$
with
$$
\chi(u) = \cases{1 & if $u\geq0$\cr
0 & if $u<0$\cr}
\eqno(2.33)
$$
we should have
$$
\int^{\infty}_{0} du [\rho_{t}(u) - \rho_{t}(\infty)] = \frac{1}{2\pi}
\int^{+\infty}_{-\infty}dk\Bigl( \tr \exp - \th_{+}(k) - \tr\chi\exp-\th(k)\chi\Bigr)
\eqno(2.34)
$$
if the trace of the operator on the right hand side is finite and integrable, as can be
checked, indeed.
If we call $E^{+}_{\lambda}(k)$ and $E_{\lambda}(k)$ the spectral projectors
of $h_{+}(k)$ and $h(k)$, in the energy range $(- \infty, \lambda)$, then we can
write
$$
\int^{\infty}_{0} du [\rho_{t}(u) - \rho_{t}(\infty)] = \frac{t}{2\pi}
\int^{\infty}_{0} d\lambda e^{-t\lambda} \int^{+\infty}_{-\infty} dk \tr\Bigl[
E^{+}_{\lambda}(k) -\chi E(k)\chi\Bigr]
\eqno(2.35)
$$
We can therefore summarise the results in two dimensions by the following
equations. The bulk density of states $n(\lambda)$ is given by
$$
n(\lambda) = \frac{1}{2\pi} \sum^{\infty}_{n=0}\theta(\lambda-n-\frac{1}{2})
\eqno(2.36)
$$
and the surface density of states is given by
$$
s(\lambda) = \frac{1}{2\pi}\int^{+\infty}_{-\infty}dk \tr \Bigl[E^{+}_{\lambda}(k)
-\chi E(k)\chi \Bigr]
\eqno(2.37)
$$
We will give later on, a more explicit expression for this quantity, in terms of
zeroes of cylinder functions. In three dimensions, we proceed in the same way.
Choosing the origin on the plane, with normal $\overrightarrow{n}$,and
calling \overrightarrow{b} the unit vector in the direction of the magnetic field and
$\theta$ the angle between the magnetic field and the normal$\overrightarrow{n}$, we choose
the axis as follows (if $\sin\theta\not=0)$
$$
\overrightarrow{e}_{1}=
\frac{\overrightarrow{b}-\cos\theta\overrightarrow{n}}{\sin\theta}\quad
\overrightarrow{e}_{2}= \frac{\overrightarrow{n}\wedge\overrightarrow{b}}{\sin\theta} \quad
\overrightarrow{e}_{3} =\overrightarrow{n}
\eqno(2.38)
$$
Since the density is gauge independent, we choose the useful gauge
$$
A_{1} =A_{3} = 0 \quad A_{2} = - \sin\theta x_{3} + \cos\theta x_{1}
$$
The appropriate hamiltonian is therefore
$$
H_{+} = - \frac{1}{2}\partial^{2}_{u} + \frac{1}{2}(\frac{1}{i}\partial_{w} + \sin\theta u
- \cos\theta v)^{2} - \frac{1}{2}\partial^{2}_{v}
\eqno(2.39)
$$
in the half-space $u \geq 0$, with Dirichlet boundary conditions on the
plane $u = 0$.
If we call $ x_{1} = u , x_{2} = w , x_{3} = v$
The density $\rho_{t}(u)$ is therefore given by
$$
\rho_{t}(u) = \int^{+\infty}_{-\infty} \frac{dk}{2\pi} (uv|\exp - tH_{+}(k)|uv)
\eqno(2.40)
$$
$H_{+}(k)$ being the same operator as the one in 2.39, but where $\frac{1}{i}\partial_{w}$
is replaced by $k$.
$\rho_{t}(u)$ does not depend on $v$ because if $V^{k}_{+}\,(u v|u' v')$ designates
the kernel of $\exp - tH_{+} (k)$ we have, for any b :
$$
V^{k+b\cos\theta}_{+}\,(u v + b|u' v' + b) = V^{k}_{+}\,(u v|u' v')
\eqno(2.41)
$$
Hence when $\cos \theta\not= 0$ we can rewrite $(2.40)$ as
$$
\rho_{t}(u) = \frac{|\cos\theta|}{2\pi} \int^{+\infty}_{-\infty}dv\,(u v|\exp - tH_{+}(o)|u
v)
\eqno(2.42)
$$
Introducing the operator $H(o)$, which is the same as $H_{+}(o)$ but defined on
the whole space, we will have
$$
\rho_{\infty}(u) = \frac{|\cos\theta|}{2\pi} \int^{+\infty}_{-\infty}dv\,(u v|\exp - tH(o)|u
v) \eqno(2.43)
$$
In fact,
$$
\rho_t(\infty) = \frac{1}{\sqrt{2\pi t}}\frac{1}{2\pi}\sum_{n=0}^{\infty}\exp
-t\left(n+\frac{1}{2}\right)
\eqno(2.44)
$$
>From $2.42$ and $2.43$, we see that we can write
$$
\int_{0}^{\infty} du [\rho_{t} (u) -\rho_{t}(\infty)] =\frac{|\cos\theta|}{2\pi} \tr
\Bigl[\exp - tH_{+}(o) - \chi\exp - t H(o) \chi\Bigr]
\eqno(2.45)
$$
There are two special cases where this expression simplifies. If $\cos \theta = 0$, we are
essentially in the two-dimensional case
then
$$
\int_{0}^{\infty} du [\rho_{t} (u) -\rho_{t}(\infty)] = \frac{1}{\sqrt{2\pi t}}\frac{1}{2\pi
} \int_{-\infty}^{+\infty} dk \tr\Bigl[ \exp - \th_{+}(k) - \chi \exp - \th(k) \chi \Bigr]
\eqno(2.46)
$$
as can be seen by using $2.40$ and $2.34$.
