Non-linear sigma models for non-Hermitian random matrices in symmetry classes AI † and AII †

Symmetry of non-Hermitian matrices underpins many physical phenomena. In particular, chaotic open quantum systems exhibit universal bulk spectral correlations classified on the basis of time-reversal symmetry † (TRS † ), coinciding with those of non-Hermitian random matrices in the same symmetry class. Here, we analytically study the spectral correlations of non-Hermitian random matrices in the presence of TRS † with signs +1 and −1, corresponding to symmetry classes AI † and AII † , respectively. Using the fermionic replica non-linear sigma model approach, we derive n-fold integral expressions for the nth moment of the one-point and two-point characteristic polynomials, valid for any matrix dimension. We also study, in the limit of large matrix dimensions, the replica limit n → 0 to derive the density of states and level-level correlations of non-Hermitian random matrices with TRS † . We compare our analytical findings with numerical results.