Universal Fluctuations in Spectra of Disordered Systems at the Anderson Transition

Details Universal Fluctuations in Spectra of Disordered Systems at the Anderson Transition Universal Fluctuations in Spectra of Disordered Systems at the Anderson Transition

1995-06-25

arxiv.org/abs/cond-mat/9506114v1

Jpn. J. Appl. Phys. 34 (1995) 4361

Physics

Condensed Matter

9 pages of Revtex, 3 figures included (tar-compressed and uuencoded using script UUFILES), to appear in Jpn.J.Appl.Phys

Scientific paper

10.1143/JJAP.34.4361

Using the level--spacing distribution and the total probability function of the numbers of levels in a given energy interval we analyze the crossover of the level statistics between the delocalized and the localized regimes. By numerically calculating the electron spectra of systems of up to $32^3$ lattice sites described by the Anderson Hamiltonian it is shown that the distribution $P(s)$ of neighboring spacings is {\em scale- independent} at the metal-insulator transition. For large spacings it has a Poisson-like asymptotic form $P(s)\propto \exp (- A\,s/\Delta )$, where $A \approx 1.9$. At the critical point we obtain a linear relationship between the variance of the number of levels $\langle [\delta n(\varepsilon)]^2 \rangle $ and their average number $\langle n(\varepsilon)\rangle $ within the interval $\varepsilon$. The constant of proportionality is less than unity due to the repulsion of the levels. Both $P(s)$ and $\langle [\delta n(\varepsilon)]^2 \rangle $ are determined by the probability density $Q_{n}(\varepsilon)$ of having exactly $n$ levels in the energy interval $\varepsilon $. The distribution $Q_{n}(\varepsilon)$ at the critical point is found to be size--independent and to obey a Gaussian law near its maximum, where $n\sim \langle n \rangle $.

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