Singularity Selector: Topological Chirality via Non-Abelian Loops around Exceptional Points

Chirality is more than a geometric curiosity; it governs measurable asymmetries across nature, from enantiomer-selective drugs and left-handed fermions in particle physics to handed charge transport in Weyl semimetals. We extend this universal concept to non-Hermitian systems by defining topological chirality, an invariant that emerges whenever an exceptional-points (EP) pair is present. Built from the non-commutative fundamental group and its braid representation, topological chirality acts as a singularity selector: clockwise EP loops occupy a homotopy class that avoids EPs, whereas counter-clockwise mirrors are equivalent only if they cross the EPs themselves. We confirm this binary rule in an optical microcavity and a non-Hermitian topological band. The same two-sheeted topology governs EP pairs in spin systems, photonic crystals and hybrid light-matter structures, where EP encirclements have already been demonstrated, so the framework transfers without alteration and confirms its experimental viability. Our findings lay the cornerstone for interpreting loop-sensitive observables such as spectral vorticity, the complex Berry phase and the non-Abelian holonomy. Finally, a gluing-of-planes construction extends the invariant to an n-sheeted surface hosting 2m EPs, unifying higher-order EP pairs.

💡 Research Summary

The manuscript “Singularity Selector: Topological Chirality via Non‑Abelian Loops around Exceptional Points” introduces a fundamentally new notion of chirality for non‑Hermitian (NH) systems that is rooted in the non‑commutative topology of loops encircling a pair of exceptional points (EPs). While previous studies have identified three distinct mechanisms of NH chirality—(i) spatial mode imbalance (e.g., CW/CCW scattering asymmetry), (ii) local spatial chirality arising from the ±π/2 phase locking of coalescing eigenmodes near an EP, and (iii) dynamic, encircling‑induced asymmetric switching—the authors argue that these are all geometry‑ or mode‑dependent and lack true topological protection.

To overcome this limitation, the authors construct a topological invariant—“topological chirality”—by treating the two‑dimensional control‑parameter space as a complex plane, puncturing it at the two EPs, and then lifting this twice‑punctured plane X = ℂ \ {EP₁, EP₂} to its two‑sheeted Riemann surface Y. The fundamental group of X is the free group on two generators, π₁(X) ≅ F₂ = ⟨a, b⟩, where a (b) winds once around EP₁ (EP₂). On the lifted surface Y the generators become a², b² (two full windings around a single EP) and a mixed generator c = ab that encircles both EPs while swapping the sheets. Crucially, the clockwise (CW) loop corresponds to the word “ab” while the counter‑clockwise (CCW) loop corresponds to “ba”. Since ab ≠ ba in the free group, the two orientations belong to distinct homotopy classes. This non‑commutativity acts as a “singularity selector”: a loop that avoids the EPs (e.g., CW) cannot be continuously deformed into its mirror (CCW) without crossing an EP. The authors formalize this using the covering map ϕ : Y → X and the induced homomorphism (ϕ₁)₍*₎ on fundamental groups, showing that lifting a loop from X to Y captures the full non‑Abelian winding behavior.

The theoretical framework is illustrated with ten representative loops (Fig. 2), demonstrating how a², b², ab, and ba appear on both the real and imaginary sheets, how ba is homotopic to the combined loop c, and how the reduced word representation uniquely characterizes any loop on Y. The analysis also reveals that two full traversals around a single EP are required for the eigenvalues λᵣ and λᵢ to return to their original values, matching the physical intuition that EPs are square‑root branch points.

Experimental relevance is established through two distinct platforms. First, an elliptical optical microcavity with semi‑axes a = 1 + χ and b = 1/(1 + χ) is studied; by varying the interior refractive index n_in and deformation χ, the complex eigenvalue surfaces display two isolated branch points (EPs) that form a pair. The Riemann‑sphere projection visualizes the two‑sheeted topology. Second, a non‑Hermitian Dirac Hamiltonian H(k) = kₓσₓ + k_yσ_y + i bₓσₓ is examined. The EP conditions reduce to kₓ = 0, k_y = ± bₓ, yielding a pair of EPs in the Brillouin zone. Both systems share the analytic form f(z) = √