Multi-EP Topology in Non-Hermitian Systems

• Multi-EP topology is the study of collective behaviors of multiple exceptional points in multiband non-Hermitian systems, defined by non-Abelian braid rules.
• It introduces a universal non-Abelian conservation rule that governs how EPs and ELs braid, merge, and constrain one another, contrasting with Abelian models.
• Practical implementations in coupled acoustic cavities and optical systems demonstrate how these non-Hermitian degeneracies can be experimentally controlled.

Searching arXiv for recent and foundational papers on multi-exceptional-point topology in non-Hermitian systems. Multi-EP topology denotes the topological organization and collective dynamics of multiple exceptional points (EPs) or exceptional lines (ELs) in multiband non-Hermitian systems. In this setting, the relevant question is not only how a single defective spectral degeneracy is stabilized, but how several such degeneracies can annihilate, coalesce, braid, reconnect, or constrain one another. The central result of the current framework is that these collective processes are governed by a universal non-Abelian conservation rule (NACR), formulated in terms of braid-group-valued invariants rather than purely Abelian charges such as winding numbers or discriminant numbers (Guo et al., 2022). In this sense, multi-EP topology is the multiband, non-Abelian extension of the familiar two-band exceptional-point picture.

1. Non-Hermitian multiband setting and the shift from Abelian to non-Abelian topology

In generic multiband non-Hermitian systems, multiple EPs or ELs emerge naturally. The distinctive feature of this regime is that collective behavior cannot, in general, be reduced to pairwise charge addition or to a simple permutation label. Instead, the topology of a closed path in parameter space depends on how eigenenergies braid as the loop is traversed, and this braiding becomes non-commutative once at least three bands are involved (Guo et al., 2022).

The mathematical structure is organized by the Hamiltonian space $X_N$. For a closed path based at $\bm{P}$ and avoiding EPs or ELs, the relevant invariant is the fundamental group

$\pi_1(\bm{P}, X_N) = B_N,$

where $B_N$ is the braid group of $N$ strands. For $N \geq 3$, $B_N$ is non-Abelian, whereas in the two-band case one has $B_2 = \mathbb{Z}$, which is Abelian (Guo et al., 2022). This distinction marks the transition from the ordinary two-band EP picture to genuine multi-EP topology.

A key implication is that complete topological classification in $N$-band non-Hermitian systems requires full braid-group invariants, not merely winding numbers or band permutations. This also explains why non-Abelian topology can arise without symmetry protection, in contrast with multiband Hermitian topological metals (Guo et al., 2022). A plausible implication is that the topology of degeneracy networks in nonconservative systems is intrinsically relational: the outcome for one EP can depend on the existence and history of others.

2. Universal non-Abelian conservation rule

The NACR assigns to each admissible closed loop $\Gamma$ a braid-group-valued invariant $\bm{P}$0, which records how eigenenergies permute and braid along the loop. Under a dynamical deformation $\bm{P}$1, as long as no EP or EL crosses $\bm{P}$2, the invariant is preserved up to conjugation by the net braid accrued at the base point during the evolution (Guo et al., 2022). Thus the conserved object is not a scalar charge but a conjugacy-sensitive braid element.

The braid generators $\bm{P}$3 satisfy the standard Artin relations

$\bm{P}$4

These relations encode the distinction between commuting and noncommuting EP processes. When two EPs act on non-adjacent sets of bands, their braid generators commute. When they are adjacent in the sense of energy levels, the associated generators need not commute, and the resulting topology becomes path dependent (Guo et al., 2022).

This formalism replaces the intuition inherited from isolated EPs. In a purely Abelian description, one expects the total charge inside a loop to determine the outcome. In the NACR framework, the ordering of braids, the base point, and the detailed path history can all matter. That is the defining structural feature of multi-EP topology.

3. Path-dependent annihilation, coalescence, and the adjacency effect

One of the central counterintuitive results is that two EPs of opposite charges, even if pairwise created, do not necessarily annihilate when brought together. Their fate depends on how they approach each other and on whether one of them has braided around other noncommuting EPs beforehand (Guo et al., 2022).

