Two-dimensional electron–hole system interacting with quantum point vortices in the frame of the Chern–Simons theory

Abstract

The Chern–Simons (C–S) theory developed by Jackiw and Pi [1] and widely used in the theory of the fractional quantum Hall effects (FQHEs) was applied to describe a two-dimensional coplanar electron–hole system in a perpendicular magnetic field interacting with quantum point vortices. The Hamiltonian of free bare conduction and valence electrons in the periodic lattice potential and the respective wave functions were subjected to the C–S unitary transformation leading to the new Schrodinger equations describing the dressed quasiparticles composed of bare electrons and holes with attached quantum point vortices. It was shown that the numbers of the attached point quantum vortices to each conduction electron and to each hole are the same. In contrast to a one-component two-dimensional electron gas, where the FQHEs were revealed, in the electron–hole system, the C–S vector potential is created together by the electron quantum vortices and by the hole quantum vortices and depends on the difference of the density operators of the two subsystems. In the mean-field approximation, the C–S vector potential and the effective magnetic field generated by it vanish if the average densities of the conduction electrons and holes coincide. Nevertheless, even in this case, the quantum fluctuation of the C–S field will lead to new branches of the collective elementary excitations..