Anderson transition in three-dimensional systems with non-Hermitian disorder

We study the Anderson transition for three-dimensional (3D) N×N×N tightly bound cubic lattices where both real and imaginary parts of on-site energies are independent random variables distributed uniformly between -W/2 and W/2. Such a non-Hermitian analog of the Anderson model is used to describe random-laser medium with local loss and amplification. We employ eigenvalue statistics to search for the Anderson transition. For 25% smallest-modulus complex eigenvalues we find the average ratio r of distances to the first and the second nearest neighbor as a function of W. For a given N the function r(W) crosses from 0.72 to 2/3 with a growing W demonstrating a transition from delocalized to localized states. When plotted at different N all r(W) cross at W_c = 6.0 ± 0.1 (in units of nearest-neighbor overlap integral) clearly demonstrating the 3D Anderson transition. We find that in the non-Hermitian 2D Anderson model, the transition is replaced by a crossover.