For EPs $\bm{P}$5 and $\bm{P}$6 created pairwise, one has $\bm{P}$7. If $\bm{P}$8 detours around a third EP $\bm{P}$9 before merging with $\pi_1(\bm{P}, X_N) = B_N,$0, the final braid invariant becomes

$\pi_1(\bm{P}, X_N) = B_N,$1

The pair annihilates only if $\pi_1(\bm{P}, X_N) = B_N,$2 and $\pi_1(\bm{P}, X_N) = B_N,$3 commute, namely only if $\pi_1(\bm{P}, X_N) = B_N,$4 (Guo et al., 2022). When they do not commute, opposite local charges are insufficient to guarantee annihilation. The pair may instead merge into a higher-order EP.

This is the source of the “adjacent” effect. Only noncommuting EPs, corresponding to adjacent energy-level permutations, affect one another’s braiding and merging outcomes. Non-adjacent EPs, for example those permuting different sets of bands, do not induce non-Abelian effects in this sense (Guo et al., 2022).

A concrete realization is provided by the three-state non-Hermitian model

$\pi_1(\bm{P}, X_N) = B_N,$5

which displays non-annihilating EP behavior and is described as easily implemented in coupled acoustic cavities or optical systems (Guo et al., 2022). The common misconception that opposite EP charges must always cancel is therefore incorrect in multiband non-Hermitian settings.

4. Exceptional-line networks and topological constraints on configuration space

The NACR also constrains which EL networks are topologically admissible. In particular, it rules out configurations that would force inequivalent braid invariants onto topologically equivalent paths. The paradigmatic forbidden example is the Hopf link between noncommuting ELs (Guo et al., 2022).

If two ELs carry braid elements $\pi_1(\bm{P}, X_N) = B_N,$6 and $\pi_1(\bm{P}, X_N) = B_N,$7, a crossing can change the effective invariant by conjugation, $\pi_1(\bm{P}, X_N) = B_N,$8. Morphing between configurations that differ by conjugacy class is impossible unless $\pi_1(\bm{P}, X_N) = B_N,$9 and $B_N$0 commute. This is why Hopf links between noncommuting ELs are excluded, whereas more intricate staggered or tangled rings are allowed provided all involved paths remain consistent with the NACR (Guo et al., 2022).

The same three-state Hamiltonian supports admissible tangled EL “staggered rings,” whose braid structures can be explicitly traced. By contrast, unlinking noncommuting ELs is forbidden because it would require changing noncommuting braids continuously to trivial ones (Guo et al., 2022).

Configuration	Permissibility	Reason
Hopf Link (Noncommuting ELs)	Forbidden	Path invariants differ
Staggered/Tangled Rings	Allowed	NACR preserved
Simple nodal loops	Allowed	No conflicting braiding
Unlinking noncommuting ELs	Forbidden	Requires changing noncommuting braids

These restrictions give multi-EP topology a network-theoretic aspect. The topology is not exhausted by local charges attached to individual ELs; it also governs the global reconnection rules for linked or knotted degeneracy manifolds.

5. Encircling multiple EPs, composite loops, and knot-like parameter-space structures

A complementary perspective arises from the topology of encircling multiple EPs. In multipartite non-Hermitian Su-Schrieffer-Heeger models, complex-energy bands can merge into composite cyclic loops that encircle single or multiple EPs in energy space. When adiabatic band labeling fails, the appropriate invariant is a nonadiabatic cyclic geometric phase

$B_N$1

with gauge invariance restored modulo $B_N$2 (Nehra et al., 2022). In this setting, two-band composite loops around a single EP are compared to Möbius strips, while four-band loops encircling three EPs are compared to Penrose triangles, reflecting $B_N$3 and $B_N$4 periodicities, respectively (Nehra et al., 2022).

The parametric topology of control loops becomes still richer in three-mode systems. For traceless $B_N$5 non-Hermitian Hamiltonians with characteristic polynomial

$B_N$6

the EP condition is the vanishing of the discriminant

$B_N$7

On hyperspheres of fixed $B_N$8, this degeneracy locus obeys $B_N$9 and forms a $N$0 torus knot, specifically a trefoil knot (Guria et al., 2023). The full EP set is a topological cone over that knot.

