Physica B 249—251 (1998) 796—800 Disordered critical wave functions of two-dimensional Dirac fermions on a lattice Y. Hatsugai*, Y. Morita, X.-G. Wen, M. Kohmoto  Department of Applied Physics, University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo 113, Japan  Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA  Institute for Solid State Physics, University of Tokyo, 7-22-1 Roppongi Minato-ku Tokyo 106, Japan Abstract We discuss critical states in random Dirac fermions on a two-dimensional lattice. They are realized on a square lattice using the -flux state with link disorder. It has a special symmetry and all states are paired which is expressed as H, "0 in an effective theory of doubled Dirac Fermions with some 44 matrix . It enables us to treat quite large systems (up to 801801) without doubling numerically. The multifractal analysis shows the zero mode is given by a critical wave function. Non-zero energy states are also investigated numerically. The density of states vanishes at zero energy as & E and the exponent  changes with the strength of the randomness. Rapid growth of the localization length near zero energy is observed.  1998 Elsevier Science B.V. All rights reserved. Keywords: Critical wave function; Dirac fermions; Quantum phase transition * Corresponding author. Tel.: #81 3 3812 2111/6809; fax: #81 3 5689 8253; e-mail: hatsugai@coral.t.u-tokyo.ac.jp. 1. Introduction A random critical point in two-dimensions is known to appear in several quantum systems, the quantum Hall system and systems with spin—orbit coupling. Recently critical behavior of random Dirac fermions is also discussed. Dirac fermions often appear in condensed matter physics, for example, a transition between different quantum Hall states [1—3], two-dimensional graphite sheets [4], a mean-field theory of the t—J model (‘-flux state’) [5] and d-wave super- conductors [6]. Recently effects of the randomness on the Dirac fermions are discussed by several groups [6—14]. Possible appearance of non-localized states, critical states, was pointed out in [7]. It was also discussed that the critical points of the Dirac fermion with random gauge fields contain infinite number of relevant directions [8]. Thus, although the theory is exactly solvable at the critical point, it seems that it can never be realized in real or numerical systems. Nevertheless, we have realized this disordered criti- cal state in a lattice model using a tight-binding 0921-4526/98/$19.00  1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 3 2 0 - 2