Non-orientable Exceptional Points in Twisted Boundary Systems

TL;DR

The paper develops a framework for non-Hermitian topology on non-orientable manifolds by studying the adiabatic motion of exceptional points on a Klein bottle Brillouin zone (KBZ) with glide symmetry. It classifies EP-induced eigenenergy/eigenstate evolution via braids and complex Berry phases, highlighting orientation-dependent invariants that can flip sign when EPs cross orientation-reversing boundaries. The authors realize these ideas in a two-band effective Hamiltonian and in a microdisk cavity with synthetic momenta, demonstrating transitions between trivial and nontrivial braiding and Berry-phase accumulation, while showing that EP-mode chirality remains robust to orientation changes. They discuss experimental feasibility in photonic microcavities and emphasize the distinction between orientation-dependent topological invariants and orientation-insensitive observables like chirality, advancing non-Hermitian topology in non-orientable momentum spaces.

Abstract

Non-orientable manifolds, such as the Möbius strip and the Klein bottle, defy conventional geometric intuition through their twisted boundary conditions. As a result, topological defects on non-orientable manifolds give rise to novel physical phenomena. We study the adiabatic transport of exceptional points (EPs) along non-orientable closed loops and uncover distinct topological responses arising from the lack of global orientation. Notably, we demonstrate that the cyclic permutation of eigenstates across an EP depends sensitively on the loop orientation, yielding inequivalent braid representations for clockwise and counterclockwise encirclement; this is a feature unique to non-orientable geometries. Orientation-dependent geometric quantities, such as the winding number, cannot be consistently defined due to the absence of a global orientation. However, when a boundary is introduced, such quantities become well defined within the local interior, even though the global manifold remains non-orientable. We further demonstrate the adiabatic evolution of EPs and the emergence of orientation-sensitive observables in a Klein Brillouin zone, described by an effective non-Hermitian Hamiltonian that preserves momentum-space glide symmetry. Finally, we numerically implement these ideas in a microdisk cavity with embedded scatterers using synthetic momenta.

Figures (8)

• Figure 1: Adiabatic evolutions (red and blue dashed arrows) of two EPs (red circles and blue rectangles) as system parameters are adiabatically varied on (a) a toroidal surface and (b) a Klein bottle surface. The light blue rectangle represents an EP located on the back side of the surface.
• Figure 2: Adiabatic evolutions of EPs in the KBZ and homotopy equivalence classes for $\gamma=0$ [(a)-(d)] and $\gamma=1$ [(e)-(h)] in the Hamiltonian, Eq. (\ref{['eq:Hamiltonian']}). (a) Real and (b) imaginary parts of complex eigenenergies of the Hamiltonian with $\gamma=0$. (c) Two EPs (red circle and blue rectangle) and encircling loop (black closed loop) in the KBZ. The colors represents arguments of complex energy difference and the white lines are real branch cuts. The jump of $\pm \pi$ shows the direction of braid operation when the closed loop across the real branch cuts. (d) Trivial braids of two bands without accumulated Berry phases as $\theta$ increases on the closed loop. (e), (f), and (g) are the same as (a), (b), and (c), respectively, but with $\gamma=1$. (h) Non-trivial braids of two bands with accumulated Berry phases $\pi$ as $\theta$ increases on the closed loop.
• Figure 3: Topological properties of energy bands of a dielectric disk with three scatterers for $\gamma=0$ [(a)-(d)] and $\gamma=1$ [(e)-(h)] in Eq. (\ref{['eq:t3']}). (a) Real parts of two selected energy bands obtained with $\gamma=0$ in the synthetic $(k_x, k_y)$ space, embedding two EPs (star symbols). (b) Arguments of the complex energy differences between the two bands. Red and blue colors indicate arguments of $\pm \pi/2$, while the white curves represent real branch cuts. The arrowed curve denotes the encircling path (with increasing $\theta$) used to examine state braiding and Berry phase accumulation along the closed loop. (c) The disk-scatterer system configuration with parameters $\varphi_i$ given in Eq. (\ref{['eq:t3']}) and spatial mode profiles $|\psi(\mathbf{r})|^2$ at the EPs of our dielectric disk system. The first scatterer $s_1$ is fixed, while the angular positions $\varphi_2$ and $\varphi_3$ of the other two scatterers $s_2$ and $s_3$ are varied with respect to the disk center. (d) Trivial braiding of the two bands, showing no Berry phase (colorbar) accumulation as $\theta$ evolves along the loop shown in (b). (e), (f), and (g) are the same as (a), (b), and (c), respectively, but with $\gamma=1$. (h) Non-trivial braiding of the two bands, showing $\pi$-Berry phase (colorbar) accumulation as $\theta$ evolves along the loop shown in (f).
• Figure 4: Exceptional classification for braid group and Berry phases associated with the number of EPs inside a closed loop in two-band Hamiltonian with multiple EPs. Braid operators, braid degrees (winding numbers), braid diagrams, knots, and Berry phases depending on the number of EPs inside a loop.
• Figure S1: Adiabatic evolutions of EPs in the KBZ and homotopy equivalence classes. (a) Real and (b) imaginary parts of complex eigenenergies of the Hamiltonian when $\gamma = 0.5$. (c) A encircling loop (black closed loop) including an EP (red circle) in the KBZ. The white lines represent real branch cuts. (d) Non-trivial braids of two bands with Berry phases as $\theta$ increases on the closed loop. The sum of accumulated Berry phases of two bands is $\pi$ as $\theta$ increases on the closed loop.