Directed Quantum Chaos

TL;DR

This paper explores directed quantum chaos by mapping quantum disordered problems with an imaginary vector potential onto a supermatrix sigma-model, revealing their equivalence to non-Hermitian matrix theories and analyzing eigenvalue distributions.

Contribution

It introduces the concept of directed quantum chaos via a sigma-model approach and connects it to the theory of non-Hermitian matrices, providing explicit eigenvalue probability calculations.

Findings

The 0D sigma-model describes directed quantum chaos phenomena.

Problems are equivalent to non-Hermitian matrix theories.

The fraction of real eigenvalues remains finite in time-reversal invariant systems.

Abstract

Quantum disordered problems with a direction (imaginary vector-potential) are discussed and mapped onto a supermatrix sigma-model. It is argued that the $0D$ version of the sigma-model may describe a broad class of phenomena that can be called directed quantum chaos. It is demonstrated by explicit calculations that these problems are equivalent to problems of theory of random asymmetric or non-Hermitian matrices. A joint probability of complex eigenvalues is obtained. The fraction of states with real eigenvalues proves to be always finite for time reversal invariant systems.