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

Abstract

We report an infinite number of orthonormal wave functions bases for the quantum problem of a free particle in presence of an applied external magnetic field. Each set of orthonormal wave functions (basis) is labeled by an integer $p$, which is the number of magnetic fluxons trapped in the unit cell. These bases are suitable to describe particles whose probability density is periodic and defines a lattice in position space. The present bases of orthonormal wave functions unveils fractional effects since the number of particles in the unit cell is independent of the number of trapped fluxons. For a single particle under $p$ fluxes in the unit cell, and confined to the lowest Landau level, the probability density vanishes in $p$ points, thus each zero is associated to a fraction $1/p$ of the particle. Remarkably the case of $n+1$ filled Landau levels, hence with a total of $N=(n+1)p$ fermions, $n$ being the highest filled Landau level, the density displays an egg-box pattern with $p^2$ maxima (minima) which means that a $(n+1)/p$ fraction of flux is associated to every one of these maxima (minima). We also consider the case of particles interacting through the magnetic field energy created by their own motion and find an attractive interaction among them in case they are confined to the lowest Landau level ($n=0$). The well-known de Haas-van Alphen oscillations are retrieved within the present orthonormal basis of wave functions thus providing evidence of its correctness.