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

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^dagger TRS$^dagger$, 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^dagger with signs +1 and -1, corresponding to symmetry classes AI^dagger and AII^dagger, 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. Performing the replica limit n→ 0, we qualitatively reproduce the density of states and level-level correlations of non-Hermitian random matrices with TRS^dagger.