If $\cos \theta = \pm 1$, the problem is that of a free particle in half-space and an
harmonic oscillator in the whole space. Therefore
$$
\int_{0}^{\infty} du [\rho_{t} (u) -\rho_{t}(\infty)] = \frac{-1}{\sqrt{2\pi t}} \frac{
1}{2} \sqrt{\frac{\pi t}{2}} \frac{1}{2\pi} \sum^{\infty}_{n=0} \exp
-t\left(n+\frac{1}{2}\right) \eqno(2.47)
$$
To summarise, in the general case we have for the bulk density of states, the
familiar Landau formula
$$
n(\lambda) = \frac{1}{\sqrt{2}}\frac{1}{\pi^{2}}\sum^{\left[\lambda-\frac{1}{2}\right]}_{n=0}
\left(\lambda - n - \frac{1}{2}\right)^{\frac{1}{2}}
\eqno(2.48)
$$
and for the surface density of states
$$
s_{\partial \Lambda} (\lambda) = \int_{\partial \Lambda} d\sigma s(\lambda ,\theta (\sigma))
\eqno(2.49)
$$
where
$$
s(\lambda,\theta) = \frac{|\cos \theta|}{2\pi} \tr\Bigl[ E^{+}_{\lambda} (\theta) -
\chi\Bigr] E_{\lambda}(\theta) \chi
\eqno(2.50)
$$
$E^{+}_{\lambda}(\theta)$ and $E_{\lambda}(\theta)$ being the spectral projectors in the energy
range $(- \infty, \lambda)$ of the hamiltonian
$$
H_{+} = - \frac{1}{2} \Delta_{+} + \frac{1}{2} (\sin \theta u - \cos \theta v)^{2}
\eqno(2.51)
$$
defined on $u \geq 0$, with Dirichlet boundary conditions on $u = 0$
and
$$
H = - \frac{1}{2}\Delta + \frac{1}{2} (\sin \theta u - \cos \theta v)^{2}
\eqno(2.52)
$$
defined on the whole space ${\Bbb R}^{2}$.
Apart from the special cases $\theta = 0, \frac{\pi}{2}, \pi$, we have not succeeded to find a
more explicit expression for $s(\lambda, \theta)$. There may be some hope to solve this
problem however, because at the classical level the hamiltonian $H_{+}$, describing
an harmonic oscillator with a wall is integrable, although not separable.
The special cases $\theta = 0, \frac{\pi}{2}, \pi$ will allow us however to find expressions
for the density of states when the volume has a cylindrical shape and the magnetic field
is directed along the axis of the cylinder or perpendicular to it. The general
formula $2.50$ can be used to compute the density of states in small magnetic fields.
\section{THE TWO-DIMENSIONAL CASE. MAGNETISATION AND SURFACE
CURRENT}
In order to analyse more thoroughly the two-dimensional case, we need to
look at the spectral properties of the hamiltonian
$$
h_{+}(k) = - \frac{1}{2} \partial^{2}_{x} + \frac{1}{2} (k + x)^{2}
\eqno(3.1)
$$
defined on the half-line $x\geq 0$, with Dirichlet boundary condition at x = 0.\newline
We denote the ordered eigenvalues by $\varepsilon_{n} (k)$, $n=0,1,2,... $ and
${\varepsilon_{n} (k) < \varepsilon_{n+1}(k)}$.To the eigenvalue $\varepsilon_{n}(k)$
corresponds the unnormalised eigenfunction:
$$
\psi_{n,k}(x) = D_{\varepsilon_{n} -\frac{1}{2}}(\sqrt{2} (x + k))
\eqno(3.2)
$$
where $D_{v}(x)$is the usual Weber cylinder function. The equation for the
eigenvalue is
$$
D_{\varepsilon_{n} -\frac{1}{2}}(\sqrt{2}k) = 0
\eqno(3.3)
$$
The following properties of the eigenvalues can be established: $\varepsilon_{n}(k)$ is
strictly increasing in $k$.
We have $\varepsilon_{n}(-\infty) = n + \frac{1}{2}$ and more precisely
$$
\varepsilon_{n}(k)-\varepsilon_{n}(-\infty)\sim \frac{(\sqrt{2}k)^{2n+1}}{n!\sqrt{2\pi}}
\exp-k^{2} \eqno(3.4)
$$
when $k \rightarrow - \infty$.
When $k \rightarrow + \infty, \varepsilon_{n}(k)$,grows quadratically in $k$ .
Finally
$$
\varepsilon_{n} (0) = 2n + \frac{3}{2}
\eqno(3.5)
$$
We can uniquely define the function $k_{n}(\lambda)$ by the relation $\varepsilon_{n}(k) =
\lambda$.\newline $\sqrt{2} k_{n} (\lambda) =x_{n} (\lambda)$ is a zero of
$D_{\lambda-\frac{1}{2}}(x)$. If $\lambda -\frac{1}{2}= N + \theta$ where $\theta \in
(0,1)$, and $N$ a positive integer, then $D_{\lambda-\frac{1}{2}}$ has $N+1$ finite zeroes,
ordered as: $x_{o} >x_{1} > ... > x_{n}$. When $\lambda - \frac{1}{2}\to N^{+}$,
$x_{N}(\lambda) \rightarrow - \infty$ and when $\lambda - \frac{1}{2}\rightarrow (N +
1)^{-}$, all the zeroes tend to the $N + 1$ finite zeroes of $D_{N+1}$, which is proportional
to a Hermite polynomial. With these preliminaries we can analyse the density $s(\lambda)$.
