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

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 $\text{ln}\text{ }L/\text{ln}\text{ }\ensuremath{\xi}(E)$, which is the ratio of the logarithm of the system width $L$ to the logarithm of the correlation length $\ensuremath{\xi}(E)$. Close to the chiral critical point and energy $|E|⪡{E}_{0}$, we find a logarithmically divergent energy dependence $\text{ln}\text{ }\ensuremath{\xi}(E)\ensuremath{\propto}{|\text{ln}({E}_{0}/|E|)|}^{1/2}$, where ${E}_{0}$ is a characteristic energy scale. Our data are in agreement with the theoretical prediction of Fabrizio and Castelliani [Nucl. Phys. B 583, 542 (2000)] and resolve an inconsistency of previous numerical work.