Spectral rigidity of non-Hermitian symmetric random matrices near the Anderson transition

We study the spectral rigidity of the non-Hermitian analog of the Anderson model suggested by Tzortzakakis, Makris, and Economou (TME). This is a $L\times L\times L$ tightly bound cubic lattice, where both real and imaginary parts of onsite energies are independent random variables uniformly distributed between $-W/2$ and $W/2$. The TME model may be used to describe a random laser. In a recent paper we proved that this model has the Anderson transition at $W=W_c\simeq 6$ in three dimension. Here we numerically diagonalize TME $L\times L\times L$ cubic lattice matrices and calculate the number variance of eigenvalues in a disk of their complex plane. We show that on the metallic side $W<6$ of the Anderson transition, complex eigenvalues repel each other as strongly as in the complex Ginibre ensemble only in a disk containing $N_c(L,W)$ eigenvalues. We find that $N_c(L,W)$ is proportional to $L$ and grows with decreasing $W$ similarly to the number of energy levels $N_c$ in the Thouless energy band of the Anderson model.

Figure 1

${\mathrm{\Sigma }}^2/\langle N\rangle$ as a function of $\langle N\rangle$ for 1000 realizations of a cube $16\times 16\times 16$ for different $W$. The blue solid line represents the analytical result for the complex Ginibre ensemble ${\mathrm{\Sigma }}^2/\langle N\rangle ={\left(\pi \langle N\rangle \right)}^{-1/2}$ [27, 28, 29]. The dashed line shows ${\mathrm{\Sigma }}^2/\langle N\rangle =1.5{\left(\pi \langle N\rangle \right)}^{-1/2}$. Numbers of strongly repelling each other eigenvalues $N_c(16,W)$ are defined by crossing points of the dashed line and extrapolated line through the data points for a given $W$.

Figure 2

The characteristic number $N_c$ as a function of $W$ for $L=8,12,16$.

Figure 3

The ratio $N_c/L$ as a function of $W$ for $L=8,12,16$.