Non-orientable Exceptional Points in Twisted Boundary Systems

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.

💡 Research Summary

The paper investigates how non‑orientable geometry fundamentally alters the topological behavior of exceptional points (EPs) in non‑Hermitian systems. Starting from the observation that EPs—degeneracies where both eigenvalues and eigenvectors coalesce—carry two kinds of topological invariants (complex winding numbers and complex Berry phases), the authors ask what happens when the parameter space is a non‑orientable manifold such as a Möbius strip or a Klein bottle. They show that, unlike the usual toroidal Brillouin zone, a momentum‑space glide‑reflection symmetry (H(k_x,k_y)=H(-k_x,k_y+\pi)) folds the two‑dimensional Brillouin zone into a Klein‑bottle topology, which they call a Klein Brillouin zone (KBZ).

A minimal two‑band non‑Hermitian Hamiltonian (H(\mathbf k)=d_x(\mathbf k)\sigma_x+d_y(\mathbf k)\sigma_y) is constructed with (d_x=\cos k_x+i\alpha) and (d_y=-\sin k_x