Equation $2.37$ tells us that is given by
$$
s(\lambda) = \frac{1}{2\pi}\int^{+\infty}_{-\infty} dk [\sum_{n}\theta
(\lambda-\varepsilon_{n}(k)) - \sum_{n} \theta (\lambda - \frac{1}{2}- n) c_{n}(k)]
\eqno(3.6)
$$
$\theta$ is the usual Heaviside function and
$$
c_{n}(k) = \int^{+\infty}_{k} dy \varphi^{2}_{n} (y)
\eqno(3.7)
$$
$\varphi_{n}(y)$ being the normalised eigenfunction of the usual harmonic oscillator
$$
h = - \frac{1}{2}\partial^{2}_{x} + \frac{1}{2} x^{2}
\eqno(3.8)
$$
Since $\varepsilon_{n}(k) \geq n + \frac{1}{2}$, we can rewrite $3.6$ as
$$
s(\lambda) = \frac{1}{2\pi} \sum^{\left[\lambda-\frac{1}{2}\right]}_{n=0}
\int^{+\infty}_{-\infty}dk[\theta (k_{n}(\lambda)-k) -c_{n}(k)]
\eqno(3.9)
$$
Using the fact that $\varphi^{2}_{n}(y)$ is even, we have :
$$
\int^{+\infty}_{-\infty}dk [\theta(k_{n}(\lambda)- k) -c_{n}(k)] = k_{n}(\lambda)
\eqno(3.10)
$$
Thus we get the desired expression
$$
s(\lambda) = \frac{1}{\sqrt{2}}\frac{1}{2\pi}\sum^{\left[\lambda-\frac{1}{2}\right]}_{n=0}
x_{n} (\lambda-\frac{1}{2})
\eqno(3.11)
$$
where $x_{n}(v)$ is the $n$'th zero of $D_{v}(x)$, the zeroes decreasing when the index $n$
increases.
It is clear that the index $n$ is nothing else than the Landau level index.
When $\lambda -\frac{1}{2}$ is not an integer, the zeroes $x_{n}(\lambda-\frac{1}{2})$ are
increasing in $\lambda$, therefore $s(\lambda)$ is increasing. Since if we start from the
value of $\lambda - \frac{1}{2} = N$ and increase it until it reaches the value $\lambda -
\frac{1}{2} = N+1$, the zeroes tend to those of $D_{N+1}(x)$, whose sum vanishes because of
the parity of $D_{N+1}(x)$, we conclude that $s(\lambda)\leq 0$. Thus we conclude
that when $\lambda - \frac{1}{2}$ is between the integers $N$ and $N+1, s(\lambda)$,is a
negative increasing function which diverges to $-\infty$ near $N$, as
$$
s(\lambda)\sim -\frac{1}{2\pi} [\sqrt{\delta}- \frac{N+\frac{1}{2}}{2}\frac{\ln
\delta}{\sqrt\delta}]
\eqno(3.12)
$$
where
$$
\delta = \ln \frac{1}{\lambda-N-\frac{1}{2}}
\eqno(3.13)
$$
This result follows from the asymptotic behaviour of $\varepsilon_{n}(k)$ as expressed
in $3.4$.
When $\lambda$ approaches $N + \frac{3}{2}$, $s(\lambda)$ vanishes.
The complicated asymptotic behaviour revealed by $3.12$ indicates that the energy
levels are packed in a very intricate way near the Landau levels in a large but
finite system.
Let us discuss now the thermodynamic properties. We consider an assembly of
fermions, of chemical potential $\mu$, in a convex box of volume $V$ and area $A$. The
pressure of the finite system is given by
$$
\beta pV = 2 \int^{\infty}_{0}\ln [1+ ze^{-tx}]dN_{\Lambda}(x) = 2t \int^{\infty}_{0}
\frac{ze^{-tx}}{1+ze^{-tx}} N_{\Lambda}(x)
\eqno(3.14)
$$
The factor $2$ comes from the spin degeneracy, since we neglect the Zeeman energy.
$t = \frac{\beta\hbar eB}{mc}$ is the inverse temperature in magnetic units,
and $z = e^{\beta\mu}$ is the fugacity. For a large sample, we have shown that
$$
\beta pV = \frac{2V}{\lambda^{2}} t \sum^{\infty}_{n=0}\ln
\Bigl[1+ze^{-t(n+\frac{1}{2})}\Bigr] + \frac{2A}{\lambda} \sqrt{2\pi t^{3}}\int^{\infty}_{0}
\frac{ze^{-tx}}{1+ze^{-tx}} s(x) dx
\eqno(3.15)
$$
where $\lambda = \sqrt\frac{2\pi \hbar^{2}\beta}{m}$ is the thermal wavelength.
The magnetisation measured in the Bohr unit $\mu_{B}= \frac{\hbar e}{2mc}$ will therefore
decomposes into a bulk contribution
$$
\frac{M_{b}}{\mu_{B}}= \frac{4V}{\lambda^{2}}\partial_{t} \sum^{\infty}_{n=0}\ln \Bigl[1 + z
e^{-t(n+\frac{1}{2})}\Bigr]
\eqno(3.16)
$$
and a surface contribution
$$
\frac{M_{s}}{\mu_{B}} = \frac{4A}{\lambda}\partial_{t} \sqrt{2\pi t^{3}} \int^{\infty}_{0}
\frac{ze^{-tx}}{1+z e^{-tx}} s(x) dx
\eqno(3.17)
$$
We will see that in the small magnetic field limit, $t << 1$, the surface
magnetisation is positive (paramagnetism) whereas the bulk magnetisation is of
course negative (diamagnetism) as is well known.
The same result holds quite generally, at high temperatures, where we can use
Boltzmann statistics, namely replace $\frac{ze^{-tx}}{1+ze^{-tx}}$ by
$ze^{-tx}$.
We will also prove an identity relating both the bulk magnetisation and the
surface one to the surface current of a semi-infinite system. This identity indicates
that the bulk and surface magnetisation tend to have opposite signs.
It is interesting however to analyse the expression for the surface magnetisation
in the zero temperature limit. It can be expressed as
$$
\frac{M_{s}}{\mu_{B}} = \frac{4A \sqrt{2\pi}}{l_{F}} v^{-\frac{1}{2}}\left[\frac{3}{2}
\int^{v}_{0}s(x)dx - vs(v)\right]
\eqno(3.18)
$$
where $v = \frac{\mu}{\in}$ is essentially the ratio of the Fermi energy to the magnetic
one,
and
$$
\l_{F}= \sqrt{\frac{2\pi \hbar^{2}}{\mu m}}
$$
is the Fermi wave length.
