Supersymmetry in the Landau Level Problem with a Disorder Potential

Abstract

We explore a supersymmetric (SUSY) theory that arises in the Landau level problem with disorder. Charged particles in a strong magnetic field and a local potential are described by small excitations around the ground state, the lowest Landau level (LLL). Around forty years ago Br\'ezin, Gross and Itzykson showed that for a disorder potential certain observables in the LLL limit are described by a chiral SUSY theory. As a consequence, the problem undergoes a dimensional reduction by two dimensions, simplifying the computations. We generalize their findings by identifying the chiral SUSY as a special limit of a `magnetic Parisi-Sourlas (PS)' theory. The latter is a modification of the theories associated to fixed points of random field models. We show that the SUSY features extend to a much larger class of observables and to higher Landau levels. Finally we identify a set of new super-Ward identities and show how observables in the disordered theory must satisfy them.