Random Matrix Theory with Non - integer β

We show that the random matrix theory with non-integer " symmetry parameter " β describes the statistics of transport parameters of strongly disordered two dimensional systems. The application of the random matrix theory (RMT) [1] to electronic transport in weakly disordered systems [2] enables us to understand the main features of the transport. The small number of parameters which enters RMT explains the universal features of transport, especially the universal conductance fluctuations as well as the simple (linear) form of the spectra of eigenvalues of the transfer matrix [3]. There are only three parameters which characterize the system in the RMT: the system size L, the mean free path l. and the symmetry parameter β. It is believed that only three integer values of β have physical meaning: β = 1, 2, 4 for orthogonal, unitary and symplectic symmetry, respectively. Recently, however, Muttalib and Klauder [4] showed that the non-integer values of β are also consistent with the DMPK equation [5]. They introduced a new parameter γ, which substitute β in the DMPK equation. γ is determined by statistical properties of eigenvectors and eigenvalues of the transfer matrix. Consequently, it could possess any positive real value, although the physical symmetry of the system remains unchanged. This observation indicates that the electronic transport in disordered systems outside the weak disorder limit could be described by the the RMT model with non-integer symmetry parameter. The same hyputhesis has been formulated in [6] on the basis of numerical studies of strongly anisotropic disordered systems This conclusion also corresponds with the hypothesis [7], that any RMT which pretends to describe strongly localized regime, should contain non-integer " symmetry parameter " β ∼ ξ/L (1) where L is the system size and ξ is scaling parameter (= localization length in the limit of strong disorder). In this Letter we study the most simple random matrix model with non-integer parameter β. To characterize the statistics of transport, we use quantities z i i = 1, 2,. .. N (N is the number of channels) which determine the eigenvalues λ i of the matrix t † t (t is the 1