0\right) $,$\ $a nontrivial background of Chern-Simons magnetic field $\left( \overline{b}=%
\overline{f}_{21}\right) $ is generated. The Chern-Simons background field is obtained as the solution of the mean field Euler-Lagrange equations derived from (2.1)

$$-\frac{N}{4\pi }\varepsilon ^{\mu \nu \rho }f_{\nu \rho }=\left\langle
j^{\mu }\right\rangle  \eqnum{2.3}$$

$$\partial _{\nu }F^{\mu \nu }=e\left\langle j^{\mu }\right\rangle
-en_{e}\delta ^{\mu 0}  \eqnum{2.4}$$

considering that the system formed by the electron fluid and the background charge $n_{e}$ is neutral

$$\left\langle j^{0}\right\rangle -n_{e}\delta ^{\mu 0}=0  \eqnum{2.5}$$

In eq. (2.5) $\left\langle j^{0}\right\rangle $ is the fermion density of the many-particle fermion system

$$\left\langle j^{0}\right\rangle =\frac{\partial \Omega }{\partial \mu }, 
\eqnum{2.6}$$

$\Omega $ is the fermion thermodynamic potential.

In this approximation it is found from (2.3)-(2.5) that the Chern-Simons magnetic background is given by

$$\overline{b}=\frac{2\pi n_{e}}{N}  \eqnum{2.7}$$

Then, the unperturbed one-particle Hamiltonian of the matter field represents a particle in the background of the Chern-Simons magnetic field $%
\overline{b\text{,}}$

$$H_{0}=-\frac{1}{2m}\left[ \left( \partial _{1}+i\overline{b}x_{2}\right)
^{2}+\partial _{2}^{2}\right]  \eqnum{2.8}$$

In (2.8) we considered the background Chern-Simons potential, $\overline{a}%
_{k}$, $(k=1,2)$, in the Landau gauge

$$\overline{a}_{k}=\overline{b}x_{2}\delta _{k1}  \eqnum{2.9}$$

The eigenvalue problem defined by the Hamiltonian (2.8) with periodic boundary conditions in the $x_{1}$-direction: $\Psi \left( x_{1}+L,\text{ }%
x_{2}\right) =$ $\Psi \left( x_{1},\text{ }x_{2}\right) $,

$$H_{0}\Psi 