This has an important consequence for the interpretation of 2D control loops. In two-mode systems, the winding around isolated EPs suffices to determine eigenvalue exchange. In three-mode systems, loops that encircle the same EPs in a two-dimensional slice can yield different eigenvalue braids if they are not homotopic in the full EP-complement, while loops that look different in the slice can yield the same braid if they are homotopic in the full space (Guria et al., 2023). This corrects another common simplification: in multi-EP systems, the topology of a projected control plane does not by itself determine the spectral outcome.

6. Multifold EP$N$1 topology and Abelian classifications

Multi-EP topology in the narrow sense concerns the collective non-Abelian behavior of several EPs or ELs. A related but distinct line of work studies the topology of individual multifold exceptional points, EP$N$2s, using Abelian invariants. These developments clarify how higher-order defective degeneracies are stabilized and why they obey doubling theorems, but they do not replace the braid-based description of collective multi-EP processes (Yoshida et al., 2024).

For generic EP$N$3s, resultant winding numbers are defined from the characteristic polynomial $N$4 through resultants of $N$5 and its derivatives. In the generic case, the resulting invariant classifies a map $N$6, giving codimension $N$7. With symmetries such as $N$8, pseudo-Hermiticity, $N$9, or chiral symmetry, the classification reduces to a map $N \geq 3$0, giving codimension $N \geq 3$1 (Yoshida et al., 2024). The associated doubling theorem states that the total topological charge of all EP$N \geq 3$2s in the Brillouin zone vanishes.

A mathematically more structural formulation interprets the resultant vector through the tenfold classification of Hermitian topological matter, the Mayer–Vietoris sequence, and the classification of vector bundles. In that language, generic EP$N \geq 3$3s are associated with integer-valued winding, while non-local symmetries can induce $N \geq 3$4-protected Fermi arcs (Stålhammar et al., 2024). This establishes Abelian charge conservation for multifold EPs in compact parameter spaces.

An additional extension introduces frequency-momentum winding numbers for nonlinear EP$N \geq 3$5s in $N \geq 3$6-band systems, with nonlinearity entering through eigenvalues. These invariants provide a unified proof of the doubling theorem for arbitrary $N \geq 3$7 and $N \geq 3$8, both without symmetry and under several symmetry constraints including $N \geq 3$9 and charge-conjugation-parity symmetries (Yoshida, 1 Apr 2026). In the linear $B_N$0-symmetric EP2 case, this framework identifies $B_N$1 topology beyond the previously reported $B_N$2 topology (Yoshida, 1 Apr 2026).

Taken together, these results suggest a layered picture. Collective multi-EP dynamics are governed by non-Abelian braid invariants, while isolated higher-order EP$B_N$3 singularities admit Abelian classifications by resultant or frequency-momentum windings. The two viewpoints are complementary rather than interchangeable.

7. Physical platforms, experimental access, and broader significance

The non-Abelian multi-EP framework is formulated for generic multiband non-Hermitian systems and is illustrated by models that could be readily implemented in coupled acoustic cavities, optical waveguides, and ring resonators (Guo et al., 2022). These platforms are especially natural because EPs and ELs are intrinsic to nonconservative systems in optics, acoustics, and related wave settings.

Direct experimental evidence for higher-dimensional multi-EP topology has been obtained in a three-mode mechanical system controlled by optomechanical interaction with a high-finesse optical cavity. In that setting, the locus of second-order EPs was mapped and found to form the predicted trefoil-knot structure, and distinct control loops were shown to produce non-commutative eigenvalue braids consistent with their topology in the full parameter space (Guria et al., 2023).

The broader significance of multi-EP topology lies in two simultaneous facts. First, it imposes strict constraints: some exceptional-line links are forbidden, some reconnections cannot occur, and opposite EP charges do not generically guarantee annihilation. Second, it enlarges the design space: permissible staggered rings, tangled EL networks, path-dependent merging, and higher-order EP formation all become available in principle (Guo et al., 2022). This suggests that manipulations and applications based on exceptional degeneracies in nonconservative systems depend not only on where degeneracies are located, but on the full braid-theoretic structure of how they coexist and evolve.