A. Bosonization for the 2DEG
We introduce a continuous formulation for the 2DEG many particles system. We replace the single particle Hamiltonian by a many electron formulation . We introduce a continuous representation, namely . Here, is the continuous form of . The coordinate and the momentum obey and The equilibrium Fermi-Dirac density is given by
One of the useful description for many electrons in two dimensions is the Bosonization method. We will modify this method  in order to introduce the effect of the vortex field and the confining potential .
B. The bosonization method in the absence of the insulating region and confining potential
In this section we will present the known [2, 10] results for a two dimensional interacting metal in the absence of the vortex field and confining potential . Our starting point is the Bosonized form of the 2DEG given in ref. [10, 16].
where is the Landau function for the two body interaction  and the notation :: represents the normal order with respect the Fermi Surface. . According to ref.[10, 16], the F.S. is described by, . The "normal" deformation to the F.S. is given by, . "s" is the polar angle on the F.S. (s), and (s) is the normal to the F.S. The commutation relations for the F.S. are,
C. The modification of the bosonization method in the presence of a confining potential
Following ref.  (see the last term of eq.10 in ref. ), we incorporate into the Bosonic hamiltonian the effect of the confining and external potentials. We parametrize the Fermi surface in terms of the polar angle s = [0 - 2π] and the coordinate . The Fermi surface momentum given by the solution, . As a result the the FERMI SURFACE (F.S.) excitations is given by, . The "normal" deformation to the F.S. is given by, and is the normal to the F.S. as a function of the polar angle s and real space coordinate .
Following refs. [11, 17], we obtain the Bosonized hamiltonian for the many particles system in the presence of the potentials, and time dependent external potential .
The new part of in the Bosonic hamiltonian is the presence of the confining and external potential . is the external microwave radiation field,
The commutation relations for the FERMI SURFACE in are given by the Kack Moody commutation relation [11, 17]
D. The bosonization method in the presence of the insulating region and confining potential
This problem can be investigated using the hamiltonian given in eq.12 supplemented by the constraints conditions imposed by the vanishing density. described by a vortex. Using the results given in eq.11 one modifies the commutation relations. This modification can be viewed as Dirac's bracket due to second class constraints . The commutator [,] is replaced by Dirac 's commutators . The region of vanishing density is described by the function for <D and zero otherwise. Using we find that the Dirac commutator replaces the commutator given in equation 13
The result given by eq. 14 due to non-commuting coordinates . We will compare this result with the one for a magnetic field B, given in ref. . The commutator for the two dimensional densities in the presence of a magnetic field B() perpendicular to the 2DEG has been obtained in ref. .
The modified commutator caused by the magnetic field (see eq. 27 in ref. ) is:
Where B() is the magnetic field and is the derivative in the tangential direction perpendicular to the vector (s) (the normal to the Fermi surface), .
Using the analogy between the vortex field an the external magnetic field (eq. 15) we can represent equation 14 in terms of the parameters given in eq. 11.
where is the derivative according to the tangential direction which is perpendicular to the vector (s) with, .
The commutator given in equation 16 allows to investigate the physics given in the hamiltonian 12. Using this formulation we will compute the rectified current.
We observe that the Dirac commutator, ≠ 0 is non-zero for s ≠ s'! The Heisenberg equation of motion will be given by the Dirac bracket. Due to the fact that different channels s ≠ s' do not commute, the application of an external electric field in the i = direction will generate a deformation for the F.S. with s ≠ , .
Using this commutation relations given in equation 16 and the hamiltonian given in equation 12, we obtain the equation of motion,
We have included in the equation of motion a phenomenological relaxation time τ for the kinetic momentum. This equation shows that the direct effect caused by the electric field is proportional to sin(s) with the maximum contribution at the polar angles, and where V
(s, ) is the Fermi velocity. The effect of the vortex is to generates a change in the kinetic momentum perpendicular to the external electric field. This part is given by the last term. The last term is restricted to <D and represents the vortex contribution. This term is maximum for the polar angles s = 0 and s = π. The maximum effect will be obtained in the region close to the classical turning point where the Fermi velocity obeys, .
The current density in the i = 1 direction is given by the polar integration of s, [0 - 2π].
We introduce the dimensionless parameter which is a function of and D the radius of the insulating region. For values of γ < 1 we can solve iteratively the equation of motion and compute the current.
In the equation for the kinetic momentum we have included a phenomenological relaxation time τ. This relaxation time will allow to perform times averages. We only keep single harmonics and neglect higher harmonics of the microwave field.
The iterative solution is given as a series in γ and the microwave amplitude E
Solving the equation of motion we determines the evolution of the Fermi surface deformation in the presence of the microwave field. We substitute the iterative solution obtained from eq. 17 into the current density formula given by eq. 18.