Relation between energy-level statistics and phase transition and its application to the Anderson model

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

Energy-level statistics at the metal-insulator transition in anisotropic systems

“…[9][10][11][12][13]17 On the insulating side of the MIT, one finds that localized states that are close in energy are usually well separated in space whereas states that are localized in vicinal regions in space have well separated eigenvalues. Consequently, the eigenvalues on the insulating side are uncorrelated, there is no level repulsion and the probability of eigenvalues to be close together is high.…”

“…ELS has been previously applied with much success at the MIT of the isotropic model and it was shown that a size-independent statistics exists at the MIT. 10,12,13 This critical statistics is intermediate between the two limiting cases of Poisson statistics for the localized states and the statistics of the Gaussian orthogonal ensemble (GOE) which describes the spectrum of extended states. 14 We check whether the critical ELS is influenced by the anisotropy.…”

“…In the present work, we focus our attention on the critical properties of this MIT. As an independent check to previous results, we employ energy level statistics (ELS) [9][10][11][12] together with the finite-size scaling (FSS) approach. ELS has been previously applied with much success at the MIT of the isotropic model and it was shown that a size-independent statistics exists at the MIT.…”

Deconfinement, chiral transition and localisation in a QCD-like model

“…In a finite volume I s 0 (λ) interpolates smoothly between the two limiting values, with the transition becoming sharper as the volume is increased. At the "mobility edge", λ c , I s 0 (λ) is volume independent [45,46], and takes the critical value I crit s 0 . In ref.…”

“…A particularly convenient observable in this respect is the integrated probability distribution function, I s 0 (λ) [45,46], 0.117. In a finite volume I s 0 (λ) interpolates smoothly between the two limiting values, with the transition becoming sharper as the volume is increased.…”