Eigenvalues in the non-Hermitian Anderson model - art. no. 165108

We study the effect of a non-Hermitian field it on the eigenvalues of a rin g 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 fluctuati ng site energies. We obtain the relation between the real and the imaginary parts of the averaged eigenvalues for arbitrary field strength. In particu lar, we identify a characteristic intermediate field value h(1), which sepa rates 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 simu lation results at high fields. At low fields the agreement is expected to b e only qualitative because of restrictions on the validity of the perturbat ion expansion for weak disorder at the energies of interest. We also compar e our results with an earlier theoretical treatment for weak disorder and l ow fields, h-->0.