gl(N|N) Super-current algebras for disordered Dirac fermions in two dimensions

We consider the non-hermitian 2D Dirac Hamiltonian with (A) real random mass, imaginary scalar potential and imaginary gauge field potentials, and (B) arbitrary complex random potentials of all three kinds. In both cases this Hamiltonian gives rise to a delocalization transition at zero energy with particle–hole symmetry in every realization of disorder. Case (A) is in addition time-reversal invariant, and can also be interpreted as the random-field XY statistical mechanics model in two dimensions. The supersymmetric approach to disorder averaging results in current–current perturbations of gl(N|N) super-current algebras. Special properties of the gl(N|N) algebra allow the exact computation of the β-functions, and of the correlation functions of all currents. One of them is the Edwards–Anderson order parameter. The theory is “nearly conformal” and possesses a scale-invariant subsector which is not a current algebra. For N=1, in addition, we obtain an exact solution of all correlation functions. We also study the delocalization transition of case (B), with broken time reversal symmetry, in the Gade–Wegner (random-flux) universality class, using a sigma model whose target space is an analytic continuation of GL(N|N;C)/U(N|N), as well as its PSL(N|N) variant, and a corresponding generalized random XY model. For N=1 the sigma model is solved exactly and shown to be identical to the current–current perturbation. For the delocalization transitions (case (A) and (B)) a density of states, diverging at zero energy, is found.