We can see that
$$
f(v) = \frac{3}{2}\int^{v}_{0} s(x) dx - vs(v)
$$
is an decreasing function of $v$ when $v$ is between $N+\frac{1}{2}$ and $N+\frac{3}{2}$,
since $s(v)$is negative and increasing in this range of $v$, for any integer $N$.
On the other hand, each time $v$ approaches $N + \frac{1}{2}$ from above $f(v)$ diverges
like $\frac{N+\frac{1}{2}}{2\pi}\sqrt{ \ln \frac{1}{v-N-\frac{1}{2}} }$ according to $3.12$,
but when $v$ approaches $N + \frac{3}{2}$ from below $f(v)$
tends to $\frac{2}{3}\int_{0}^{ N+\frac{3}{2} }s (x) dx$ which is negative. Thus we see that
the surface magnetisation will be positive (paramagnetic) when $v \sim N + \frac{1}{2}$ and
even divergent like
$$
\frac{M_{s}}{\mu_{B}} \sim \frac{4A}{l_{F}}
\sqrt{ \frac{ N+\frac{1}{2} }{2\pi} \ln\frac{1}{v-N-\frac{1}{2}} }
\eqno(3.19)
$$
and decrease to a negative value (diamagnetism) when $v$ approaches $N + \frac{3}{2}$ from
below.
At very low temperatures, the singularity will be washed out and $\frac{M_{s}}{\mu_{B}} \sim
\frac{4A}{l_{F}}\sqrt{ \frac{ N +\frac{1}{2} }{2\pi} \ln t }$ but one will see very strong
oscillations opposite in sign to those of the bulk magnetisation. These oscillations like the
de Haas-van Alphen one are the remnants of the Landau level structure.
Let us finally look at the relation between the magnetisation and surface currents.
Consider the semi-infinite system $x \geq 0$ and a particle submitted to a magnetic
field, whose dynamics is described by the hamiltonian
$$
h_{+} = -\frac{1}{2} \partial^{2}_{x}+ \frac{1}{2} (\frac{1}{i}\partial_{y}-x)^{2}
\eqno(3.20)
$$
There will be a current $j(x)$ flowing along the $y$ axis, induced by the magnetic
field. This current will be given by
$$
j(x) = - z \int^{+\infty}_{-\infty} dk (k+x) (x|\exp - \th_{+}(k)|x)
\eqno(3.21)
$$
where the hamiltonian $h_{+}(k)$ is given in equation $3.1$. We have used
Boltzmann distribution. In the more interesting case of Fermi statistics the
operator at the right hand side of 3.21 should be replaced by
$$
\frac{z \exp -\th_{+}(k)}{1+ z\exp -\th_{+}(k)}
$$
If we denote by
$$
L_{k} = \exp - \th_{+}(k) - \chi\exp -\th (k) \chi
\eqno(3.22)
$$
then we can see that
$$
j (x) = \frac{-z}{2\pi} \int^{+\infty}_{-\infty} dk (k+x) L_{k} (x|x)
\eqno(3.23)
$$
and we have
$$
\int^{\infty}_{0} dx j(x) = -\frac{z}{2\pi}\int^{+\infty}_{-\infty} dk \tr (x + k) L_{k}
\eqno(3.24)
$$
and
$$
\int^{\infty}_{0} dx xj(x) = -\frac{z}{2\pi} \int^{+\infty}_{-\infty} dk \tr x(x+k) L_{k}
\eqno(3.25)
$$
On the other hand
$$
\partial_{k} \tr L_{k}= - t \tr (k+x) L_{k}- t \tr (k+x)\chi V_{k} \chi -\partial_{k} \tr
\chi V_{k} \chi \eqno(3.26)
$$
where
$$
V_{k} = \exp - \th (k)
\eqno(3.27)
$$
but
$$
\int^{+\infty}_{-\infty} dk \partial_{k} \tr L_{k} = 0
\eqno(3.28)
$$
and
$$
\int^{+\infty}_{-\infty} dk \partial_{k} \tr \chi V_{k}\chi=\left.\sum^{\infty}_{n=0}
e^{-t(n+\frac{1}{2})} \int^{+\infty}_{k}
dy \varphi^{2}_{n}(y) \right|_{-\infty}^{+\infty} = -
\sum^{\infty}_{n=0}e^{-t(n+\frac{1}{2})} \eqno(3.29)
$$
$$
\int^{+\infty}_{-\infty} dk \tr(k+x) \chi V_{k}\chi = \sum^{\infty}_{n=0}
e^{-t(n+\frac{1}{2})}\int^{+\infty}_{-\infty} dk \int^{+\infty}_{k}\varphi^{2}_{n}(y)
= \sum^{\infty}_{n=0} (n+\frac{1}{2}) e^{-t(n+\frac{1}{2})}
\eqno(3.30)
$$
Thus we see that integrating equation $3.26$ and using equations $3.28$, $3.29$,
$3.30$ and the definition $3.23$, we get
$$
t \int^{\infty}_{0} j(x) dx = \frac{z}{2\pi}\sum^{\infty}_{n=0} [- 1+t(n+\frac{1}{2})]
\exp - t(n+\frac{1}{2})
\eqno(3.31)
$$
or
$$
t \int^{\infty}_{0} j(x) dx = - \frac{z}{2\pi}\partial_{t} t \sum^{\infty}_{n=0} \exp
- t(n+\frac{1}{2})
\eqno(3.32)
$$
It can be seen from equation $3.16$ that the right hand side of this equality is
$\frac{-\lambda^{2}}{4V} \frac{M_{b}}{\mu_{B}}$ where $M_{b}$ is the bulk magnetisation
with Boltzmann statistics (first order in $z$).
The case of Fermi statistics can be treated by expanding the Fermi operator in $z$,
when $|z| < 1$
$$
\frac{z \exp - \th}{1 + z \exp - \th} = - \sum^{\infty}_{j=1} (- z)^{j}\exp -
t jh
$$
using the identity $3.32$ for each term in the sum we get
$$
\frac{M_{b}}{\mu_{B}}=- \frac{4V}{\lambda^{2}} t \int^{\infty}_{0} j (x) dx
\eqno(3.33)
$$
The result remains valid when $z > 1$ by analytic continuation. Such a result was
first obtained by Macris, Martin, Pul\'e [7] by a different technique and generalised
to some interacting situations.
