Critical non-Hermitian skin effect in a cross-coupled Hermitian chain

In this work, we demonstrate an anomalous non-Hermitian skin effect which decays from both ends of a system consisting of two coupled Hermitian chains induced by nonreciprocal interchain cross-coupling. An intriguing feature of the system is that its eigenenergy spectrum in the 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 in the system. In this case, the system exhibits a critical non-Hermitian skin effect (CNHSE) 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 interchain coupling in tuning non-Hermitian skin modes. Finally, we model the implementation of the model based on the topolectrical platform, using realistic circuit parameter values, to demonstrate the experimental feasibility of the model, thus suggesting a new avenue for the modulation of the skin mode characteristics of open systems.