General theory for geometry-dependent non-Hermitian bands

Abstract

In two- and higher-dimensional non-Hermitian lattices, systems can exhibit geometry-dependent bands, where the spectrum and eigenstates under open boundary conditions depend on the bulk geometry even in the thermodynamic limit. Although geometry-dependent bands are widely observed, the underlying mechanism for this phenomenon remains unclear. In this work, we address this problem by establishing a higher-dimensional non-Bloch band theory based on the concept of "strip generalized Brillouin zones" (SGBZs), which describe the asymptotic behavior of non-Hermitian bands when a lattice is extended sequentially along its linearly independent axes. Within this framework, we demonstrate that geometry-dependent bands arise from the incompatibility of SGBZs and, for the first time, derive a general criterion for the geometry dependence of non-Hermitian bands: non-zero area of the complex energy spectrum or the imaginary momentum spectrum. Our work opens an avenue for future studies on the interplay between geometric effects and non-Hermitian physics, such as non-Hermitian band topology.

Figures (4)

• Figure 1: Illustration of geometry-dependent non-Hermitian bands. For the same system, different geometries ($G_{1}$ and $G_{2}$) yield distinct spectra ($\sigma_{1}$ and $\sigma_{2}$) and eigenstates ($\psi_{1}$ and $\psi_{2}$), even in the thermodynamic limit.
• Figure 2: Definition of the SGBZ. (a) Schematic diagram of the strip geometry of a 2D non-Hermitian lattice, where the cyan region represents the periodic unit of the strip. (b) Definition of the function $\mu_{2,0}$, which is a periodic function between $\mu_{2}^{(M_{2})}$ and $\mu_{2}^{(M_{2}+1)}$. (c) Definition of the base manifold $X(E,\mu_{1})$, where the blue solid lines and cyan dashed lines represent the $M_{2}$-th and $(M_{2}+1)$-th solutions of the eigenvalue equations, respectively, and the orange solid curves represent the winding loops for $w_{1}(\theta_{2};E,\mu_{1})$. (d) SGBZ bands and the strip winding number. When $W(E,\mu_{1})$ exhibits a plateau, the reference energy lies outside the SGBZ bands (panel i). Otherwise, the reference energy belongs to the SGBZ bands (panel ii).
• Figure 3: Relation between SGBZs and geometry-dependent bands. (a) Illustration of the 2D HN model. (b) Illustration of the [10]-strip, [01]-strip, and [11]-strip, where the major axes are $\mathbf{a}_{x}=(1,0)$, $\mathbf{a}_{y}=(0,1)$, and $\mathbf{a}_{[11]}=(1,1)$, respectively. (c, d) Spectra of (c) $[10]$-SGBZ or $[01]$-SGBZ, and (d) $[11]$-SGBZ. (e, f) Comparison between the SGBZ bands and the finite-size OBC spectra in parallelogram regions with different aspect ratios, where (e) illustrates the region with compatible SGBZs, and (f) illustrates the region with incompatible SGBZs. The coupling coefficients are $J_{x1}=1+\mathrm{i}$, $J_{x2}=1.5+1.2\mathrm{i}$, $J_{y1}=-1+\mathrm{i}$, and $J_{y2}=-1.2-0.5\mathrm{i}$. In numerical calculations, the total number of sites is set to be $12800$ (or the nearest integer to $12800$).
• Figure 4: Criterion for uniform or geometry-dependent bands. (a) Cases with or without uniform bands, where the blue solid curve and cyan dashed curve represent $\mu_{2}^{(M_{2}+1)}(\theta_{1};E,\mu_{1,0})$ and $\mu_{2}^{(M_{2})}(\theta_{1};E,\mu_{1,0})$, respectively, and the black dotted lines in ii and iii are reference lines. In cases i--iii, uniform bands are not allowed, while in case iv, uniform bands are allowed. (b) Requirements on the winding numbers when a system exhibits a uniform band. The blue curve represents the solutions of $\det[E-h(\beta_{1},\beta_{2})]=0$ on the plane $(\mu_{1},\mu_{2})=(\mu_{1,0},\mu_{2,0})$, and the orange curve represents the winding loop. (c) Illustration of the criterion for uniform or geometry-dependent bands. For an arbitrary SGBZ, if the complex energy spectrum and all the imaginary momentum spectra have zero area, the bands are uniform. Otherwise, if the energy spectrum or at least one of the imaginary momentum spectra has nonzero area, the bands are geometry-dependent.