Topological Phases in Non-Hermitian Nonlinear-Eigenvalue Systems

TL;DR

This work addresses the challenge of defining bulk-boundary correspondence and topological invariants for non-Hermitian nonlinear-eigenvalue systems. It develops an auxiliary-system framework and a non-Bloch, generalized Brillouin-zone theory, linking the nonlinear eigenproblem $P(ω)=H_0-ωS(ω)$ to a tractable linear pencil. The key finding is the coexistence of real-band and complex-band topological phases, including a novel complex-band topology, with nonreciprocal hoppings and higher-order nonlinearities enriching the phase diagram. It provides a practical toolkit for designing nonlinear topological metamaterials in photonics, acoustics, and elastic systems.

Abstract

The discovery of topological phases has ushered in a new era of condensed matter physics and revealed a variety of natural and artificial materials. They obey the bulk-boundary correspondence (BBC), which guarantees the emergence of boundary states with non-zero topological invariants in the bulk. A wide attention has been paid to extending topological phases to nonlinear and non-Hermitian systems. However, the BBC and topological invariants of non-Hermitian nonlinear systems remain largely unexplored. Here, we establish a complete BBC and topological characterization of the topological phases in a class of non-Hermitian nonlinear-eigenvalue systems by introducing an auxiliary system. We restore the BBC broken by non-Hermiticity via employing the generalized Brillouin zone on the auxiliary system. Remarkably, we discover that the interplay between non-Hermiticity and nonlinearity creates an exotic complex-band topological phase that coexists with the real-band topological phase. Our results enrich the family of nonlinear topological phases and lay a foundation for exploring novel topological physics in metamaterial systems.

Figures (3)

• Figure 1: (a) Real-$\omega$ bands (red lines), real part of complex-$\omega$ bands (gray lines), and winding number $\mathcal{W}$ (cyan dashed lines) in different $w$. Band structure of $\lambda$ when (b1) $w=2v$ and (b2) $0.875v$. (c) Probability distribution of the edge states, with the inset being the eigenvalues. (d) Real-$\omega$ bands (red lines), real part of complex-$\omega$ bands (gray lines), and $\mathcal{W}$ in different $U_0$ when $w=2v$. (e) Phase diagrams in the (e) $U_0$-$w$ and (f) $U_0$-$M$ spaces when $w=0.75v$. We use $U_0=0.5v$ in (a)-(c) and $M=-0.5v^{-1}$ and $L=80$ for all.
• Figure 2: Real-$\omega$ bands (red lines), real part of complex-$\omega$ bands (gray lines) under the (a) OBC and (b) PBC. Winding numbers $\mathcal{W}$ (cyan dashed lines) defined in the (a) generalized BZ and (b) BZ. The black solid line in (b) shows the Fermi level $\omega=U_0$. (c) Eigenvalues of $\lambda$. Band structures of $\text{Re}(\lambda)$ under (b1) the OBC (blue) and (b2) PBC (brown) when $U_0=0$. Phase diagrams in the (e) $U_0$-$\delta$ and (f) $U_0$-$w$ spaces. We use $w=1.1v$ in (a)-(d) and $0.75v$ in (e) and $\delta=0.3v$, $M=-0.5v^{-1}$, and $L=50$ for all.
• Figure 3: (a) Band structure of $\omega$ in different $w$. The real and complex edge-mode eigenvalues are marked by red and blue lines, respectively. (b) Band structures of the real part of complex $\lambda$ (blue dots) and real $\lambda$ (orange dots) when $w=4v$. (c) Winding numbers in different $w$. Phase diagrams for the real (d) and complex (e) edge modes. We use $\mathcal{M}=-0.5 v^{-2}$, $\gamma=v^{-1}$, and $L=100$.