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The Hamiltonian describing a composite fermion system is usually presented in a phenomenological way. By using a classical nonrelativistic U(1) × U(1) gauge field model for the electromagnetic interaction of electrons, we show how to obtain the mean-field Hamiltonian describing composite fermions in 2 + 1 dimensions. In order to achieve this goal, the Dirac Hamiltonian formalism for constrained systems is used. Furthermore, we compare these results with the ones corresponding to the inclusion of a topological mass term for the electromagnetic field in the Lagrangian.

The study of low-dimensional electron systems is a topic of great current interest in condensed matter. Under certain conditions, three-dimensional solids can behave as pseudo two or one-dimensional quantum systems. Among other things, this is because these systems have special characteristics which lead to phenomena such as superconductivity of high critical temperature and some magnetic properties, with potential technological applications.

Since a long time ago, the phenomenon of high-temperature superconductivity is being considered with increasing interest. The current status of knowledge on this subject does not allow establishing on firm foundation this phenomenology. Different approaches are used to address this issue. A promising approach is based on models using composite particles [

A manner of studying these systems is based on quantum field theory. Another manner is to use quantum many-body theory [

In Refs. [

To deal with models that consider the electromagnetic interaction of composite fermions, a phenomenological Hamiltonian is commonly used. In this paper, by using the standard methods of field theory, we show that the same Hamiltonian can be reached.

In this context, our purpose is to relate the results of Refs. [

Besides, we show that the inclusion of a topological mass term for the electromagnetic field in the Lagrangian density introduces interaction terms absent in the original canonical Hamiltonian.

The paper is organized as follows. In Section 2, we show how to obtain the canonical Hamiltonian from the starting Lagrangian. Then, in Section 3, we see what happens when we add a topological mass term for the electromagnetic field in the Lagrangian. Finally, in Section 4, we display our conclusions.

We consider a classical nonrelativistic field theory with U(1) × U(1) gauge symmetry in 2 + 1 dimensions for the electromagnetic interaction of electrons. This model uses a Chern-Simons (CS) U(1) gauge auxiliary field

In order to describe this interaction, we propose the following singular Lagrangian density:

The Greek indices take the values

The matter field

The independent dynamical field variables are

conjugate momenta, defined by

dices take the values

The canonical Hamiltonian density, defined as

where the Latin indices take the values

As usual, the relations between the fields and the momenta not depending on the velocities lead to primary constraints. Thus, the momentum

Finally, classifying the total set of constraints, we find the first and second-class constraints. The first-class constraints are

while the second-class ones are

We choose the following gauge-fixing conditions, which are consistent with the equations of motion:

The algebraic development corresponding to these results were performed in Ref. [

As it is well known, one can define the Dirac brackets from the Bose-Fermi ones, and then all the equations of the theory are formulated taking into account the former brackets, as it can be seen in Ref. [

It is worth to mention that, in order to obtain Equation (15) from Equation (13), we have substituted the logarithmic interaction [

By replacing the momenta of Equations (19) and (20) in Equation (4), we get

The term

Taking the spatial derivative of the momentum

In Ref. [

The curl of the CS field is linked with a magnetic field in the form

Therefore, the Lagrangian density (1) describes a composite fermion system, where

Replacing all previous results in Equation (2), we obtain

where

Since the field

with

where

Besides, following Ref. [

where B is perpendicular to the motion plane.

Finally, we can write

Usually, when only an external uniform magnetic field is considered, the canonical Hamiltonian reduces to

As it is well known, the addition of the CS term to the Maxwell action leads to the topologically massive (2 + 1)-dimensional electrodynamics [

In this section, we analyze the differences that appear with respect to the original model when a topological mass term for the electromagnetic field is added to the Lagrangian density

where

In this case, the canonical Hamiltonian density becomes

where

The only first-class constrains that change with respect to the previous ones are

In the same way, the only gauge-fixing condition that changes is

These results have been formally demonstrated in Ref. [

Following the same steps like in the previous section, it can be proved that the constraint (22) remains valid.

In contrast to Equation (15),

Finally, the topological massive canonical Hamiltonian density is

By replacing

where

Once we integrate Equation (37), the first two terms give the same contributions to the Hamiltonian. By considering the constraint (22), these terms can be associated to the interaction between the external and CS magnetic fields mediated by the distance. In the limit of infinite topological mass, this interaction vanishes. This is consistent with the fact that in the usual formulation, where the external magnetic field is uniform, this interaction is not present. The last term in Equation (37) represents the self interaction of the external magnetic field in a higher order in the topological mass.

By means of a classical nonrelativistic U(1) × U(1) gauge field model for the electromagnetic interaction of electrons, we have shown how to find the mean-field Hamiltonian describing composite fermions in 2 + 1 dimensions. For this purpose, the Dirac Hamiltonian formalism for constrained systems was considered.

Furthermore, in Section 2, we showed how the Coulomb interaction naturally appears while in the usual formulation it is introduced ad hoc. In the usual formulation, Equation (22) is taken as a constraint of the theory. In this paper, we derived its validity from the constraint structure.

Finally, in Section 3, we saw that the inclusion of a topological mass term for the electromagnetic field in the Lagrangian density introduces interaction terms absent in the canonical Hamiltonian (29).

The authors acknowledge Dr. R. Id Betan for his invaluable contribution and helpful discussions.

Edmundo C.Manavella,Carlos E.Repetto, (2015) Alternative Derivation of the Mean-Field Equations for Composite Fermions. Journal of Modern Physics,06,1737-1742. doi: 10.4236/jmp.2015.612175