Near Critical States of Random Dirac Fermions

Abstract

Random Dirac fermions in a two-dimensional space are studied numerically. We realize them on a square lattice using the $\pi$-flux model with random hopping. The system preserves two symmetries, the time-reversal symmetry and the 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 of states vanishes at zero energy as $\sim E^{\alpha}$ and the exponent $\alpha$ changes with strength of the randomness, which implies the existence of the critical line. Rapid growth of the localization length near zero energy is suggested and the eigenstates near zero energy exhibit anomalous behaviour which can be interpreted as a critical slowing down in the available finite-size system. The level-spacing distributions close to zero energy deviate from both the Wigner surmise and the Poissonian, and exhibit critical behaviour which reflects the existence of critical states at zero energy.Comment: latex209, REVTEX, 6 figure

Narrow depression in the density of states at the Dirac point in disordered graphene

“…A disorder dependent vanishing of the density of states was previously reported for massless random Dirac fermions on a two-dimensional square lattice 43 and on a honeycomb lattice, 24 where non-diagonal disorder was introduced by real random hopping terms. These model systems preserve time reversal symmetry and therefore belong to the (chiral) orthogonal universality class.…”

Density of states of disordered graphene

“…20) seem to indicate a non-linear dependence of the value of the minimum as a function of the strength of the disorder potential, and is at least not inconsistent with Eq. (22).…”

“…In this paper we do not consider in this type of disorder, which can result from height fluctuations (ripples) in the graphene sheet and lead to qualitatively different results from the ones presented in this paper. 14,19,22,23 The rest of the paper is organized as follows: In Sec. II, we describe the approximation schemes that we use.…”

Electron-electron interactions in graphene sheets

“…A variety of analytical 15 and numerical techniques 16 has been used to study this problem. We will follow the renormalization group scheme presented in Ref.…”