Let us call
$$
q(t) = t \int^{\infty}_{0} e^{-tx} s(x)
\eqno(3.34)
$$
we have
$$
q(t) = \int^{+\infty}_{-\infty} \frac{dk}{2\pi}\tr L_{k}
\eqno(3.35)
$$
We rewrite this as
$$
\sqrt{t} q(t) =\int^{+\infty}_{-\infty} \frac{dk}{2\pi}\tr \Bigl[U^{+}_{k} (1) - \chi
U_{k}(1)\chi \Bigr]
\eqno(3.36)
$$
where
$$
U^{+}_{k} (s) = \exp - \frac{s}{2}[- \partial^{2}_{x} + (k + xt)^{2}]
\eqno(3.37)
$$
the operator in the exponential being defined on the half-line $x\geq 0$. $U_{k} (s)$
being the same operator, but defined on the whole line.
>From this equation follows that
$$
\partial_{t}\sqrt{t} q(t) =\int^{+\infty}_{-\infty} \frac{dk}{2\pi} \{-\tr
x(x+k)[U^{+}_{k}(1) - \chi U_{k}(1)\chi] + a(k)\}
\eqno(3.38)
$$
where
$$
a(k) = \int^{1}_{0}ds \tr \chi [U_{k} (1-s), x(k+x)] U_{k}(s) \chi
$$
It is easily seen that $a(k)$ is integrable and odd in $k$. Thus we have the
identity
$$
\partial_{t}\sqrt{t} q(t) =- \sqrt{t}\int^{+\infty}_{-\infty} \frac{dk}{2\pi} \tr x (x +
k) L_{k}
\eqno(3.39)
$$
and therefore, from $3.29$ and $3.17$ follows that
$$
\frac{M_{s}}{\mu_{B}} = \frac{4A\sqrt{2\pi t}}{\lambda}\int^{\infty}_{0} xj (x) dx
\eqno(3.40)
$$
for the Boltzmann distribution. The case of Fermi statistics can be handled
similarly and the identity $3.40$ still holds. Equations $3.33$ and $3.40$ giving the
bulk magnetisation as $M_{b}\sim - \int^{\infty}_{0} j(x) dx$ and the surface magnetisation
as $M_{s}\sim \int^{\infty}_{0} x j(x) dx $ should remain valid even in the interacting case,
at least when the system doesn't show a phase transition. It would be particularly
interesting to see if they keep their validity in the quantum Hall regime, when the
filling factor is a fraction associated to a plateau in the Hall conductivity. If not,
they could characterise the nature of these incompressible states. We can also remark
that these identities explain qualitatively that the surface magnetisation tends to be
opposite to the bulk magnetisation, favouring paramagnetism at low fields.
We have been unable to generalise the identity for the surface magnetisation, to
the three dimensional case. There is a term proportional to $\sin\theta
\int^{\infty}_{0} xj (x) dx$ but to which is added another one, whose physical
interpretation remains elusive.
\section{SAMPLES WITH A CYLINDRIC SHAPE}
For a large sample of volume $V$ and area $A$, the pressure of a three-dimensional system is
given by:
$$\eqalign{
\beta p V & = \frac{2V}{\lambda^{3}} (2\pi t)^{3/2}t \int^{\infty}_{0} \frac{z e^{-tx}}{1+z
e^{-tx}} n(x) dx\cr
& + \frac{2A}{\lambda^{2}} 2\pi t^{2} \int^{\infty}_{0} \frac{z e^{-tx}}{1+z
e^{-tx}}\overline{s}_{\partial\Lambda} (x) dx\cr
}\eqno(4.1)
$$
where
$$
n(x) = \frac{1}{\sqrt{2}\pi^{2}}
\sum^{\left[x -\frac{1}{2}\right]}_{n=0}(x-n-\frac{1}{2})^{1/2} \eqno(4.2)
$$
and
$$
\overline{s}_{\partial\Lambda} (x) = \frac{1}{|\partial\Lambda|}\int_{\partial\Lambda}
d\sigma s (x, \theta(\sigma))
\eqno(4.3)
$$
The bulk pressure is of course the familiar Landau expression.
The surface magnetisation is given by
$$
\frac{M_{s}}{\mu_{B}}=\frac{4A2\pi}{\lambda^{2}} \partial_{t} t^{2}\int^{\infty}_{0}
\frac{z e^{-tx}}{1+ze^{-tx}}\overline{s}_{\partial\Lambda} (x) dx
\eqno(4.4)
$$
Since we can compute explicitly $s(x,\theta)$ only for special values of $\theta$, we
need to turn to special types of shapes in order to get an explicit expression for the
magnetisation, for arbitrary magnetic fields.
Let us assume therefore from now on that the sample has the shape of a cylinder
with an arbitrary convex base. The magnetic field will be assumed to be either
parallel or perpendicular to the axis of the cylinder.
