Critical generalized inverse participation ratio distributions

TL;DR

This study numerically investigates the fluctuations of generalized inverse participation ratios at criticality, revealing scale-invariance and algebraic decay of finite size corrections, with implications for self-averaging properties.

Contribution

It provides new insights into the size dependence and fluctuation behavior of generalized IPRs at critical points, including symmetry and dimensionality effects.

Findings

Variances of IPR logarithms are scale-invariant at large sizes.

Finite size corrections decay algebraically with nontrivial exponents.

Large-q variances grow as q^2, supporting self-averaging of generalized dimensions.

Abstract

The system size dependence of the fluctuations in generalized inverse participation ratios (IPR's) $I_{\alpha}(q)$ at criticality is investigated numerically. The variances of the IPR logarithms are found to be scale-invariant at the macroscopic limit. The finite size corrections to the variances decay algebraically with nontrivial exponents, which depend on the Hamiltonian symmetry and the dimensionality. The large-$q$ dependence of the asymptotic values of the variances behaves as $q^2$ according to theoretical estimates. These results ensure the self-averaging of the corresponding generalized dimensions.