Orthonormal wave functions for periodic fermionic states under an applied magnetic field

We report an infinite number of orthonormal eigenfunction bases for the quantum problem of a free electron in presence of an applied external magnetic field, suitable to describe doubly periodic electronic densities. The completeness of these bases is shown here and so, a single basis, labeled by the number of magnetic fluxons trapped in the unit cell (integer p), expands any function in the unit cell. The present framework unveils for the electronic density an egg-box pattern that displays fractional charge and magnetic flux. In case of electrons confined to the lowest Landau level we obtain an analytic expression for the local magnetic field created by their own motion and find that it yields an attractive magnetic interaction. The well-known de Haas-van Alphen oscillations are retrieved, thus showing the correctness of the present theoretical framework.

Automated summary

This paper reports an infinite number of orthonormal eigenfunction bases suitable for describing doubly periodic electronic densities in a quantum system of a free electron in the presence of an external magnetic field. The completeness of these bases is demonstrated, with a single basis expanding any function in the unit cell based on the number of magnetic fluxons trapped in the unit cell. The framework reveals an egg-box pattern for the electronic density displaying fractional charge and magnetic flux, and for electrons confined to the lowest Landau level, an analytic expression for the local magnetic field created by their own motion is obtained, showing an attractive magnetic interaction. The well-known de Haas-van Alphen oscillations are also retrieved, confirming the correctness of the theoretical framework presented.