The surface of the sample parallel to the magnetic field, of area $A''$, will
give a contribution to the magnetisation given by
$$
\frac{M^{\parallel}_{s}}{\mu_{B}} = \frac{4A^{\parallel}.2\pi}{\lambda^{2}} \partial_{t}
t^{2} \int^{\infty}_{0}\frac{z e^{-tx}}{1+ze^{-tx}} s_{\parallel} (x) dx
\eqno(4.5)
$$
and the part perpendicular to the magnetic field of area $A^{\perp}$ will give another
contribution
$$
\frac{M^{\perp}_{s}}{\mu_{B}} = \frac{4A^{\perp}.2\pi}{\lambda^{2}} \partial_{t}
t^{2} \int^{\infty}_{0}\frac{z e^{-tx}}{1+ze^{-tx}} s_{\perp}(x) dx
\eqno(4.6)
$$
and the surface magnetisation will be
$$
M_{s} = M_{s}^{\parallel} + M_{s}^{\perp}
\eqno(4.7)
$$
According to equation 2.47, $s_{\perp} (x)$ will be such that
$$
t \int^{\infty}_{0} s_{\perp} (x) \exp - tx = - \frac{1}{8\pi}\sum^{\infty}_{n=0}\exp -
t(n+\frac{1}{2})
\eqno(4.8)
$$
which gives
$$
s_{\perp}(x) = - \frac{1}{8\pi} [x + \frac{1}{2}]
\eqno(4.9)
$$
therefore
$$
\frac{M^{\perp}_{s}}{\mu_{B}} = - \frac{A^{\perp}}{\lambda^{2}}\partial_{t} t
\sum^{\infty}_{n=0}\ln \Bigl[1 + z e^{-t(n +\frac{1}{2})} \Bigr]
\eqno(4.10)
$$
This expression shows that
$$
M^{\perp}_{s} = - \frac{1}{4}M_{b}(2 d)
\eqno(4.11)
$$
where $M_{b}(2 d)$ is the bulk magnetisation of a two-dimensional system of
volume $A^{\perp}$. The presence of the minus sign shows that at weak field or high
temperatures this contribution is paramagnetic. It also indicates that at low
temperatures $M^{\perp}_{s}$ will show anti de Haas-van Alphen type oscillations.
Equation $2.46$ gives for $s_{\parallel} (x)$
$$
t \int^{\infty}_{0} dx e^{-tx} s_{\parallel}(x) = \frac{t}{\sqrt{2\pi t}}
\int^{\infty}_{0} dx e^{-tx} s(x)
$$
and therefore
$$
s_{\parallel} (x) = \frac{1}{ \sqrt{2}\pi } \int^{x}_{0} dy \frac{1}{\sqrt{x-y}} s(y)
\eqno(4.12)
$$
We see that $s_{\parallel}(x)$ is negative. Thus both surface contributions give a
negative surface pressure. Equation $4.12$ shows that, contrary to the two-dimensional
case, the density of states $s_{\parallel}(x)$ doesn't diverge at a Landau level, but
rather vanishes, like $\sqrt{\varepsilon \ln \frac{1}{\varepsilon}}$, where $\varepsilon = x
- N - \frac{1}{2}$.
The precise dependence of $M^{\parallel}_{s}$ in the magnetic field is not very easy to
analyse from equations $4.6$ and $4.12$, even at zero temperature. It should however be
possible to compute numerically this function from the expression we derived for $s(x)$.
We would expect oscillations, as in the case of $M^{\perp}_{s}$.
Finally, we would like to remark that often the magnetisation is needed for a
system in which the density and not the chemical potential is fixed. One should
in this case study surface corrections in the canonical ensemble. But one expects
equivalence of ensembles to hold with an $L^{3/2}$ accuracy in three dimensions.
Since in this case the surface term is of order $L^{2}$,it is possible to express
the magnetisation in the canonical ensemble $M_{\rho}$ as
$$
M_{\rho} = M_{\mu} (\rho, B) + \left.\frac{\partial\mu}{\partial B}\right)_{\!\!\!\rho}
\rho_{s}(\rho,B) \eqno(4.13)
$$
where $M_{\mu}(\rho, B)$ is the grand-canonical magnetisation that we computed, in which the
chemical potential $\rho$ has been expressed in terms of the density $\mu$, by using
the bulk relationship between them.$\rho_{s} (\rho, B)$ is the surface contribution to
the density in the grand-canonical ensemble, but with $\mu$ expressed in terms
of $\rho$.
\section{WEAK MAGNETIC FIELDS}
In the general case, it is possible to compute the zero field susceptibility,
defined as
$$
\chi_{s} = \lim_{B\to0} \frac{M_{s}}{B}
\eqno(5.1)
$$
>From the equation 4.4 we see that this quantity is given by
$$
\chi_{s} = \frac{2 c^{2}}{m c^{2}} \frac{z}{z +
1}\frac{A}{|\partial\Lambda|}\int_{\partial\Lambda} d\sigma a (\theta(\sigma))
\eqno(5.2)
$$
where
$$
a (\theta) = \lim_{t\to0}\frac{1}{t}
\partial_{t}\, t \gamma_{\theta}(t)
\eqno(5.3)
$$
with
$$
\gamma_{\theta}(t) = t \int^{\infty}_{0} e^{-tx} s(x, \theta) dx
\eqno(5.4)
$$
>From equation 2.50, we have
$$
\gamma_{\theta} (t) = \frac{|\cos\theta|}{2\pi}\tr [\exp - t H_{+} - \chi \exp - t H
\chi] \eqno(5.5)
$$
It appears useful to express this quantity as a path integral $[7]$
$$
\gamma_{\theta} (t) = \frac{|\cos\theta|}{(2\pi)^{2}}\int^{\infty}_{0} dx
\int^{+\infty}_{-\infty} dy \int D\alpha F_{x} (\alpha) \exp -
\frac{t^{2}}{2}\int^{1}_{0} (x + R(s))^{2} ds
\eqno(5.6)
$$
In this expression
$$\eqalign{
&X = \sin \theta x + \cos \theta y\cr
&R(s) = \sin \theta \alpha_{x} (s) + \cos \theta \alpha_{y} (s)\cr
}
\eqno(5.7)
$$
and
$$
F_{x} (\alpha) =\cases {-1 & if $x + \alpha_{x} (s) \leq 0$. for some $s$\cr
0& otherwise\cr}
\eqno(5.8)
$$
expressing the fact that the paths are constrained to the half-space $x\geq0$.
$\alpha_{x} (s), \alpha_{y} (s)$ are independent brownian bridges, i.e. gaussian processes
with covariance
$$
\overline{\alpha(s)\alpha(s')} = s(1-s') \quad (0\leq s \leq s' \leq 1)
\eqno(5.9)
$$
and zero mean.
The introduction of these processes is a useful way to extract the spatial
dependence of the paths in the Wiener integral.
