In the first part of the talk I will discuss the complex eigenvalue statistics of a XXZ spin chain with imaginary disorder, where we find an interplay between Hermiticity and integrability breaking at different scales of the disorder strength. We compare with the symmetry class of AI† and Poisson statistics in 1 ≤ D ≤ 2 dimension. In particular, the nearest- and next-to-nearest-neighbour spacing distributions of classes AI† and AII† are very well approximated by a two-dimensional Coulomb gas at β = 1.4 and 2.6, respec-tively.
In the second part of the talk, I will turn to the general study of the local bulk statistics of non-Hermitian random matrices. Based on numerically generated nearest-neighbour spacing distributions, it is conjectured that among all 38 non-Hermitian symmetry classes, only 3 different local bulk statistics exist. The simplest representatives are com-plex Ginibre matrices (class A), complex symmetric matrices (class AI†) and complex self-dual matrices (class AII†). While class A is well understood, only very few is known about the latter two classes. I will present the first analytic results for the expectation value of two characteristic polynomials in classes AI† and AII†. This includes results at finite matrix size as well as global and local edge and bulk asymptotics.