Diffusion of electrons in three dimensional disordered systems is investigated numerically for all the three universality classes, namely, orthogonal, unitary and symplectic ensembles. The second moment of the wave packet < r 2 (t) > at the Anderson transition is shown to behave as ∼ t a (a ≈ 2/3). From the temporal autocorrelation function C(t), the fractal dimension D 2 is deduced, which is almost half the value of space dimension for all the universality classes.Metal-insulator transitions are one of the most extensively investigated subjects in condensed matter physics. Especially interesting is the quantum phase transition, where the transition is driven by changing the parameter of quantum systems instead of temperature.

The boundary condition dependence of the critical behavior for the three dimensional Anderson transition is investigated. A strong dependence of the scaling function and the critical conductance distribution on the boundary conditions is found, while the critical disorder and critical exponent are found to be independent of the boundary conditions.

The Anderson transition in a 3D system with symplectic symmetry is investigated numerically. From a one-parameter scaling analysis the critical exponent ν of the localization length is extracted and estimated to be ν = 1.3±0.2. The level statistics at the critical point are also analyzed and shown to be scale independent. The form of the energy level spacing distribution P (s) at the critical point is found to be different from that for the orthogonal ensemble suggesting that the breaking of spin rotation symmetry is relevant at the critical point. 71.30.+h, 71.55.J, 72.15.Rn, Since the original work by Anderson [1] there has been considerable interest in the metal-insulator transition in disordered electron systems [2,3]. Critical phenomena at the Anderson transition are conventionally classified according to the universality class into which the system falls: orthogonal, unitary or symplectic [4,5]. The Anderson transition has been studied intensively numerically, analytically and experimentally. Nevertheless, in spite of this effort, we believe it is fair to say that some aspects of the Anderson transition remain puzzling. For example, the critical exponent ν of the localization length has been estimated for three-dimensional (3D) systems with orthogonal [3] and unitary symmetry [6][7][8] by means of finite-size scaling. The estimated numerical values of the critical exponent for these two universality classes have turned out to be rather close to each other. At present it is unclear whether or not this is an accidental coincidence. An obvious question immediately presents itself; does this coincidence also occur for the symplectic symmetry class? And if so does this coincidence also hold for other characteristics of the critical point?Systems belonging to the symplectic universality class exhibit the Anderson transition even in two-dimensions (2D) [9][10][11][12], while systems belonging to the unitary and orthogonal classes do not, in general, exhibit any Anderson transition in 2D. Thus we also have the opportunity, when studying symplectic systems, to see how the critical behavior at the Anderson transition depends on the dimensionality of the system.The statistical properties of energy levels [4,13,14] in the vicinity of the Anderson transition, and at the critical point in particular, have attracted considerable attention recently [15][16][17][18][19][20][21][22][23][24]. On the metallic side of the transition it has been demonstrated that the energy level statistics can be described by random matrix theory [4,13,14]; for example, the spacing s between successive energy levels is well described by a distribution function P (s) which is quite close to the Wigner surmise. In contrast, on the insulating side of the transition, energy levels are uncorrelated and the spacing distribution is Poissonian. At the critical point, where the metal-insulator transition takes place, it has been claimed [15,16] that the energy level statistics are also universal but different from those predicted by random matrix theor...

Diffusion of electrons in a two-dimensional system in static random magnetic fields is studied by solving the time-dependent Schrödinger equation numerically. The asymptotic behaviors of the second moment of the wave packets and the temporal auto-correlation function in such systems are investigated. It is shown that, in the region away from the band edge, the growth of the second moment of the wave packets turns out to be diffusive, whereas the exponents for the power-law decay of the temporal auto-correlation function suggest a kind of fractal structure in the energy spectrum and in the wave functions. The present results are consistent with the interpretation that the states away from the band edge region are critical.

We investigate how the criticality of the quantum Hall plateau transition in disordered graphene differs from those in the ordinary quantum Hall systems, based on the honeycomb lattice with ripples modeled as random hoppings. The criticality of the graphene-specific n = 0 Landau level is found to change dramatically to an anomalous, almost exact fixed point as soon as we make the random hopping spatially correlated over a few bond lengths. We attribute this to the preserved chiral symmetry and suppressed scattering between K and K' points in the Brillouin zone. The results suggest that a fixed point for random Dirac fermions with chiral symmetry can be realized in free-standing, clean graphene with ripples.PACS numbers: 73.43.-f, 72.10.-d, 71.23.-k After the seminal observation of the anomalous quantum Hall effect (QHE) in graphene, [1,2,3] fascination expands with the graphene QHE. One crucial question that is not fully explored is: what exactly is the role of the chiral symmetry in the problem? This has to do with a most significant feature of double Dirac cones (at K and K' in the Brillouin zone) in graphene. Although a single Dirac cone would already imply a characteristic Landau level structure with the zero-energy level, if we really want to look at the effect of disorder on the graphene Landau levels, we have to go back to the honeycomb lattice for which the chiral (A-B sub-lattice) symmetry [4,5] and the associated valley (K and K') degrees of freedom enter as an essential ingredient. The effect of disorder should then be sensitive to the nature of disorder, i.e., bond disorder or potential disorder, which determines the presence or otherwise of the chiral symmetry [6,7,8], and whether the disorder is short-ranged or long-ranged, which controls the scattering between K and K' points.For Dirac fermions, effects of random gauge fields induced by ripples in the two-dimensional plane have been discussed [9,10,11], and the stability of zero modes has been argued in terms of the index theorem and the chiral symmetry [1,4,5,12]. More recently, the plateau-toplateau transition for random Dirac fermions has been discussed, where the particle-hole symmetry is shown to make the zero-energy Landau level robust [13]. As for the criticality, however, the result [13] shows nothing special about the n = 0 Landau level, but this is obtained for a model of the Dirac fermions for which the randomness is introduced as a scalar random potential, so the chiral symmetry is degraded.On the other hand, the actual randomness in graphene, even when atomically clean, is known to have ripples, i.e, long-ranged corrugation of the graphene plane [9]. In fact, while a monolayer graphene naively contradicts with the well-known theorem that two-dimensional crystals should be thermodynamically unstable, one explanation attributes the stability to the ripples [14]. In this sense, we can take the disorder coming from ripples in graphene as an intrinsic disorder. Since the ripples consist of random bending of the honeycomb lattice, its ma...