Optimal Fluctuations and Tail States of Non-Hermitian Operators

Abstract

A statistical field theory is developed to explore the density of states and spatial profile of 'tail states' at the edge of the spectral support of a general class of disordered non-Hermitian operators. These states, which are identified with symmetry broken, instanton field configurations of the theory, are closely related to localized sub-gap states recently identified in disordered superconductors. By focusing separately on the problems of a quantum particle propagating in a random imaginary scalar potential, and a random imaginary vector potential, we discuss the methodology of our approach and the universality of the results. Finally, we address potential physical applications of our findings.

Extended tail states in an imaginary random potential

“…This transition is usually referred as the mobility edge in 1d problem (which is already surprising [12]). In this paper we consider another example, where the localized(delocalized) eigenfunctions of non-Hermitean Hamiltonian behave in the "inverted", compared to the Hermitean case, way.Two recent papers [10,11] deals with the non-Hermitean quantum mechanical Hamiltonian with an imaginary random potentialThis study was motivated by the observation that the Euclidean evolution operator r| exp[−tH]|0 with H (1) after averaging over δ-correlated disordered potential V coincides with the probability distribution Z(r, t) for the Edwards self-repulsing polymer [13]. Many applications of non-Hermitean operators (like both our examples) come from the statistical physics and naturally deal with the imaginary time(Euclidean) evolution.…”

“…Two recent papers [10,11] deals with the non-Hermitean quantum mechanical Hamiltonian with an imaginary random potential…”

Random matrices close to Hermitian or unitary: overview of methods and results

“…All this makes random matrix models a very useful tool for getting insights into gross universal features of the analytical structure of the QCD partition function. In turn, information on such a structure is based on the knowledge of statistics of complex eigenvalues of the Dirac operator with non-vanishing chemical potential which stimulated a lot of research of such eigenvalues [38,39,40,41,42,43,44].Finally, it deserves to be briefly mentioned that objects related to weakly nonHermitian random matrices emerge also in the context of interesting problems of motion and localization of a quantum particle in disordered media subject to an imaginary vector potential [45] or a random scalar potential [46]. The first of these problems is related to the motion of superconductor flux lines in a sample with columnar defects, the second to the motion of a self-repelling polymer chain and other interesting problems (see the cited references).…”

“…Finally, it deserves to be briefly mentioned that objects related to weakly nonHermitian random matrices emerge also in the context of interesting problems of motion and localization of a quantum particle in disordered media subject to an imaginary vector potential [45] or a random scalar potential [46]. The first of these problems is related to the motion of superconductor flux lines in a sample with columnar defects, the second to the motion of a self-repelling polymer chain and other interesting problems (see the cited references).…”

Optimal fluctuations and tail states of non-Hermitian operators

“…As a result, we can immediately deduce the homogeneous mean-field solution (14) as well as the inhomogeneous instanton solution (13) of the saddle-point equation. Thus, from the homogeneous mean-field solution, one obtains the expression (16) for the DoS, i.e. the complex DoS is flat and non-vanishing over the interval…”