BRAID Parity Effect in Topology & Quantum Systems

• BRAID Parity Effect is a phenomenon where parity-labeled braid groups extend classical braid theory to capture refined topological invariants in both algebraic and physical systems.
• It employs discrete labels (e.g., Z2) and modified Reidemeister moves to enforce parity constraints, leading to richer invariants and classification compared to classical braids.
• The effect manifests physically through robust features such as unpaired exceptional points in non-Hermitian systems and parity-protected Majorana braiding for quantum computation.

The BRAID Parity Effect is a multifaceted phenomenon arising across topological phases of matter, non-Hermitian band theory, knot theory, and Majorana-based quantum systems. At its core, “braid parity” codifies the ways in which parity information—understood algebraically as discrete labels or symmetries, or physically as topological invariants—modifies the structure, classification, and protected behavior of braids, exceptional points, and related topological defects. This effect sharply distinguishes between systems and classifications that are fundamentally Abelian (where parity assignment or winding-number suffices) and those that are non-Abelian, where full braid-group invariance, marked parity, or noncommuting defects stabilize unpaired or otherwise robust features.

1. Algebraic Structures of Braids with Parity

The classical $n$-strand Artin braid group $\mathrm{Br}^n$ is generated by elementary overcrossings $\sigma_i$ ($i=1,\ldots,n-1$), subject to far-commutativity and the third Reidemeister move: $$\sigma_i \sigma_j = \sigma_j \sigma_i \quad (|i-j|\geq2),\qquad \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}.$$ Braid parity extends these relations by associating a label (typically $\mathbb{Z}_2$ or a more general group $G$) to each crossing. The $\mathbb{Z}_2$-braid group $Br_2^n$ is generated by $\sigma_{i,0}$ (even) and $\sigma_{i,1}$ (odd), with the parity-constrained relation for the third Reidemeister move: $$\sigma_{i,\varepsilon} \sigma_{i+1,\eta} \sigma_{i,\xi} = \sigma_{i+1,\xi} \sigma_{i,\eta} \sigma_{i+1,\varepsilon},\quad \varepsilon + \eta + \xi \equiv 0\ \mathrm{mod}\ 2$$ (Fedoseev et al., 2015). This structure prevents cancellation/interchanges unless parity is conserved around each affected region, resulting in a finer equivalence than for classical braids.

The concept generalizes to $G$-braid groups, where crossings are decorated with group elements $g\in G$ and group product constraints replace the above parity sum (Fedoseev et al., 2015).

Parity can also be attached to strands (“dots”), producing the dotted braid group $Br_d^n$, in which the operation

$\sigma_{i,1} = \gamma_i\,\sigma_i\,\gamma_{i+1}$

realizes an odd crossing using strand-level marks (Fedoseev et al., 2015). The group $G_{n,p}^2$ further extends free braid groups to encode parity, with explicit geometric realization via dots or even as the linking number with an added “virtual” strand (Kim, 2016).

2. Parity Effects in Braid Topology and Invariants

The presence of parity and its algebraic encoding leads to sharp distinctions in allowable moves and equivalence classes. In $Br_2^n$, second Reidemeister “lune” cancellations are restricted to pairs of identical parity; “mixed” parity crossings persist and endow the group with richer structure. Third Reidemeister moves are only permitted when the parity condition is satisfied, leading to the possibility of new invariants unobservable in the purely classical case.

Embedding the classical group using only even crossings is injective: “If two classical braids (all crossings even) become equal in $Br_2^n$, then they are equal in the ordinary Artin group” (Theorem 3.5) (Fedoseev et al., 2015). This establishes that parity distinctions are not an artifact but represent genuine extra structure.

In the free braid context, parity is encoded algebraically and geometrically, leading to a robust classification whereby a crossing marked as “odd” is indistinguishable from a standard crossing in an $(n+1)$th strand, up to mod~2 linking (Kim, 2016). This demonstrates a deep connection between parity and higher-dimensional linking invariants.

3. Braid Parity in Topological Phases and Non-Hermitian Systems

Parity-affiliated braid invariants arise in non-Hermitian band theory, where the spectrum’s eigenvalues braid as momentum-space paths are traversed. In $N$-band systems, the energy eigenvalues form points in $\mathrm{Conf}_N(\mathbb{C})$; as a closed loop is traced in the Brillouin zone, the eigenvalues undergo a permutation described by a word in the braid group $B_N$ (König et al., 2022, Li et al., 2023).

In the non-Hermitian case, a “braid parity effect” emerges: an odd (“parity-odd”) combination of exceptional points (EPs)—for example, a single unpaired EP of order $n=3$—cannot be locally pairwise annihilated. Instead, full non-Abelian braid invariants such as the commutator $[\sigma_1,\sigma_2]$ in $B_3$ protect the existence of unpaired exceptional points (“non-Abelian monopoles”). Abelian invariants (winding numbers) are insufficient, as they cannot capture commutator information nor stability against pairwise fusion/annihilation unless nontrivial global (non-Abelian) braid invariants vanish (König et al., 2022).

