Logarithmic scaling of Lyapunov exponents in disordered chiral two-dimensional lattices

TL;DR

This study investigates the scaling behavior of Lyapunov exponents in disordered 2D chiral lattices, confirming a logarithmic energy dependence and validating theoretical predictions through numerical analysis.

Contribution

It provides the first numerical confirmation of the logarithmic scaling of Lyapunov exponents near the chiral critical point in disordered 2D lattices.

Findings

Lyapunov exponents depend on a single parameter ln L/ln xi(E)

Energy dependence of xi(E) diverges logarithmically as |E| approaches zero

Results agree with theoretical predictions by Fabrizio and Castelliani

Abstract

We analyze the scaling behavior of the two smallest Lyapunov exponents for electrons propagating on two-dimensional lattices with energies within a very narrow interval around the chiral critical point at E=0 in the presence of a perpendicular random magnetic flux. By a numerical analysis of the energy and size dependence we confirm that the two smallest Lyapunov exponents are functions of a single parameter. The latter is given by ln L/ln xi(E), which is the ratio of the logarithm of the system width L to the logarithm of the correlation length xi(E). Close to the chiral critical point and energy |E| << E_0, we find a logarithmically divergent energy dependence lnxi(E)proporitonal to |\ln(E_0/|E|)|^{1/2}, where E_0 is a characteristic energy scale. Our data are in agreement with the theoretical prediction of M. Fabrizio and C. Castelliani [Nucl.\Phys.B 583, 542 (2000)] and resolve an inconsistency of previous numerical work.