Anderson transition in three-dimensional systems with non-Hermitian disorder
Yi Huang, B. I. Shklovskiǐ
Abstract
We study the Anderson transition for three-dimensional (3D) $N\ifmmode\times\else\texttimes\fi{}N\ifmmode\times\else\texttimes\fi{}N$ tightly bound cubic lattices where both real and imaginary parts of on-site energies are independent random variables distributed uniformly between $\ensuremath{-}W/2$ 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 ${W}_{c}=6.0\ifmmode\pm\else\textpm\fi{}0.1$ (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.