Eigenvalues in the non-Hermitian Anderson model

Abstract

We study the effect of a non-Hermitian field h on the eigenvalues of a ring of one-orbital tight-binding sites for weak disorder of the site energies. The eigenvalue equation is expressed in terms of transfer matrices in the site representation and is solved exactly to second order in the fluctuating site energies. We obtain the relation between the real and the imaginary parts of the averaged eigenvalues for arbitrary field strength. In particular, we identify a characteristic intermediate field value ${h}_{1},$ which separates domains $h<{h}_{1}$ and $h>{h}_{1}$ in which the energy thresholds beyond which complex eigenvalues disappear from the spectrum have quite different forms. Our high-field threshold is in good agreement with earlier numerical simulation results at high fields. At low fields the agreement is expected to be only qualitative because of restrictions on the validity of the perturbation expansion for weak disorder at the energies of interest. We also compare our results with an earlier theoretical treatment for weak disorder and low fields, $\stackrel{\ensuremath{\rightarrow}}{h}0.$