Phase-space Generalized Brillouin Zone for spatially inhomogeneous non-Hermitian systems

The generalized Brillouin zone (GBZ) has been highly successful in characterizing the topology and band structure of non-Hermitian systems. However, its applicability has been challenged in spatially inhomogeneous settings, where the non-locality of non-Hermitian pumping competes with Wannier-Stark localization and quantum interference, potentially leading to highly non-exponential state accumulation. To transcend this major conceptual bottleneck, we develop a general phase-space GBZ formalism that encodes non-Bloch deformations in both position and momentum space, such as to accurately represent spatially inhomogeneous non-Hermitian pumping. A key new phenomenon is the bifurcation of the phase-space GBZ branches, which allows certain eigenstates to jump abruptly between different GBZ solutions at various points in real space. The freedom in the locations of such jumps opens up an emergent degree of freedom that protects the stability of real spectra and, more impressively, the robustness of a new class of topological zero modes unique to GBZ bifurcation.The response from these novel spectral and GBZ singularities can be readily demonstrated in mature metamaterial platforms such as photonic crystals or circuit arrays, where effective real-space hoppings can be engineered in a versatile manner.Our framework directly generalizes to more complicated unit cells and further hoppings, opening up a vast new arena for exploring unconventional spectral and topological transitions as well as GBZ fragmentation in spatially inhomogeneous non-Hermitian settings.