The parity (odd/even site index) can control the order and symmetry properties of EPs: in the non-Hermitian SSH model, a defect away from the midpoint yields an EP$_3$ (odd parity), but at the symmetric midpoint, parity reflection symmetry plus chiral symmetry makes for an EP$_4$ (even parity) (Li et al., 2023).

System Type	Role of Parity	Key Consequence
$\mathbb{Z}_2$-braid ($Br_2^n$)	Crossing label	Restrictions on moves
Non-Hermitian band, $N\geq 3$	Braid commutators	Unpaired EPs, non-Abelian
Free braids ($G_{n,p}^2$), points on strands	Strand parity	Algebra–geometry link

4. Physical Manifestations and Experimental Signatures

The braid-parity effect produces concrete and observable phenomena:

• Parity-protected robust states: In non-Hermitian band systems with $N\geq3$, unpaired EPs are stabilized by non-Abelian braid invariants; only braiding around large loops in the torus can remove them. Their presence is marked by Fermi arcs in the complex spectrum, which may be closed or noncontractible (König et al., 2022, Li et al., 2023).
• Superconducting diode effect: A “parity-protected” version of the superconducting diode effect arises in topological Josephson junctions with Majorana zero modes, where the enhancement in nonreciprocity is robust against quasiparticle poisoning and independent of the parity sector (Legg et al., 2023).
• Dynamical phase transitions: In non-Hermitian stochastic models, transition between trivial and braid phases is marked by nonanalyticity of the cumulant generating function at specific points, e.g., the appearance of cusps at $\chi = \pi$ and decaying oscillations of parity probabilities—directly traceable to nontrivial braid monodromy (Ren et al., 2012).
• Majorana braiding/fusion outcomes: The even/odd number of Majorana exchanges (braids) defines the action on the ground-state parity. Even numbers return the system to the initial parity up to a phase; odd numbers result in superpositions, observable in parity measurement statistics (e.g., after two braids, vanishing return probability is a topological indicator) (Clarke et al., 2016).

5. Topological Quantum Computation and Limitations

The non-Abelian braid-parity framework underpins proposals for topological quantum computation using Majorana zero modes. The unitary corresponding to a single braid, $U_{ij} = \exp(\pi/4\,\gamma_i\gamma_j)$, rotates the ground-state manifold; braid parity determines the sequence of quantum gates enacted. In systems with Kramers pairs of Majorana states, local Berry phase accumulation (parity-encoded) can render braid operations path-dependent, undermining topological protection unless additional symmetries enforce “no local mixing” (Wölms et al., 2015). In such cases, purely topological, path-independent gate operation is restored only when parity defects are trivial in all possible projections.

A key experimental implication is that signatures of true non-Abelian statistics require statistics over multiple successive braids; fusion alone, or an odd number of exchanges, may mimic topological signals in trivial systems (Clarke et al., 2016).

6. Mathematical Encoding and Open Problems

Braid parity encapsulates a range of algebraic and geometric constructions:

• Parity labels correspond to $\mathbb{Z}_2$ or arbitrary group elements per crossing (decorated braid groups) (Fedoseev et al., 2015).
• The geometric realization of parity arises via points or dots on strands; algebraically, the mapping $a_{ij}^1 \mapsto \tau_i a_{ij} \tau_i$ encodes parity by counting dots (i.e., parity is “pointed”) (Kim, 2016).
• Embedding into higher-strand braid groups demonstrates that parity is equivalent to mod~2 linking with an auxiliary strand, which can never be removed by classical Reidemeister-type moves alone (Kim, 2016).

Open questions include the realization of virtual or dotted braids inside classical braid groups, the search for new invariants and embedding theorems, and the extension of parity constructions to more complex tangles and free group settings (Fedoseev et al., 2015).

7. Summary and Outlook

The BRAID Parity Effect is a structurally intrinsic feature of various algebraic, topological, and physical models wherever parity-marked or non-Abelian braid group structure is present. It enables the stabilization and classification of topological objects, such as unpaired exceptional points and robust Majorana-based operations, that evade detection by Abelian invariants or winding numbers. Its mathematical encoding in terms of marked braid groups, dotted braids, or parity-augmented free groups offers a unifying algebraic framework, while its physical signatures in condensed matter, quantum computation, and stochastic dynamics highlight its broad applicability and significance in contemporary research (König et al., 2022, Ren et al., 2012, Legg et al., 2023, Fedoseev et al., 2015, Kim, 2016, Li et al., 2023, Clarke et al., 2016, Wölms et al., 2015).