than the energy gap ($T\gg \omega _{c}$). Sec. 6 contains the summary and discussion.

MCS Many-Particle Model
=======================

The Lagrangian density of the 2+1 dimensional non-relativistic charged MCS system is

$${\cal L}=-\frac{1}{4}F_{\mu \nu }^{2}-\frac{N}{4\pi }\varepsilon ^{\mu \nu
\rho }a_{\mu }\partial _{\nu }a_{\rho }+en_{e}A_{0}+i\psi ^{\dagger
}D_{0}\psi -\frac{1}{2m}\left| D_{k}\psi \right| ^{2}+\psi ^{\dagger }\mu
\psi  \eqnum{2.1}$$

where $A_{\mu }$ and $a_{\mu }$ represent the electromagnetic and the Chern-Simons fields respectively. The role of the Chern-Simons fields is simply to change the quantum statistics of the matter field, thus, they do not have an independent dynamics. $\psi $ represents non-relativistic spinless fermions. $N\ $ is a positive integer that determines the magnitude of the Chern-$%
%TCIMACRO{\func{Si}}
%BeginExpansion
\mathop{\rm Si}%
%EndExpansion
$mons coupling constant. The charged character of the system is implemented by introducing a chemical potential $\mu $; $n_{e}$ is a background neutralizing ‘classical’ charge density, and $m$ is the fermion mass. We will consider throughout the paper the metric $g_{\mu \nu }$=$(1,-%
\overrightarrow{1})$. The covariant derivative $D_{\nu }$ is given by

$$D_{\nu }=\partial _{\nu }+i\left( a_{\nu }+eA_{\nu }\right) ,\qquad \nu
=0,1,2  \eqnum{2.2}$$

It is known that to guarantee the system neutrality in the presence of a different from zero fermion density $\left( n_{e}\neq 