Edge theory of non-Hermitian skin modes in higher dimensions

In this paper, we establish an effective edge theory to characterize non-Hermitian edge-skin modes in higher dimensions. We begin by proposing a bulk projection criterion to straightforwardly identify the localized edges of skin modes. Through an exact mapping, we show that the edge-skin mode shares the same bulk-boundary correspondence and localization characteristics as the zero-energy edge states in a Hermitian semimetal under open-boundary conditions, bridging the gap between non-Hermitian edge-skin effect and Hermitian semimetals. Another key finding is the introduction of "skewness," a term we proposed to describe the characteristic decay direction of skin mode from the localized edge into the bulk. Remarkably, we demonstrate that skewness is an intrinsic quantity of the skin mode and can be analytically determined using the corresponding cylinder-geometry bulk Hamiltonian, without requiring any boundary details. Furthermore, we reveal that, in the edge-skin effect, the spectrum exhibits anomalous spectral sensitivity to weak local disturbances, a feature that crucially distinguishes it from the corner-skin effect.