Pseudo-Hermitian Topology of Multiband Non-Hermitian Systems

TL;DR

The paper addresses non-Hermitian multiband topology beyond exceptional points (EPs) by introducing pseudo-Hermitian lines (PHLs) as lines in 2D parameter spaces where pseudo-Hermiticity holds. It develops a homotopy/permutation-group framework to classify state exchanges and shows that non-contractible PHLs on a torus can produce nontrivial band topology without EPs, while EPs can still drive topological phase transitions. The work generalizes to multiband systems, analyzes how different 2D space topologies constrain EP configurations, and demonstrates a photonic-crystal realization where PHLs govern braiding of bands, offering a route to robust non-Hermitian topology without fine-tuning EPs. Overall, the results expand the toolkit for non-Hermitian topological phases by highlighting PHLs as a fundamental structural element with practical realizations.

Abstract

The complex eigenenergies and non-orthogonal eigenstates of non-Hermitian systems exhibit unique topological phenomena that cannot appear in Hermitian systems. Representative examples are the non-Hermitian skin effect and exceptional points. In a two-dimensional parameter space, topological classifications of non-separable bands in multiband non-Hermitian systems can be established by invoking a permutation group, where the product of the permutation represents state exchange due to exceptional points in the space. We unveil in this work the role of pseudo-Hermitian lines in non-Hermitian topology for multiple bands. In particular, the non-separability of non-Hermitian multibands can be topologically non-trivial without exceptional points in two-dimensional space. As a physical illustration of the role of pseudo-Hermitian lines, we examine a multiband structure of a photonic crystal system with lossy materials. Our work builds on the fundamental and comprehensive understanding of non-Hermitian multiband systems and also offers versatile applications and realizations of non-Hermitian systems without the need to consider exceptional points.

Figures (12)

• Figure 1: Real energy surfaces on a two-dimensional Brillouin zone (torus). (a) Two energy surfaces, inner red and outer blue surfaces, cross along the branch cut (black arc) that ends at two exceptional points (EPs, two black dots). The blue and red portions of loops represent the trajectories for adiabatic parameter changes on the blue and red energy surfaces, respectively. Conventional state exchange occurs along the path encircling an EP, path 1. The state exchange also occurs along path 2, without encircling an EP. (b) The EPs merge and vanish, leaving the pseudo-Hermitian line (black circle). Despite the absence of EPs in the parameter space, state exchange still occurs along path 2.
• Figure 2: Complex eigenenergies with PHLs. (a) Real and (b) imaginary parts of the complex eigenenergies of the Hamiltonian in Eq. (\ref{['eq:ham1']}) when $(s_1 , s_2) = (0.5, 0)$. The red, green, and gray bands are the first, second, and third bands, according to the order of real parts of complex eigenenergies. The green and gray bands are not separable and are mapped on Fig. \ref{['fig:fig_EP_PHL']}(b). The yellow line represents a PHL. The black spheres and red cubes represent the initial and final states when $k_y$ changes from $-\pi$ to $\pi$. The arrows form closed loops because of the periodicity of the 2D Brillouin zone.
• Figure 3: Complex eigenenergies with a pair of EPs. (a) Real parts of the complex eigenenergies of the Hamiltonian in Eq. (3) when $s_2 = 0.3$. The two lower bands are mapped on Fig. \ref{['fig:fig_EP_PHL']}(a). The two yellow points and lines represent EPs and corresponding branch cuts. The black circle and red rectangle are initial states when $k_y = - \pi$, and the black and red lines denote closed loops when $k_y$ changes from $-\pi$ to $\pi$ with $k_x = \pi / 2$. (b) Complex eigenenergies of the two lower bands on the complex plane when $k_x = \pi$. The black circle and red rectangle are initial states when $k_y = - \pi$, and the black lines denote closed loops when $k_y$ changes from $-\pi$ to $\pi$ with $k_x = \pi$. (c) Complex eigenenergies of two lower bands on the complex plane when $k_x = \pi/2$. The black circle and red rectangle correspond to those in (a).
• Figure 4: Three kinds of two EPs. (a) Paired EPs with a shared branch cut, (b) intersected EPs, and (c) disjointed EPs with independent branch cuts. Combinations of EPs in (d) 2D infinite and (e) periodic space. Black and blue represent paired EPs, and red represents an unpaired EP. Unpaired EPs (a single EP, or intersected or disjointed EPs) cannot exist in (e) 2D periodic space because of the compactness of the space.
• Figure 5: Complex eigenenergies in two different given spaces. (a) Real parts of the complex eigenenergies of the Hamiltonian in Eq. (3) when $(s_1, s_2) = (0.25, 0)$, comprising a PHL (orange line) and two EPs (red circles) with two associated branch cuts (red lines). An initial state (blue circle) returns to its origin after a three-cycle rotation on the Brillouin zone boundary. (b) A PHL, pair of EPs, and associated branch cuts on a torus as a 2D periodic space. (c) A PHL, pair of EPs, and associated branch cuts on a Riemann sphere with one point at infinity (violet) as a 2D infinite space. The PHL is a non-contractible loop due to the presence of an EP in the loop, while the line is contractible if there is no EP in the loop.