No abstract

A general method to describe a second-order phase transition is discussed. sider a yet undefined characteristic quantity α(M, W ) which shall be chosen in such a way that:whereα M e ,α In andα Cr are different constants for the metallic side, the insulating side and at the critical point, respectively.With respect to the W dependence α(M, W )is singular for M = ∞, but analytical otherwise because all the singularities connected with the phase transition are rounded for a finite size. As α(M, W ) describes a system which exhibits a second order phase transition with characteristic length ξ ∞ , Eq.(1) suggests generally, following the work of Brézin, 13 a finite-size scaling be-But as α(M, W ) is analytical for finite M, f can be expanded around the critical point asfor any quantity α which fulfills the condition (2) with a singularity at W c for M → ∞ and which obeys the scaling relation α(M, W ) = f (M/ξ ∞ (W )). We note that one has to consider a finite-size scaling 14 procedure so that the size of the system is kept finite, suitable for numerical work, and not usual scaling 15for which an infinite system has to be considered.Before defining α(M, W ) we recall how P (s) and ∆ 3 were computed. 7 The energy spectra were obtained from the Anderson To analyse the scaling of the statistical properties of the eigenvalue spectrum, starting from P (s) an evident choice for α, which satisfies the expected finite-size behaviour, would be α = ∞ 0 P (s)ds. But because P (s) is normalized this makes no sense. A solution is provided by the appearance of a fixed point at s 0 ≃ 2 when one plots P (s) for different disorders W and sizes M. 6,7 So one can choose α P (s) (M, W ) = ∞ s 0 P (s)ds, with s 0 ≃ 2 as already quoted. 6 An equivalent choice 7 would be α P (s) (M, W ) = s 0 0 P (s)ds.While these choices are formally possible it is much better from a numerical point of view to consider the cumulative level-spacing distribution I(s) = s 0 P (s ′ )ds ′ as plotted in Fig.1 instead of P (s) because the most significant changes which occur in P (s) for small s are emphasized and because I(s) is much smoother. Using I(s) we chooseThe results in Fig

The Thouless conjecture states that the average conductance of a disordered metallic sample in the diffusive regime can be related to the sensitivity of the sample's spectrum to a change in the boundary conditions.Here we present results of a direct numerical study of the conjecture for the Anderson model. They were obtained by calculating the Landauer-Büttiker conductance g L for a sample connected to perfect leads and the distribution of level curvatures for the same sample in an isolated ring geometry, when the ring is pierced by an Aharonov-Bohm flux. In the diffusive regime (L ≫ l e ) the average conductance g L is proportional to the mean absolute curvature |c| : g L = π |c| /∆, provided the system size L is large enough, so that the contact resistance can be neglected. l e is the elastic mean free path, ∆ is the mean level spacing. When approaching the ballistic regime, the limitation of the conductance due to the contact resistance becomes essential and expresses itself in a deviation from the above proportionality. However, in both regimes and for all system sizes the same proportionality is recovered when the contact resistance is subtracted from the inverse conductance, showing that the curvatures measure the conductance in the bulk. In the localized regime, the mean logarithm of the absolute curvature and the mean logarithm of the Landauer-Büttiker conductance are proportional.

-We present a new method for the numerical treatment of second order phase transitions using the level spacing distribution function P (s). We show that the quantities introduced originally for the shape analysis of eigenvectors can be properly applied for the description of the eigenvalues as well. The position of the metal-insulator transition (MIT) of the three dimensional Anderson model and the critical exponent are evaluated. The shape analysis of P (s) obtained numerically shows that near the MIT P (s) is clearly different from both the Brody distribution and from Izrailev's formula, and the best description is of the form P (s) = c 1 s exp(−c 2 s 1+β ), with β ≈ 0.2. This is in good agreement with recent analytical results.

The critical behaviour of 3-dimensional disordered systems with magnetic field is investigated by analyzing the spectral fluctuations of the energy spectrum. We show that in the thermodynamic limit we have two different regimes, one for the metallic side and one for the insulating side with different level statistics. The third statistics which occurs only exactly at the critical point is independent of the magnetic field. The critical behaviour which is determined by the symmetry of the system at the critical point should therefore be independent of the magnetic field.