The phase diagram of the multi-dimensional Anderson localization via analytic determination of Lyapunov exponents

Abstract

The method proposed by the present authors to deal analytically with the problem of Anderson localization via disorder [J.Phys.: Condens. Matter 14 (2002) 13777] is generalized for higher spatial dimensions D. In this way the generalized Lyapunov exponents for diagonal correlators of the wave function, ψ 2 n,m , can be calculated analytically and exactly. This permits to determine the phase diagram of the system. For all dimensions D > 2 one finds intervals in the energy and the disorder where extended and localized states coexist: the metal-insulator transition should thus be interpreted as a first-order transition. The qualitative differences permit to group the systems into two classes: low-dimensional systems (2 ≤ D ≤ 3), where localized states are always exponentially localized and high-dimensional systems (D ≥ Dc = 4), where states with non-exponential localization are also formed. The value of the upper critical dimension is found to be D0 = 6 for the Anderson localization problem; this value is also characteristic of a related problem -percolation.

Anderson localization: 2‐D system in an external magnetic field and the generalized diffusion approach

“…In other words, this is nothing but a generalization of our dispute on phase transitions [2][3][4][5][6] for the case of an external magnetic field. We demonstrate below that the model can be exactly solved using the analytical approach [2][3][4] for a certain range of magnitude of the control parameter (magnetic flux). As a result, the generally-accepted viewpoint on the metal-insulator transitions should be revised.…”

“…This is why the magnetic field problem solution requires a generalization of the mathematical formalism [2][3][4][5], to be discussed below.…”

“…In our previous papers [2][3][4] the Anderson localization was considered as the particular case of the generalized diffusion (see pp. 253-256 in Ref.…”

“…In the present paper, we apply our approach [2][3][4][5][6] to a 2-D system in an external magnetic field. The field could be included into a 2-D Hamiltonian by means of a Peierls factor [12].…”

“…The calculation of the joint correlators is discussed which permits to extract analytically information on the Lyapunov exponents and to draw the phase diagram. Since we extend our approach [2][3][4] for the case of magnetic field, we discuss mostly the necessary modifications of our approach. This is why knowledge of our previous papers is prerequisite here.…”