Low dimensional disordered systems

Anderson localization remains one of the main problems in the physics of disordered systems. In the series of our recent papers [1-3] an exact analytic solution for the Anderson localization was presented. Note that under the exact solution we mean only the calculation of the phase diagram with the localization length. To solve this problem, we use the recursive (Schrödinger) equation for the Cauchy problem with fixed initial conditions. There are some limitations of the analytical theory. An exact solution is only possible for the conventional Anderson model with a diagonal disorder, where on-site potentials are independently and identically distributed. The transition from delocalized to localized states is treated as a generalized diffusion with a noise-induced first-order phase transition. The generalized diffusion manifests itself in the divergence of averages of wavefunctions (correlators). This divergence is controlled by the Lyapunov exponent γ, which is the inverse of the localization length, ξ=1/γ. For example, Fig.1 and Fig.2, 2-D system in an external magnetic field (q – number of Landau subbands).

Most of our results contradict the generally-accepted rules of the scaling theory and numerical modelling. Two particular results were highly unexpected in the Anderson community: (i) existence of the Anderson transition in two dimensional (2-D) disordered systems of noninteracting electrons, and (ii) the Anderson transition in N-D (N > 2) dimensions is of first-order, and localized and conducting states could co-exist (Fig.1). The result (i) formally does not contradict experiment where the unexpected presence of a metallic phase in 2-D was observed. Experiments of the last decade do not support the prevailing point of view that there can be no metallic state or metal-insulating transition in a 2-D system. The physics behind these observations at present is not understood.

 

References:
  1. V.N. Kuzovkov, W. von Niessen, V. Kashcheyevs, and O. Hein, Exact analytic solution for the generalized Lyapunov exponent of the two-dimensional Anderson localization. - J. Phys.: Cond. Matter, 2002, 14, p. 13777-13797.
  2. V.N. Kuzovkov and W. von Niessen, The phase diagram of the multi-dimensional Anderson localization via analytic determination of Lyapunov exponents. - Europ. Phys. J. B, 2004, 42, p. 529-542.
  3. V.N. Kuzovkov and W. von Niessen, Random walk approach to the analytic solution of random systems with multiplicative noise - The Anderson localization problem. - Physica A, 2006, 369, p. 251-265.