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

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