Spectral and Entanglement Transitions from Non-Hermitian Skin Pumping

Non-Hermitian physics has unveiled unconventional spectral, topological and critical phenomena, challenging traditional band theories. This thesis advances its understanding in three aspects. First, the non-Hermitian skin effect (NHSE) is shown to protect real spectra in some ansatz models, not relying on crystal symmetries. Second, we discover the phenomenon of scaling-induced non-Hermitian exceptional criticality (SIEC), marked by a unconventional negative dip in entanglement entropy scaling, deviating from the well-established logarithmic behavior. A scaling-dependent generalized Brillouin zone (GBZ) is developed to analytically predict this SIEC. Third, we formulate a theoretical framework for phase-space GBZs, extending the concept of the GBZ to position-dependent systems, particularly for those with spatially inhomogeneous NHSE hoppings. Unprecedented phenomena, including GBZ bifurcation which also protects the stability of real spectra, are revealed, introducing new forms of topological robustness governed by phase-space GBZ bifurcations. This thesis also explores the NHSE in Bethe lattices, where the hyperbolic-like lattice geometry gives rise to a hierarchy of loop sizes that leads to new forms of critical NHSE behavior.