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

TL;DR

The paper establishes a non-Hermitian, topological notion of chirality that arises whenever an EP pair exists. By leveraging the non-commutative orbifold fundamental group and its braid-group representation, it shows that CW and CCW encirclements belong to distinct, non-homotopic classes, endowing loop-induced effects with topological protection. The authors demonstrate the framework in two platforms—a optical microcavity and a NH Dirac lattice—and discuss measurable consequences for spectral vorticity, complex Berry phase, and non-Abelian holonomy, with a gluing-of-planes construction that extends to n-sheeted surfaces hosting 2m EPs. This work unifies EP topology across spin systems, photonic crystals, and hybrid light-matter structures, providing a rigorously grounded foundation for robust NH devices and loop-sensitive observables.

Abstract

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.

Figures (6)

• Figure 1: Universal appearance of an EP pair in two disparate open systems. (a, b) Real and imaginary parts of the eigenvalue surface for an elliptical optical microcavity as the deformation parameter $\chi$ and interior refractive index $n_{\mathrm{in}}$ are varied. The two coloured sheets touch at precisely two isolated degeneracies (blue dots), forming an EP pair. (c) Same data mapped to the Bloch (Riemann) sphere: See the details in supplemantary. (d, e, f) Real and imaginary parts of the spectrum for a non-Hermitian Dirac model$H = k_x\sigma_x + k_y\sigma_y + i\sigma_x$; the control parameters are the crystal momenta $(k_x,k_y)$. Again two EPs appear in the parameter space. (f) Parameter space view of the lattice model: the loci defined by $d_R^{2}-d_I^{2}=0$ (circular curve) and $d_R\!\cdot\!d_I = 0$(elliptic curve); their intersections are the EP pair. In regions where one condition holds and the other does not, the spectrum exhibits “Fermi‐arc” behavior: when $d_R\!\cdot\!d_I = 0$ and $d_R^2 - d_I^2 < 0$ (shaded area), the real part of the energy vanishes (pink curve); conversely, when $d_R\!\cdot\!d_I = 0$ and $d_R^2 - d_I^2 > 0$ (outside shaded area), the imaginary part vanishes (sky blue curve). Panels (a)-(c) and (d)-(f) thus visualise the EP pairs described by $f(z)=\sqrt{(z-z_{1})(z-z_{2})}$ realised in an optical microcavity and non-Hermitian Dirac model, underscoring the abundance and universality of EP pairs.
• Figure 2: Upper panels (a)-(h): Panels (a)-(d) depict representative loops $[\tilde{\alpha}]\in\pi_{1}(Y,y_{0})$ on the Riemann surface encircling $RP_1$, $RP_2$, both points individually, and both simultaneously. Panels (e)-(h) show their projected loops $[\alpha]=(\phi^{\circ})_{*}[\tilde{\alpha}]\in\pi_{1}(X,x_{0})$ around the corresponding exceptional points $\textsf{EP}_1$, $(\textsf{EP})_2$, etc. Lower panels (i),(j): Homotopy-lifting diagrams for CW encirclement (i) and CCW encirclement (j), illustrating the bijection between loops on $X$ and their lifts on $Y$ as time t increase.
• Figure 3: (a-c) Hopping picture: two copies of the punctured plane $\mathbb{C}\setminus\{-i,i\}$ arranged side-by-side. Two hops across the branch cut at $-i$ realise the loop $a^{2}$ (a); two hops across $i$ realise $b^{2}$ (b); one hop across each cut realises $ab$ (c). (d-f) Gluing construction that corresponds to the hopping moves in (a-c), demonstrating their one-to-one correspondence. Colored arrows indicate sheet changes along the loop from $t=0$ to $t=1$.
• Figure 4: Topological chirality for a pair of EPs. (a) Braid for the loop $w_{0}=ab$ concatenated with its mirror-image inverse. When the crossings in the concatenated braid are pulled straight, they cancel out and reduce to the trivial braid (two parallel strands). Loops (b) and (c) on the Riemann surface are homotopically trivial, since they shrink to the identity once the cone relations $a^2 = b^2 = e$ are imposed. . Panel (d) shows the clockwise loop $w_{0} = ab$ and its braid, while panels (e), (f), and (g) display three counter-clockwise mirror loops $w_{1} = ba$, $w_{2} = b^{-1}a^{-1}$, $w_{3} = b^{-1}a$ and their braid, respectively. More generally, given any reference loop $w_{0}$ (or its corresponding braid), one obtains two natural categories: the set of all loops homotopic to $w_{0}$, and the topological chirality set$\{w_{\alpha}\}$, whose mirror-image elements satisfy $w_{0}w_{\alpha} = e$ in the capped fundamental group.
• Figure S1: Geometry of the two prototype loops in the complex parameter plane. The blue curve $\alpha(t)$ represents the composite loop $[a][b]$, winding clockwise around $\mathrm{EP}_2=-1$ (inner arc) and then $\mathrm{EP}_1=1$ (outer arc). The orange curve $\beta(t)$ is a single clockwise circuit enclosing both EPs at once. Despite their different shapes, these loops are homotopic in $\mathbb{C}\setminus\{\pm1\}$.