Phase diagram of disordered fermion model on two-dimensional square   lattice with $\pi$-flux
  A fermion model with random on-site potential defined on a two-dimensional square lattice with $\pi$-flux is studied. The continuum limit of the model near the zero energy yields Dirac fermions with random potentials specified by four independent coupling constants. The basic symmetry of the model is time-reversal invariance. Moreover, it turns out that the model has enhanced (chiral) symmetry on several surfaces in the four-dimensional space of the coupling constants. It is shown that one of the surfaces with chiral symmetry has Sp(n)$\times$Sp(n) symmety whereas others have U(2n) symmetry, both of which are broken to Sp(n), and the fluctuation around a saddle point is described, respectively, by Sp($n)_2$ WZW model and U(2n)/Sp(n) nonlinear sigma model. Based on these results, we propose a phase diagram of the model. 
T. Fukui