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

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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×L×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=Wc≃6 in three dimension. Here we numerically diagonalize TME L×L×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 Nc(L,W) eigenvalues. We find that Nc(L,W) is proportional to L and grows with decreasing W similarly to the number of energy levels Nc in the Thouless energy band of the Anderson model.

Original languageEnglish (US)
Article number064212
JournalPhysical Review B
Issue number6
StatePublished - Aug 1 2020

Bibliographical note

Funding Information:
We are grateful to A. Kamenev and I. K. Zharekeshev for useful discussions. Calculations by Y.H. were supported primarily by the National Science Foundation through the University of Minnesota MRSEC under Award Nos. DMR-1420013 and DMR-2011401. The authors acknowledge the Minnesota Supercomputer Institute (MSI) at the University of Minnesota for providing resources that contributed to the research results reported within this paper.

Publisher Copyright:
© 2020 American Physical Society.

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