Non-Hermitian systems exhibit unique spectral properties, including the non-Hermitian skin effect and exceptional points, often influenced by boundary conditions. The modulation of these phenomena by generalized boundary conditions remains unexplored and not understood. Here, we analyze the Hatano-Nelson model with generalized boundary conditions induced by complex hopping amplitudes at the boundary. Using similarity transformations, we determine the conditions yielding real energy spectra and skin effect, and identify the emergence of exceptional points where spectra transition from real to complex. We demonstrate that tuning the boundary hopping amplitudes precisely controls the non-Hermitian skin effect, i.e., the localization of eigenmodes at the lattice edges. These findings reveal the sensitivity of spectral and localization properties to boundary conditions, providing a framework for engineering quantum lattice models with tailored spectral and localization features, with potential applications in quantum devices.
Non-Hermitian systems exhibit unique spectral properties, including the non-Hermitian skin effect and exceptional points, often influenced by boundary conditions. The modulation of these phenomena by generalized boundary conditions remains unexplored and not understood. Here, we analyze the Hatano-Nelson model with generalized boundary conditions induced by complex hopping amplitudes at the boundary. Using similarity transformations, we determine the conditions yielding real energy spectra and skin effect, and identify the emergence of exceptional points where spectra transition from real to complex. We demonstrate that tuning the boundary hopping amplitudes precisely controls the non-Hermitian skin effect, i.e., the localization of eigenmodes at the lattice edges. These findings reveal the sensitivity of spectral and localization properties to boundary conditions, providing a framework for engineering quantum lattice models with tailored spectral and localization features, with potential applications in quantum devices.
Quasiperiodic lattices can host critical eigenstates, whereas nonreciprocal hopping in non-Hermitian lattices can induce non-Hermitian skin effect. In this work, we investigate localization phenomena in a Hatano--Nelson model with quasiperiodically modulated hopping amplitudes, where nonreciprocity arises from unequal modulation strengths of the right and left hoppings. Using a non-unitary gauge transformation, we map the non-Hermitian system into a Hermitian quasiperiodic system and obtain an exact analytical expression for the Lyapunov exponent in the thermodynamic limit. Under periodic boundary conditions, inverse participation ratios and finite-size scaling analysis are used to identify the quasiperiodic critical regimes. The comparison shows that parameter regimes hosting quasiperiodic critical states under periodic boundary conditions can exhibit the non-Hermitian skin effect under open boundary conditions. Furthermore, the non-Hermitian skin effect associated with quasiperiodic critical regimes is also observed in representative long-range hopping models and multiband extensions. Our results provide an analytically controlled perspective on how quasiperiodicity, modulated nonreciprocity, and boundary conditions jointly shape the non-Hermitian skin effect in critical regimes.
Symmetry-protected ideal flat bands in one-dimensional (1D) Hermitian lattices are populated by compact localized states (CLS) - a special class of localization with wavefunctions confined within a small region. In this work, we discover that the non-Hermitian skin effect (NHSE) can appear in a flat band. Unlike conventional NHSEs for dispersive bands that are protected by nontrivial point-gap topology, the flat band remains a point on the complex-energy plane and is therefore always topologically trivial. We found that, intriguingly, the flat-band skin effect (FBSE) is associated with the non-trivial spectral topology of the dispersive bands enclosing the flat band on the complex-energy plane, so it only emerges within a finite range of non-Hermitian parameters and can counterintuitively disappear at large non-Hermiticity. Moreover, the gaps between the flat and the dispersive bands can close at higher-order exceptional points under both periodic and open boundary conditions. The flat-band wavefunctions are discontinuous in quantum distance across these exceptional points, signifying that the gap-closing is singular. The FBSE was experimentally observed in a non-Hermitian mechanical lattice. Our work reveals flat-band phenomena unique to non-Hermitian systems and highlights new possibilities in quantum geometry and localization control.
Non-Hermitian skin effect, which is a unique feature of non-Hermitian systems, exhibits the formation of an extensive number of boundary modes under open boundary conditions. However, its manifestation in higher dimensions remains elusive. In our work, we demonstrate a hybrid skin-topological effect arising from the interplay between first-order band topology and non-reciprocal hopping in an engineered two-dimensional brick-wall geometry. The non-Hermitian brick-wall lattice under open boundary conditions in both directions exhibits several unconventional spectral features. Notably, the eigenvalues associated with the corner skin modes do not exhibit non-trivial windings in the complex energy plane; instead, they exhibit dynamically stable exceptional point-like features that do not originate from eigenvector coalescence. In contrast, the remaining modes accumulate at the opposite pair. Of all the corner skin modes, only the two that originate from the topological corner states of the Hermitian brick-wall lattice remain localized at individual corners, while the rest accumulate at the pair of opposite corners. This spatial distribution contrasts sharply with the second-order skin effect, where corner skin modes are more uniformly distributed. Finally, for the non-Hermitian Brick-wall lattice, we design and implement the corresponding topolectrical circuit (circuit for a square lattice is included for comparison) to directly visualize the hybrid skin-topological modes.
The non-Hermitian skin effect is nonreciprocity-induced localization phenomena in which a macroscopic number of eigenstates accumulate anomalously at the boundary, accompanied by the extreme sensitivity to boundary conditions. Here, we develop a geometric characterization of the non-Hermitian skin effect. We demonstrate that the localization length scale associated with the skin effect is encoded in the quantum metric defined solely from right eigenstates, but not in the biorthogonal quantum metric. Moreover, we show that the quantum metrics can exhibit the power-law divergences at gapless points that depend on the different boundary conditions. We also reveal that cusps of the generalized Brillouin zone in non-Bloch band theory are signaled by discontinuities in the quantum metrics. We illustrate these behavior using prototypical non-Hermitian models, such as the Hatano-Nelson model and the non-Hermitian, nonreciprocal Su-Schrieffer-Heeger model.
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.
Non-Hermitian skin effects are conventionally manifested as boundary localization of eigenstates in linear systems. In nonlinear settings, however, where eigenstates are no longer well defined, it becomes unclear how skin effects should be faithfully characterized. Here, we propose a Koopman-based characterization of nonlinear skin effects, in which localization is defined in terms of Koopman eigenfunctions in a lifted observable space, rather than physical states. Using a minimal nonlinear extension of the Hatano-Nelson model, we show that dominant Koopman eigenfunctions localize sharply on higher-order observables, in stark contrast to linear skin effects confined to linear observables. This lifted-space localization governs the sensitivity to boundary amplitude perturbations, providing a distinct dynamical signature of the nonlinear skin effect. Our results establish the Koopman framework as a natural setting in which skin effects unique to nonlinear non-Hermitian systems can be identified.