Topological origin of non-Hermitian skin effect in higher dimensions and uniform spectra

Abstract

The non-Hermitian skin effect is an iconic phenomenon characterized by the aggregation of eigenstates near the system boundaries in non-Hermitian systems. While extensively studied in one dimension, understanding the skin effect and extending the non-Bloch band theory to higher dimensions encounters a formidable challenge, primarily due to infinite lattice geometries or open boundary conditions. This work adopts a point-gap perspective and unveils that non-Hermitian skin effect in all spatial dimensions originates from point gaps. We introduce the concept of uniform spectra and reveal that regardless of lattice geometry, their energy spectra are universally given by the uniform spectra, even though their manifestations of skin modes may differ. Building on the uniform spectra, we demonstrate how to account for the skin effect with generic lattice cuts and establish the connections of skin modes across different geometric shapes via momentum-basis transformations. Our findings highlight the pivotal roles point gaps play, offering a unified understanding of the topological origin of non-Hermitian skin effect in all dimensions.

Figures (5)

• Figure 1: Sketch of lattice cuts and open boundary conditions (OBC). (a) 1D case: OBC is achieved by cutting off the lattice across a single bond. (b) 2D case: Open boundary can be oriented in any direction, leading to diverse geometric shapes (in different colors) defined by the lattice cuts.
• Figure 2: Different types of energy spectra. (a) The 1D case with model $H(\beta)=\beta+\beta^{-1}+\frac{2}{5}\beta^{-2}$. It displays the PBC spectra $\sigma_{\rm PBC}$ (blue loop), OBC spectra $\sigma_{\rm OBC}$ (red arcs), and the rescaled spectra $Sp(0.8)$ (green region), $Sp(1)$ (blue region), $Sp(2)$ (orange region). (b) The 2D case with model $H(\beta_1,\beta_2)=\beta_1+\frac{1}{2}\beta_1^{-1}+\beta_2+\frac{1}{2}\beta_2^{-1}+2\beta_1\beta_2$. It displays the PBC spectra $\sigma_{\rm PBC}$ (blue region), rescaled spectra $Sp(1,1)$ (blue region and shading region) and $Sp(0.8,0.8)$ (green region), and the uniform spectra $\sigma_{\rm U}$ (red region).
• Figure 3: Analysis of skin modes on generic lattice geometries. (a) Sketch of the decomposition of an $N$-polygon into $N$ simple parallelograms at the corners. Each parallelogram is formed by two neighboring cuts of the $N$-polygon. (b) Density distributions of the eigenstates for the 2D Hatano-Nelson model. The lattice cut is along the $(\hat{x}+2\hat{y})$ direction. The parameters $(t_1,t_{-1},r_1,r_{-1})$ for ①②③④ are $(1,2,2,1); (1,2,2,\sqrt2); (1,2,2,3); (1,1/2,2,5/3)$, respectively.
• Figure 4: Skin modes on different lattice geometries. The model is taken as $H(\beta_x,\beta_y)=1.2\beta_x+0.8\beta_x^{-1}+1.2\beta_y+0.4\beta_y^{-1}+0.2(\beta_x+\beta_x^{-1})(\beta_y+\beta_y^{-1})$ for (a1--a4) and $H(\beta_x,\beta_y)=\beta_x+0.5\beta_x^{-1}+\beta_y+0.2\beta_y^{-1}+2\beta_x\beta_y$ for (b1)--(b4). (a1), (b1) Energy spectra for square (red dots) and rhombus (blue dots) shaped lattices. (a2), (a3), (b2), (b3) Spatial distributions of the eigenmodes near $E=3$ and $E=1.3+0.25i$, respectively. (a4), (b4) Fitting deviations $\Delta(\kappa_x',\kappa_y')$ as described by Eq. (\ref{['fitt']}).
• Figure S1: Amoeba for the 2D model $H(\beta_1,\beta_2)=\beta_1+\frac{1}{2}\beta_1^{-1}+\beta_2+\frac{1}{2}\beta_2^{-1}+2\beta_1\beta_2$ with reference energy chosen as (a) $E_1=-2+2i$ (outside the OBC spectra region); and (b) $E_2=2+0.2i$ (inside the OBC spectra region).