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\begin{document}
\begin{flushright}
Preprint DFPD/97/TH/15
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\begin{center}
{\large \bf On the Torus Quantization of Two Anyons \\
with Coulomb Interaction in a Magnetic Field}
\end{center}
\vskip 1. truecm
\begin{center}
{\bf Luca Salasnich}\footnote{Electronic address:
salasnich@math.unipd.it}
\vskip 0.5 truecm
Dipartimento di Matematica Pura ed Applicata \\
Universit\`a di Padova, Via Belzoni 7, I 35131 Padova, Italy \\
and\\
Istituto Nazionale di Fisica Nucleare, Sezione di Padova,\\
Via Marzolo 8, I 35131 Padova, Italy
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\begin{center}
{\bf Abstract}
\end{center}
\vskip 0.5 truecm
\par
We study two anyons with Coulomb interaction in a uniform
magnetic field $B$. By using the torus quantization
we obtain the modified Landau and Zeeman formulas for the two anyons.
Then we derive a simple algebraic equation for the full spectral
problem up to the second order in $B$.
\vskip 0.5 truecm
\begin{center}
To be published in {\it Modern Physics Letters} B
\end{center}
\newpage
\par
In 1977 Leinaas and Myrheim$^{1)}$ suggested
the possible existence of arbitrary statistics
between the Bose and the Fermi case in
space dimensions less than three.
A model of particles with
arbitrary, or fractional, statistics was introduced
by Wilczek$^{2)}$, who called such objects as {\it anyons}.
Anyons are thought to be good
candidates for the explanation of such important
collective phenomena in condensed matter physics as the
fractional quantum Hall effect and high temperature
superconductivity (for recent reviews see Ref. 3--5).
\par
In this paper we study two anyons with Coulomb interaction in a uniform
magnetic field. The eigenvalue problem for this system
was solved numerically by
Myrheim, Halvorsen and Vercin$^{6)}$ by using a finite difference
method to discretize the Schr\"odinger equation.
Here, instead, we obtain some analytical formulas of the
energy spectrum by using the torus quantization$^{7),8)}$.
\par
The Lagrangian of two anyons with Coulomb
interaction in two dimensions is given by
\beq
L_0={m\over 2}({\dot {\bf r}_1}^2+{\dot {\bf r}_2}^2)
+ {e^2 \over ||{\bf r}_2 - {\bf r}_1||}
+ \alpha \hbar {\dot \theta}
\eeq
where ${\bf r}_1=(x_1,y_1)$, ${\bf r}_2=(x_2,y_2)$ and
$\alpha$ is the statistical parameter: Bose statistics is recover for
$\alpha =0$ and Fermi statistics for $\alpha =1$. The azimuthal
angle is defined by $\theta = \arctan ({y_2 - y_1 \over x_2 - x_1})$.
\par
The separation of the center of mass motion can be achieved through
the change of variables ${\bf r} = {\bf r}_2 - {\bf r}_1$ and
${\bf r}_G = {\bf r}_2 + {\bf r}_1$. In this way we obtain
\beq
L_0= {\mu \over 2} {\dot {\bf r}_G}^2
+ {\mu \over 2} {\dot {\bf r}}^2 + {e^2\over r}
+ \alpha \hbar {\dot \theta} \; ,
\eeq
where $\mu = m/2$ is the reduced mass and $r=||{\bf r}||$.
Neglecting the motion of the center of mass ${\bf r}_G$,
the Lagrangian can be written in polar coordinates
${\bf r}=(r \cos{\theta}, r\sin{\theta})$ as
\beq
L_0={\mu \over 2} ({\dot r}^2 + r^2 {\dot \theta}^2 ) + {e^2\over r}
+ \alpha \hbar {\dot \theta} \; .
\eeq
\par
Let us introduce a uniform magnetic field ${\bf B}$ along the
$z$--axis. Choosing the symmetric gauge,
the Lagrangian of interaction is given by
\beq
L_I = -{e B\over 2 c} r^2 {\dot \theta} \; ,
\eeq
so that the total lagrangian reads
\beq
L = L_0 + L_I = {\mu \over 2} ({\dot r}^2 + r^2 {\dot \theta}^2 )
+ {e^2\over r} + (\alpha \hbar -
{eB\over 2c}r^2 ){\dot \theta} \; .
