Spectral crossover in non-hermitian spin chains: comparison with random matrix theory

We systematically study the short range spectral fluctuation properties of three non-hermitian spin chain hamiltonians using complex spacing ratios. In particular we focus on the non-hermitian version of the standard one-dimensional anisotropic XY model having intrinsic rotation-time-reversal ($\mathcal{RT}$) symmetry that has been explored analytically by Zhang and Song in [Phys.Rev.A {\bf 87}, 012114 (2013)]. The corresponding hermitian counterpart is also exactly solvable and has been widely employed as a toy model in several condensed matter physics problems. We show that the presence of a random field along the $x$-direction together with the one along $z$ facilitates integrability and $\mathcal{RT}$-symmetry breaking leading to the emergence of quantum chaotic behaviour indicated by a spectral crossover resembling Poissonian to Ginibre unitary ensemble (GinUE) statistics of random matrix theory. Additionally, we consider two $n \times n$ dimensional phenomenological random matrix models in which, depending upon crossover parameters, the fluctuation properties measured by the complex spacing ratios show an interpolation between 1D-Poisson to GinUE and 2D-Poisson to GinUE behaviour. Here 1D and 2D Poisson correspond to real and complex uncorrelated levels, respectively.