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

We study the stability of nonergodic but extended (NEE) phases in non-Hermitian systems. For this purpose, we generalize the so-called Rosenzweig-Porter random-matrix ensemble, 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 localized phases are stable against the non-Hermitian nature of matrix entries. However, the stability of the fractal phase depends on the choice of the diagonal elements. For purely real or imaginary diagonal potential, the fractal phase is intact, while for a generic complex diagonal potential the fractal phase disappears, giving way to a localized one.