\eeq
Introducing the generalized momenta
\beq
p_r = {\partial L\over \partial {\dot r} } = \mu {\dot r} \; ,
\eeq
\beq
p_{\theta}={\partial L\over \partial {\dot \theta} } =
\mu r^2 {\dot \theta} + (\alpha \hbar -{eB\over 2c}r^2 ) \; ,
\eeq
the Hamiltonian $H=p_r {\dot r} + p_{\theta} {\dot \theta} - L$
can be written as
\beq
H = {p_r^2\over 2 \mu} + {(p_{\theta}-\alpha \hbar )^2\over 2 \mu r^2}
- {e^2\over r} + {e^2 B^2 \over 8 \mu c^2}r^2
+ {eB\over 2 \mu c} (p_{\theta} - \alpha \hbar ) = E \; .
\eeq
The angular momentum $p_{\theta}$ is a constant of motion
because the Hamiltonian is independent of $\theta$.
It is readily verified that $H$ and $p_{\theta}$ are in involution,
so this two--degrees of freedom system is integrable.
We observe that ${eB\over 2 \mu c} (p_{\theta} - \alpha \hbar )$
is constant and we study the reduced Hamiltonian
\beq
{\bar H} = {p_r^2\over 2 \mu} + W(r) = {\bar E} ,
\eeq
where
\beq
W(r)= {(p_{\theta}-\alpha \hbar )^2\over 2 \mu r^2}
- {e^2\over r} + {e^2 B^2 \over 8 \mu c^2}r^2 \; .
\eeq
Also the Hamiltonian ${\bar H}$ is integrable and each classical
trajectory is confined on a 2--dimensional torus.
The action variables on the torus are given by
\beq
I_{\theta}={1\over 2\pi}\oint p_{\theta} d\theta = p_{\theta} \; ,
\eeq
\beq
I_r = {1\over 2\pi} \oint \sqrt{2\mu ({\bar E} - W(r))} dr \; ,
\eeq
and by inverting the last relation
we have ${\bar E}={\bar H}(I_r, I_{\theta})$.
\par
The energy spectrum of the system can be obtained by using
the {\it torus quantization}.
The torus quantization goes back to the early days of quantum mechanics
and was developed by Bohr and Sommerfeld for
separable systems, it was then generalized for integrable
(but not necessarily separable) systems by Einstein$^{7)}$.
In fact, Einstein's result was corrected for the phase changes
due to caustics by Maslov$^{8)}$.
The torus quantization
is just the first term of a certain $\hbar$-expansion
of the Schr\"odinger equation$^{8),9)}$ but it is
of extreme importance since in many cases this is the only
approximation known in the form of an explicit formula$^{10),11)}$.
For our system the torus quantization reads
\beq
I_{\theta} = n_{\theta} \hbar \; ,
\eeq
\beq
I_r = (n_r +{1\over 2}) \hbar \; ,
\eeq
where the radial quantum number $n_r$ is a non--negative integer,
whilst the angular quantum number $n_{\theta}$ can be also negative.
\par
In this way the quantized spectrum is given by
\beq
E(n_r, n_{\theta})= {\bar E}(n_r, n_{\theta}) +
{eB\hbar \over 2\mu c}(n_{\theta} - \alpha ) \; ,
\eeq
where ${\bar E}={\bar E}(n_r ,n_{\theta})$ is obtained by solving the equation
\beq
{1\over 2\pi}\oint \sqrt{ - F + {2 G\over r} - {M \over r^2} - N r^2 } dr
= (n_r+{1\over 2})\hbar \; ,
\eeq
with $F=2\mu (-{\bar E})$, $G=\mu e^2$, $M=(n_{\theta} - \alpha )^2 \hbar^2$
and $N=e^2B^2/(4c^2)$.
\par
This integral equation can be evaluated analytically
in some limit cases. In particular, if $B$ is very strong then
the Coulomb interaction is negligible ($G=0$). In this case,
if $r$ is represented in the complex plane, the integrand can be pictured
on a Riemann surface of two sheets with branch points at the
roots $r_1$ and $r_2$ ($r_1