Assessment of spectral phases of non-Hermitian quantum systems through complex and singular values

Chaotic behavior or lack thereof in non-Hermitian systems is often diagnosed via spectral analysis of associated complex eigenvalues. Very recently, singular values of the associated non-Hermitian systems have been proposed as an effective measure to study dissipative quantum chaos. Motivated by the rich properties of non-Hermitian power-law banded random matrices and its promise as a platform to study localized and delocalized phases in non-Hermitian systems, we make an in-depth study to assess different spectral phases of these matrices through the lens of both complex eigenvalues and singular values. Remarkably, the results from complex spectra and singular value analysis are seemingly different, thereby necessitating caution while identifying different phases. We also exemplify our findings by studying a non-Hermitian Hamiltonian with a complex on-site disorder. Our work indicates that systems, where disorder is present both in the Hermitian and non-Hermitian segments of a Hamiltonian, are sensitive to the specific diagnostic tool that needs to be employed to study quantum chaos.