Anomalous Non-Hermitian Skin Effects in Coupled Hermitian Chains with Cross-Coupling

In this work, we demonstrate the presence of an anomalous non-Hermitian skin effects which decay from both ends of a system consisting of two coupled Hermitian chains induced by non-reciprocal inter-chain cross-coupling. Another intriguing feature of the system is that its eigenenergy spectrum in thermodynamic limit deviates from the generalized Brillouin zone (GBZ), contrary to that of conventional non-Hermitian systems. The thermodynamic-limit energy spectrum is however restored to the GBZ by the presence of even an infinitesimal amount of gain and/or loss term to the system. In this case, the system exhibits a critical phenomena similar to that of coupled non-Hermitian chains, whereby the eigenspectrum starts to approach the GBZ beyond some critical size. Furthermore, the non Hermitian skin effect becomes less pronounced as the system size exceeds this critical size. We analytically explain these peculiar features which highlight the important role of gain and loss terms as well as inter-chain coupling in tuning non-Hermitian skin modes, thus suggesting a new avenue for the modulation of the skin mode characteristics of open systems.