Geometry-adaptive formulation of non-Bloch bands in arbitrary dimensions and spectral instability

Non-Hermitian skin effect (NHSE) is a distinctive phenomenon in non-Hermitian systems, marked by the accumulation of eigenstates at system boundaries. While well understood in one dimension, unraveling the NHSE in higher dimensions is challenging due to the diversity of lattice geometries. Here, we present a geometry-adaptive non-Bloch band theory in arbitrary dimensions, through the lens of spectral potential. Our formulation precisely determines the energy spectra and generalized Brillouin zone in the thermodynamic limit, unveiling their geometric dependency. We establish exact spectral relations that elucidate the geometric dependence of non-Bloch spectra and their connection to amoeba spectra. Moreover, we show that critical skin modes in higher dimensions exhibit scale-free localization along the boundary, thereby making the spectrum highly sensitive to the system size and boundary ratios. We demonstrate that these critical skin modes lead to spectral non-convergence and instability. Our findings pave the way toward a unified understanding of NHSE and non-Bloch bands in arbitrary dimensions. In non-Hermitian quantum systems, the skin effect causes states to cluster at the edges, which is straightforward in one dimension but gets complicated in higher dimensions because of different lattice shapes. Here, the authors report a flexible theory that adapts to any dimension and shape, revealing how these factors influence energy patterns and lead to unstable edge states sensitive to system size.