Exact calculation of multifractal exponents of the critical wave function of Dirac fermions in a random magnetic field

The multifractal scaling exponents are calculated for the critical wave function of a two-dimensional Dirac fermion in the presence of a random magnetic field. It is shown that the problem of calculating the multifractal spectrum maps into the thermodynamics of a static particle in a random potential. The multifractal exponents are simply given in terms of thermodynamic functions, such as free energy and entropy, which are argued to be self-averaging in the thermodynamic limit. These thermodynamic functions are shown to coincide exactly with those of a generalized random energy model, in agreement with previous results obtained using Gaussian field theories in an ultrametric space.