Exceptional Topology on Nonorientable Manifolds

Abstract

We classify gapped and gapless phases of non-Hermitian band structures on two-dimensional nonorientable parameter spaces. Such spaces arise in a wide range of physical systems in the presence of non-symmorphic parameter space symmetries. For gapped phases, we find that nonorientable spaces provide a natural setting for exploring fundamental structural problems in braid group theory, such as torsion and conjugacy. Gapless phases, which host exceptional points (EPs), explicitly violate the fermion doubling theorem, even in two-band models. We demonstrate that EPs traversing the nonorientable parameter space exhibit non-Abelian charge inversion. These braided phases and their transitions leave distinct signatures in the form of bulk Fermi arc degeneracies, offering a concrete route toward experimental realization and verification.

Figures (4)

• Figure 1: Exceptional phases on nonorientable manifolds: Under the boundary identification shown in the top-left insets, a free parameter space becomes (a) the Klein bottle, (b) the real projective plane, illustrated as an immersion into $\mathbb{R}^3$ in the bottom-right insets. The red and blue paths become non-contractible loops in the space, and generate its fundamental group. The total black loop enclosing the fundamental domain is a combination of these fundamental loops. Non-Hermitian exceptional phases are identified by the braiding of spectra along the fundamental loops. Statements about their separation gap can be made from the total braid along the black loop, following \ref{['eq:fermion-doubling']}. Background shading denotes the orientation of the neighboring unit cells.
• Figure 2: Fermi arcs as experimentally accessible signatures. Real (imaginary) Fermi arcs as defined in \ref{['eq:fermi-arc-condition']} are drawn in red (blue), with the orientation defined in the main text. (a) the gapped three-band model on $\mathbf{K}^2$ given in \ref{['eq:klein-model-gapped']}, with lighter and darker tones denoting Fermi arcs of different band pairs. The exceptional phase, identified by the braids $(\sigma_2\sigma_1^{-1},\Delta^{-1})$ along the $p-$ and $q-$directions, has three Fermi arcs encircling the fundamental domain in the positive $p$-direction, originating from the Abelianization of $\Delta$. This clearly distinguishes this phase from the trivial flat-band phase $(1,1)$. (b) the two-band model in \ref{['eq:model-two-band']} at $\ell=0$, perturbed to $e^{0.02i} H^{\mathbf{K}^2}_\text{EP}$ to better showcase the Fermi arcs. A real and an imaginary Fermi arc emerge from each EP (star marker), with a direction given by the eigenvalue crossing behavior. This direction is inverted on the nonorientable boundary, allowing for a total flux out of the fundamental domain that matches the total braid degree, $2$. (c) the two-band model in \ref{['eq:model-two-band']}, perturbed as in (b) and tuned to $\ell=1$ exhibits a characteristic monopole. Two real and two imaginary Fermi arcs emerge from the monopole, accounting for its total braid degree $A_\text{tot}^{\mathbf{K}^2}=2$. Such an EP cannot exist alone on an orientable space, where outgoing Fermi arcs cannot be compensated.
• Figure 3: Rules for braided phase transitions. An EP (star marker) with braid invariant $B_\text{EP}$, measured along the light blue loop, crosses an (a) periodic ((b) anti-periodic) boundary. This leads to a phase transition from braid $B_p$ (light red) to $\tilde{B}_p = B_\text{EP} B_p$ (dark red) in both cases. The EP's braid invariant changes as well, transitioning according to \ref{['eq:phase-transition-rule-ep']}. This can be understood by observing that an EP from the (anti-)periodic continuation moves into the fundamental domain as the original EP exits. This new EP carries the same (inverse) invariant, measured from the base point in the neighboring domain base point. The change in invariant to $\tilde{B}_\text{EP}$ corresponds to a change of base point associated with the encircling trajectory. Potential other degeneracies (shaded black) can change this rule by additional conjugations if the EP encircles them before exiting the fundamental domain (dashed blue line in (b)); this follows the established non-Abelian braiding rules.
• Figure 4: The braided phase transition on the Klein bottle described in \ref{['eq:ham-transition']}. Increasing parameter $t$ through critical $t_0\approx 0.29$, the system transitions from phase $(B^{\mathbf{K}^2}_p,B^{\mathbf{K}^2}_q) = (\sigma_2\sigma^{-1}_1,\Delta^{-1})$ to $(\sigma^{-1}_1,\Delta^{-1})$. (a) The parameter space for $t=\frac{1}{4}$, before the phase transition. Colored lines denote loops that determine the braids relevant to the transition, in line with \ref{['fig:encircling-rules']}. Real Fermi arcs (black lines) terminate in EPs (stars). The light blue EP crosses the boundary, turning into the dark blue EP after the transition. (b) Spectral braiding along the loops shown in (a) in matching colors, parameterized along parameter $\ell$. Avoided crossings are highlighted. The light blue EP carries braid $B_\text{EP}=\sigma_1^{-1}\sigma_2^{-1}\sigma_1^{-1}\sigma_2\sigma_1 = \sigma_2^{-1}$. It moves across the $p$-boundary, changing the boundary braid from $B_p = \sigma_1^{-1}\sigma_2^{-1}\sigma_1\sigma_2 = \sigma_2\sigma_1^{-1}$ determined along the light red path to $\tilde{B}_p = \sigma_1^{-1}\sigma_2^{-1}\sigma_2 = \sigma_1^{-1}$ measured along the dark red path. In the transition, the EP charge changes to $\tilde{B}_\text{EP} = \sigma_2 \sigma_1 \sigma_2^{-1}$ given in dark blue.