Near Critical States of Random Dirac Fermions

Two-dimensional random Dirac fermions are studied numerically. They are realized on a square lattice by the $\pi$-flux model with random hopping. It preserves a symmetry denoted by $\{H,\gamma \}=0\mathrm{}$ in an effective field theory. Although it belongs to the orthogonal ensemble, the zero-energy states do not localize but become critical. The density of states vanishes as $\sim E^{\alpha }$ and the exponent $\alpha$ changes with strength of the randomness (the critical line). Rapid enhancement of the Thouless number is observed near the zero energy. The level-spacing distribution is also investigated, which is consistent with the existence of the critical states at $E=0\mathrm{}$.