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Random Dirac fermions in a two-dimensional space are studied numerically. Werealize them on a square lattice using the $\pi$-flux model with randomhopping. The system preserves two symmetries, the time-reversal symmetry andthe symmetry denoted by ${{\cal H},\gamma}=0$ with a $4\times 4$ matrix $\gamma$ in an effective field theory. Although it belongs to the orthogonal ensemble,the zero-energy states do not localize and become critical. The density ofstates vanishes at zero energy as $\sim E^{\alpha}$ and the exponent $\alpha$changes with strength of the randomness, which implies the existence of thecritical line. Rapid growth of the localization length near zero energy issuggested and the eigenstates near zero energy exhibit anomalous behaviourwhich can be interpreted as a critical slowing down in the availablefinite-size system. The level-spacing distributions close to zero energydeviate from both the Wigner surmise and the Poissonian, and exhibit criticalbehaviour which reflects the existence of critical states at zero energy.Comment: latex209, REVTEX, 6 figure

Near Critical States of Random Dirac Fermions