Pure real energy spectrum and exact anomalous mobility edges in a non-Hermitian flat band geometry

The interplay between quasi-periodicity and non-Hermiticity can give rise to rich localization phenomena. In this work, we investigate the localization transition in a one-dimensional nonreciprocal cross-stitch flat band lattice with diagonal quasi-periodic mosaic modulation, which incorporates both constant and quasi-periodic potentials. In general, non-reciprocity induces the skin effect in non-Hermitian systems through nonreciprocal transitions. However, in this work, we find that in a non-reciprocal flat band lattice, when the constant potential is zero, the skin effect does not exist in the system, and the energy spectrum remains purely real and well-defined. In the presence of a non-zero constant potential, we derive analytical solutions for a class of anomalous mobility edges (AMEs) under periodic boundary conditions (PBCs), revealing the systems localization and critical properties. Through analytic results, we demonstrate that the system is fundamentally equivalent to a generalized non-Hermitian Aubry–André (AA) model. Importantly, this equivalence implies the existence of a transition from a critical phase to a localized phase, as predicted by the non-Hermitian AA model. However, our analysis reveals that the transition point from the critical phase to the localized phase is energy-dependent, which fundamentally accounts for the emergence of AMEs. Furthermore, we design a classical electrical circuit to experimentally realize our system. This work provides new insights into localization transitions in non-Hermitian flat band systems.