Non-Hermitian Rosenzweig-Porter random-matrix ensemble: Obstruction to the fractal phase

Understand

We study the stability of non-ergodic but extended (NEE) phases in non-Hermitian systems.

• For this purpose, we generalize a so-called Rosenzweig-Porter random-matrix ensemble (RP), known to carry a NEE phase along with the Anderson localized and ergodic ones, to the non-Hermitian case.
• We analyze, both analytically and numerically, the spectral and multifractal properties of the non-Hermitian case.
• We show that the ergodic and the localized phases are stable against the non-Hermitian nature of matrix entries.