If we integrate over the $y$ coordinate, we get
$$
t \gamma_{\theta}(t) = (\frac{1}{2\pi})^{3/2}\int^{\infty}_{0} dx \int D\alpha
F_{x}(\alpha) \exp - \frac{t^{2}}{2} \int^{1}_{0}
(R(s) - \overline{R})^{2} ds
\eqno(5.10)
$$
where
$$
\overline{R} = \int^{1}_{0} ds R(s)
\eqno(5.11)
$$
We can deduce an interesting consequence of this representation, namely
$$
\partial_{t} t \gamma_{\theta}(t) \geq 0
\eqno(5.12)
$$
The meaning of this inequality is the following. If we were using the Boltzmann
distribution instead of the Fermi one, the surface magnetisation would be given
by
$$
\frac{M_{s}}{\mu_{B}} = \frac{4A}{\lambda^{2}}2\pi z \partial_{t}t \int_{\partial \Lambda}
\gamma_{\theta(\sigma)} (t) \frac{d\sigma}{|\partial\Lambda|}
\eqno(5.13)
$$
we can therefore conclude that in the case of the {\em Boltzman distribution\/} (i.e. at
high temperatures in the Fermi case), {\em the surface magnetisation is paramagnetic\/}
for all fields, i.e.$M_{s}\geq 0$.
We can also use the path integral to simplify the computation at weak fields.
Indeed, from $5.10$ follows that :
$$
a(\theta) =- \int^{\infty}_{0} \frac{dx}{(2\pi)^{3/2}}
\int D\alpha F_{x}(\alpha) \int^{1}_{0}(R(s) - \overline{R})^{2} ds
\eqno(5.14)
$$
an expression which shows immediately that the weak field magnetisation will be
also paramagnetic.
But in fact, $5.14$ shows that
$$
a(\theta) = a(o) \cos^{2}\theta + a (\frac{\pi}{2}) \sin^{2}\theta
\eqno(5.15)
$$
because the mixed term $\int D\alpha (\alpha_{x} (s) -\overline{\alpha}_{x}) (\alpha _{y}
(s) - \overline{\alpha}_{y}) = 0 $ in Eq. $5.14$. We are left to compute only $a(0)$ and
$a (\frac{\pi}{2})$. We have seen in equation $2.47$, that
$$
\gamma_{0}(t) = -\frac{1}{8\pi} \sum^{\infty}_{n=0} \exp - t(n + \frac{1}{2})
\eqno(5.16)
$$
This gives easily
$$
a(0) = \frac{1}{8\pi} . \frac{1}{12}
\eqno(5.17)
$$
>From equations $2.46$ and $3.34$ follow that
$$
\gamma_{\frac{\pi}{2}} (t) = \frac{1}{\sqrt{2\pi t}} q (t)
\eqno(5.18)
$$
and therefore
$$
a(\frac{\pi}{2}) = \frac{1}{\sqrt{2\pi}} \lim_{t\to0} \frac{1}{t}
\partial_{t}\sqrt{t} q (t) \eqno(5.19)
$$
The computation of this quantity is a bit more tricky. We will use the relationship
between surface magnetisation and the surface current established in paragraph 3. After
some rescaling of the variables, equation $3.39$ can be written as
$$
\frac{1}{t} \partial_{t} \sqrt{t} q(t) = - \int^{+\infty}_{-\infty} \frac{dk}{2\pi} \tr x(x
+ \frac{k}{t}) \overline{L}_{k} (1)
\eqno(5.20)
$$
where
$$
\overline{L}_{k} (s) = V^{+}_{k} (s) - \chi V_{k}(s) \chi
$$
with
$$
V^{+}_{k} (s) = \exp - \frac{s}{2}[- \partial^{2}_{x} + (k + xt)^{2}]
\eqno(5.22)
$$
the operator being defined on the half-line with Dirichlet boundary conditions.
$V_{k}(s)$ is the same operator but defined on the whole line.
If in the second term of the expression appearing at the right hand side of
equation $5.20$ we make an integration by parts on the $k$ variable, we can put
equation $5.20$ in the more useful form
$$
\frac{1}{t} \partial_{t} \sqrt{t} q(t) = - \int^{+\infty}_{-\infty}
\frac{dk}{2\pi}\int^{1}_{0}ds\tr x [x, V^{+}_{k}(1-s)] V^{+}_{k}(s)
\eqno(5.23)
$$
the term corresponding to $V_{k}$ disappearing because it gives a contribution odd
in $k$.
In this form, it is possible to take the limit $t \rightarrow 0$.In this limit the
kernel of
$ V^{+}_{k}(s)$ becomes $e^{- \frac{k^{2}}{2}} U_{s} (x|y)$, with
$$
U_{s}(x|y) = \frac{1}{\sqrt{2 \pi s}} [\exp - \frac{(x-y)^{2}}{2s} - \exp -
\frac{(x+y)^{2}}{2s}] \eqno(5.24)
$$
Inserting this expression into $5.23$, we get after some lengthy computation
$$
\lim_{t\to0} \frac{1}{t}\partial_{t} \sqrt{t} q (t) = \frac{1}{\sqrt{2\pi}}
\frac{3}{2^{7}}
\eqno(5.25)
$$
and therefore
$$
a (\frac{\pi}{2}) = \frac{1}{2\pi} \frac{3}{2^{7}}
\eqno(5.26)
$$
Grouping all these results we get finally
$$
\chi_{s} = \frac{Ae^{2}}{mc^{2}} \frac{z}{z+1} \frac{1}{\pi 2^{7}} [3 -
\frac{1}{3|\partial\Lambda|} \int_{\partial\Lambda} d\sigma
(\overrightarrow{n}(\sigma) \,. \overrightarrow{b})^{2}]
\eqno(5.27)
$$
where we have denoted by $\overrightarrow{b}$ the unit vector directed along the
magnetic field, and we recall that $\overrightarrow{n}(\sigma)$ is the normal at the
boundary point $\sigma$. In the case of a cyclindric shape, this formula becomes
$$
\chi_{s} = \frac{e^{2}}{mc^{2}} \frac{z}{z+1} \frac{1}{\pi 2^{7}}[2 A_{1} (3 -
\frac{\cos^{2}\theta_{1}}{3}) + A_{2} (3 - \frac{\cos^{2}\theta_{2}}{3})]
\eqno(5.28)
$$
where $A_{1} (A_{2})$ is the area respectively of the base and of the lateral face of the
cylinder, and $\theta_{1} (\theta_{2})$ designates the angle made by the magnetic field with
the respective normals to theses faces.
