Wiener-Hopf factorization and non-Hermitian topology for Amoeba formulation in one-dimensional multiband systems

TL;DR

This work develops a Wiener-Hopf factorization (WHF) framework for non-Hermitian, multiband 1D systems to extend the Amoeba formulation beyond single-band limits. By combining WHF with Hermitian doubling, it links edge-mode physics to the asymptotics of open-boundary determinants and identifies precise criteria—via residual partial indices—for when the generalized Szegő limit theorem applies, and when explicit corrections are required. For class AII$^\dagger$, the WHF naturally yields symmetry-decomposed Ronkin functions, providing a rigorous foundation for TRS$^\dagger$-protected and nonprotected edge phenomena and a concrete route to generalize Szegő-type results to this symmetry class. Numerical tests in classes A and AII$^\dagger$ validate the need for correction terms in multiband regimes (including a fragile $\kappa=2$ phase) and demonstrate the predictive power of the WHF-based formulation for OBC spectra and density of states. Overall, the WHF approach unifies topology, spectral potentials, and symmetry constraints, enabling systematic generalizations to broader symmetry classes and higher dimensions.

Abstract

The non-Hermitian skin effect (NHSE), characterized by the extensive localization of bulk modes at the boundaries, has attracted significant attention as a hallmark feature of non-Hermitian topology. This localization invalidates the conventional Bloch band theory, necessitating an analysis under open boundary conditions even in the thermodynamic limit. The Amoeba formulation addresses this challenge by computing the spectral potential rather than the spectrum itself. Based on the (strong) Szegö limit theorem and its topological generalization, this approach reduces the evaluation of the potential to an optimization problem involving the Ronkin function. However, while the generalized Szegö limit theorem is formally applicable in arbitrary dimensions, its implementation is limited to single-band systems, and its applicability to multiband systems remains unclear. In this paper, we establish the Wiener-Hopf factorization (WHF) of the non-Bloch Hamiltonian as a powerful framework, providing a unified and rigorous foundation for Amoeba analysis in multiband systems. By combining the WHF with Hermitian doubling, we first elucidate the applicability criteria for the generalized Szegö limit theorem in multiband systems. We then show that the WHF provides the natural mathematical origin for the symmetry-decomposed Ronkin function in symmetry class AII$^\dagger$, leading to a rigorous proof of the generalized Szegö limit theorem for these systems and opening a path toward systematic generalizations to other symmetry classes.

Figures (13)

• Figure 1: Comparison of OBC and PBC spectra for the two uncoupled Hatano-Nelson models defined in Eq. (\ref{['eq: two-HN']}). Blue dotted lines show the PBC spectra of the individual blocks $h_1$ and $h_2$, while red lines denote the full OBC spectrum for $N=70$. Shaded regions in the complex energy plane indicate where the partial indices are nonzero. The blue-shaded region corresponds to $\kappa_1 = W[\sigma_1] = -1$ with $\kappa_2 = 0$, and the red-shaded region corresponds to $\kappa_2 = W[\sigma_2] = 1$ with $\kappa_1 = 0$. Their overlap (purple) represents the regime $\mathcal{K}[\sigma] = (-1, 1)$, where the conventional Amoeba optimization breaks down and the generalized formulation of Sec. \ref{['sec: Amoeba multiband']} must be applied.
• Figure 2: Ronkin functions for two representative reference energies $E$, selected from distinct topological domains in Fig. \ref{['fig: Two HN spectrum and pi']}. The total Ronkin function $R_\sigma$ is shown in black, while the partial Ronkin functions $R_{\sigma_1}$ and $R_{\sigma_2}$ are plotted in red and orange, respectively. (a) Trivial case ($E = -2.0 + 1.0i$): The reference point lies in the white region, where $\mathcal{K}[\sigma] = (0, 0)$. Both $R_{\sigma_1}$ and $R_{\sigma_2}$ can be minimized simultaneously, and the conventional optimization scheme applies. (b) Nontrivial case ($E = 0.0 + 0.2i$): The reference point lies in the purple region, where $\mathcal{K}[\sigma] = (-1, 1)$. Here, $R_{\sigma_1}$ and $R_{\sigma_2}$ cannot be minimized simultaneously, and the correction term in Eq. (\ref{['eq: generalized Szego for class A with (+1, -1)']}) is required to obtain the correct OBC potential.
• Figure 3: Ronkin functions and $\mu$-dependent partial indices for two reference energies taken from the red region in Fig. \ref{['fig: Two HN spectrum and pi']}. Each top panel shows the Ronkin functions, similar to Fig. \ref{['fig: Two HN partial indices for white and purple']}. Each bottom panel displays the $\mu$-dependent partial indices $\mathcal{K}[\sigma_\mu] = (\kappa_1^{(\mu)}, \kappa_2^{(\mu)})$. (a) $E = 2.2 + 0.1i$: The partial indices vanish after optimization, i.e., the residual partial indices are $\mathcal{K}^*[\sigma] = (0, 0)$, yielding the correct OBC potential through the standard optimization scheme. (b) $E = 0.5 + 0.6i$: The residual partial indices are nonzero even after optimization, $\mathcal{K}^*[\sigma] = (+1, -1)$, indicating that the correction in Eq. (\ref{['eq: generalized Szego for class A with (+1, -1)']}) is required to obtain the correct potential.
• Figure 4: Comparison of the DOS evaluated by (a) the conventional formulation and (b) our WHF-based formulation. Values are scaled by the area of the energy cell $\Delta \mathrm{Re}~E \times \Delta \mathrm{Im}~E \simeq 2.5 \times 10^{-4}$ due to the discretizations of the Hessian in Eq. (\ref{['eq: DOS and Potential']}). (a) The conventional simple optimization fails to reproduce the OBC spectrum (red line), as evident from the mismatch with the calculated DOS (grayscale). (b) The WHF-based formulation, which includes the correction for nonvanishing partial indices, accurately reproduces the OBC spectrum.
• Figure 5: Validation of the partial index $\kappa$ as the topological invariant. The Hamiltonian is defined in Eq. (\ref{['eq: Numerics_nonBloch']}), with parameters $t_1 = 0.3$, $t_2 = 0.8$, $g_1 = 0.5$, $\Delta_1 = 0.3$, and $\Delta_2 = 0.2$. Blue and red points indicate the PBC and OBC spectra, respectively ($N = 400$). The shaded region denotes the domain where the WHF-derived partial index is $\kappa = 1$. This region coincides with the $\mathbb{Z}_2$-nontrivial phase ($\nu[\tau] = 1$) obtained independently, confirming that the parity of $\kappa$ correctly captures the $\mathbb{Z}_2$ topology.