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Communications Physics

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

Ze-Yu Xing, Yuncheng Xiong, Haiping Hu

Published Mar 2, 2026• 1 citation

Abstract

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.

DOI: 10.1038/s42005-026-02546-2

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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.

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.

We formulate quantum geometry for non-Hermitian systems under open boundary conditions. By defining quantum-geometric quantities in both real-space and non-Bloch representations, we establish a unified framework beyond conventional Bloch band theory. Our central result is an exact equivalence between the real-space integrated quantum metric and a non-Bloch integrated quantum metric defined on the generalized Brillouin zone. We further introduce localized non-Bloch Wannier functions in the presence of the non-Hermitian skin effect and show that the non-Bloch integrated quantum metric gives the gauge-invariant part of their spread functional. These results establish quantum geometry as a natural framework for characterizing open-boundary non-Hermitian band structures and the localization properties encoded in skin modes.

The non-Hermitian skin effect (NHSE), characterized by a macroscopic accumulation of eigenstates at the edge of a system with open boundaries, is often ascribed to a non-trivial point-gap topology of the Bloch Hamiltonian. We revisit this connection and separate the question of the NHSE as a spectral reconstruction effect in clean systems from the question of stable topological protection. For a Hatano-Nelson ladder, where point-gap winding and non-normality can be varied independently, we demonstrate that, in a clean translationally invariant multiband setting, the NHSE can occur without point-gap winding and, conversely, that point-gap winding can persist without the NHSE. These results establish that in the clean case the connection between point-gap winding and the NHSE only holds in the scalar one-band case but, in general, not in the multiband case. Even more importantly, the eigenspectrum of non-normal operators is generically highly sensitive to boundary conditions and perturbations, and therefore does not constitute a stable object encoding topological information. Instead, topological properties are reflected in the splitting of the singular-value spectrum for finite systems and, in the semi-infinite limit, correspond to boundary-localized kernel modes implied by the index of the corresponding Toeplitz operator.

The interplay between the non-Hermiticity and dimensionality gives rise to exotic characteristics in higher dimensions, with one representative phenomenon known as the geometry-dependent skin effect (GDSE), which refers to that the localization of extensive eigenstates depends on the system's geometry under open boundary conditions. In this paper, we demonstrate the emergence of GDSE in a two-dimensional

sp

optical ladder lattice with on-site atom loss, which can be manifested by anisotropic dynamics of Bloch oscillations in the bulk of the system. By applying a static force in different directions, the wave-packet dynamics retrieves the complex energy spectra with either nonzero or zero spectral winding number, indicating the presence or absence of skin accumulation in the corresponding directions, respectively. We find that this bulk dynamical characterization can be extended to interacting multiparticle systems, offering a method for detecting GDSE and other types of skin effect in quantum many-body systems, even without a well-defined open boundary.

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.

The non-Hermitian skin effect (NHSE) is a hallmark of non-Hermitian system, yet its generalized Brillouin zone (GBZ) description is restricted to periodic systems. We develop a site-resolved theory via a local scaling transformation (LST), introducing local twisting

T_n

to quantify metric operator

ξ

nontriviality. This elucidates the NHSE's origin and uncovers the generalized multiple-channel skin effect (MCSE). Exploiting

T_n

's translational independence, we define the Zahlen-Brillouin Zone (ZBZ), extending non-Hermitian band theory to nonperiodic and disordered lattices. Furthermore, we unify the

ξ

with GBZ Riemannian geometry, establishing the metric and state correspondence (MSC) as the principle for real-space localization. With a global skin index

\mathbfΓ

for phase transitions, our results provide a universal paradigm for non-Hermitian physics in both crystalline and amorphous media.