Beyond characteristic equations: A unified one-dimensional non-Bloch band theory via wavefunction data

Abstract

Non-Hermitian systems play a central role in nonequilibrium physics, where determining the energy spectrum under open boundary conditions is a fundamental problem. Non-Bloch band theory, based on the characteristic equation $\text{det}[E - H(β)] = 0$, has emerged as a key tool for this task. However, we show that this framework becomes insufficient in systems with certain symmetries, where identical characteristic equations can yield different spectra. To resolve this, we develop a unified theory that incorporates additional wavefunction information beyond the characteristic equation. Our framework accurately captures spectral properties such as the energy spectrum and the end-to-end signal response in a broad class of systems, particularly those with high symmetry. It reveals the essential role of wavefunction information and symmetry in shaping non-Hermitian band theory.

Figures (3)

• Figure 1: The open boundary spectra and GBZs of two models, $H_1(k)$ and $H_2(k)$, which share the same characteristic function. The parameters are set as $t_0=1$ (energy unit) and $a=b=c=d=1$ (dimensionless). Panels (a), (c), and (e) correspond to properties of $H_1$, while panels (b), (d), and (f) correspond to those of $H_2$. (a), (b) Numerically computed spectra for a finite chain of length $L=100$. The energy $E_0 = -1/2$ lies within the spectrum of $H_1$ but not within that of $H_2$. Faint red vertical lines represent Re($E$) = -1/2 in both panels. (c), (d) The GBZs of two models, shown as blue curves. Both models share the same auxiliary GBZ, plotted as faint red curves for reference. Notably, the GBZ in panel (d) includes $\beta=0$. In the GBZ of $H_1$, the mode with generalized momentum $\beta_0$ corresponds to the energy $E_0$, whereas $\beta_0$ does not lie on the GBZ of $H_2$. (e), (f) Spectra reconstructed from the GBZs (c) and (d), respectively, representing the open boundary spectra in the limit $L\rightarrow \infty$.
• Figure 2: (a) Workflow of the original non-Bloch band theory algorithm. Only the most common scenario is shown; in systems with symmetry, the GBZ condition may require case-specific treatment. (b) Workflow of the modified non-Bloch band theory algorithm. The application of this algorithm to specific models $H_1$ and $H_2$ is illustrated in Tables \ref{['tab: 1']} and \ref{['tab: 2']}.
• Figure 3: Comparison between the solutions of the characteristic equation $\text{det}[\omega-H(\beta)]=0$ and the end-to-end exponential factor $\alpha_{\leftarrow}$. Parameters are set as $t=1.01,u=1,\gamma=1.2$, and $\omega=0.01i$. (a) Magnitudes of analytical solutions $\beta_i$ of the characteristic equation plotted as a function of the parameter $\Delta$. (b) End-to-end exponential factor $\alpha_{\leftarrow}$ as a function of the parameter $\Delta$, obtained via a linear fit of the logarithm of the end-to-end Green's function vs the system length $L$, with $L\in[150,170]$.