This formula reproduces the result of Angelescu, Nenciu and Bundaru $[1]$ in the
case of a parallelepiped.
In the case of a sphere, equation 5.27 gives
$$
\chi_{s} = \frac{Ae^{2}}{mc^{2}} \frac{z}{z+1} \frac{13}{9\pi . 2^{6}}
\eqno(5.29)
$$
We may note that the zero field limit is a subtle one and quite probably the
development in magnetic field is only asymptotic.
\section*{CONCLUSION}
We can briefly summarize the new results we have obtained. First of all, the computation of
the surface density of states, for convex bodies, is reduced to the solution of the
Schr\"odinger equation for a particle, confined to a tilted half--plane and submitted to a
harmonic potential in the $x$ direction. An expicit solution of this last problem in special
cases allows us to determine completly the surface density of states in two dimensions and
for cylindrical shapes in three dimensions for arbitrary magnetic fields.
In the most general case, the best we could do was to compute the zero field magnetic
susceptibility.
A more explicit solution of the Schr\"odinger equation alluded to, would allow a complete
determination of the surface density of states, for arbitrary magnetic fields in the three
dimensional case. This is the most important remaining problem to be solved in our opinion.
\section*{APPENDIX}
We would like here to briefly discuss the problem of the asymptotic
behaviour of the density of states.
If we consider a box made of the disjoint union of two boxes $\Lambda_{1}$, and
$\Lambda_{2}$, then for Dirichlet boundary conditions, the density of
states $N_{\Lambda}(x)$ satisfies the inequality
$$
N_{\Lambda,u\Lambda_{2}}(x) \geq N_{\Lambda_{1}}(x) + N_{\Lambda_{2}}(x)
\eqno(A.1)
$$
On the other hand, Colin de Verdi\`ere $[8]$ has proven that for a cube
$$
N_{\Lambda} (x) \leq |\Lambda| n (x)
\eqno(A.2)
$$
where $n(x)$ is the bulk density of states $2.48$. This inequality can be extended to
parallelepipeds. From these two set of inequalities it can be proven by standard
techniques that
$$
\lim_{\Lambda P R^{d}} \frac{N_{\Lambda}(x)}{|\Lambda|} = n (x)
$$
for a large class of boxes.
The problem of the surface correction is however much more subtle. It is known
in the case of the Laplacian, i.e. for the problem without a magnetic field, that
convergence of the difference
$$
s_{\Lambda}(x) = \frac{N_{\Lambda}(x) - |\Lambda| n (x)}{|\partial\Lambda|}
$$
cannot hold pointwise, generally. Some assumptions must be made about the
density of periodic orbits of the corresponding classical system $[9]$. But in statistical
mechanics a much weaker kind of convergence is needed at positive
temperatures, the so-called weak convergence of measures. It guarantees for
example that the pressure
$$
p = z \int^{\infty}_{0} \frac{e^{-\beta\lambda}}{1+z e^{-\beta\lambda}}
\frac{N_{\Lambda}(\lambda)}{|\Lambda|} d \lambda
\eqno(A.3)
$$
will have the asymptotic behaviour that we derived. Indeed theorem 2a, XIII 1,
volume II of W. Feller's book $[10]$ guarantees that this will hold, provided
$s_{\Lambda}(x)$ is negative or positive, so that the convergence of its Laplace
transform (that we proved) will imply convergence of the measure $s_{\Lambda}(x)
dx$. In our case Colin de Verdi\`ere inequality $(A.2)$. guarantees that $s_{\Lambda}(x)$ is
negative for parallelepipeds. We would expect such a result for large enough convex boxes,
but have not proved it.
The pointwise type of convergence would be needed if we would consider the
problem at zero temperatures, and then look at the asymptotic behaviour for large
samples. Fortunately, from the physical point of view, in most situations, one
needs to consider the other order of limits, first large samples and then low
temperatures.
\section*{REFERENCES}
\begin{itemize}
\item[{[1]}] N.\ Angelescu, G.\ Nenciu, M.\ Bundaru
Comm.\ Math.\ Phys.\ {\bf 42}, 9 (1975)
\item[{[2]}] J.~M.\ van Ruitenbeek, D.\ van Leeuwen
Modern Physics Letters~{\bf B 7}, 1053 (1993)
\item[{[3]}] M.~Robnik
J.\ Phys.\ {\bf A 19}, 3619 (1986)
\item[{[4]}] B.~L.~Altschuler, Y.~Gefen, Y.~Imry
Phys.\ Rev.\ Letters~{\bf 66}, 88 (1991)
\item[{[5]}] M.\ Kac
Amer.\ Math.\ Monthly~{\bf 73}.~1--23 (1966)
(Slaught Mem.\ Papers No.~11)
\item[{[6]}] B.\ Simon
{\em Functional Integration and Quantum Physics\/}
Academic Press (1979)
\item[{[7]}] N.\ Macris, Ph.\ A.\ Martin, J.\ V.\ Pul\'e
Commun.\ Math.\ Phys.~{\bf 117}, 215 (1988)
\item[{[8]}] Y.\ Colin de Verdi\`ere
Commun.\ Math.\ Phys.~{\bf 105}, 327 (1986)
\item[{[9]}] V.\ Petrov, L.\ Stoyanov
{\em Geometry of reflecting rays and inverse spectral problems\/}
J.\ Wiley \& Sons (1992)
\item[{[10]}] W.\ Feller
{\em An introduction to probability theory and its applications\/}, Volume II
J.\ Wiley \& Sons (1971).
\end{itemize}
\end{document}