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arXiv:2604.00492

Braiding and exchange statistics of liquid crystalline Majorana quasiparticles

A. I. Tóth, G. Negro, A. D. Huxley, D. Marenduzzo

cond-mat.softquant-phSubmitted Apr 1, 2026• Announced Apr 2, 2026
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Abstract

Liquid crystalline defects in 3D can be viewed as geometric spinors, whose emergent properties are reminiscent of those of topological excitations in quantum condensed matter, such as Majorana quasiparticles. However, it is unclear how deep this analogy is, and whether this is a purely mathematical mapping, or it extends to key physical features, such as the exchange statistics or braiding behaviour. To address this question, here we consider a simple pattern made up of four nematic Majorana-like defect profiles, and ask how the defect profiles change as we braid them repeatedly around each other. Surprisingly, we find that in a large range of parameter space the defect profiles behave as classical analogues of non-Abelian anyons, which can be described in our case by defect bivectors moving on a Bloch-like hemisphere. Elastic interactions and dynamical effects enhance the complexity of the gates which can be performed by braiding these quasiparticles, making these liquid crystalline spinors promising candidates as components of topological computers.

Comments
5 pages, 4 figures plus 9 pages of supplementary material + 6 figures

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Liquid crystalline defects in 3D can be viewed as geometric spinors, whose emergent properties are reminiscent of those of topological excitations in quantum condensed matter, such as Majorana quasiparticles. However, it is unclear how deep this analogy is, and whether this is a purely mathematical mapping, or it extends to key physical features, such as the exchange statistics or braiding behaviour. To address this question, here we consider a simple pattern made up of four nematic Majorana-like defect profiles, and ask how the defect profiles change as we braid them repeatedly around each other. Surprisingly, we find that in a large range of parameter space the defect profiles behave as classical analogues of non-Abelian anyons, which can be described in our case by defect bivectors moving on a Bloch-like hemisphere. Elastic interactions and dynamical effects enhance the complexity of the gates which can be performed by braiding these quasiparticles, making these liquid crystalline spinors promising candidates as components of topological computers.
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We study symmetries and defects of a wide class of two-dimensional Abelian topological phases characterized by Lie algebras. We formulate the symmetry group of all Abelian topological field theories. The symmetries relabel quasiparticles (or anyons) but leave exchange and braiding statistics unchanged. Within the class of ADEADEADE phases, in particular, these anyonic symmetries have a natural origin from the Lie algebra. We classify one-dimensional gapped phases along the interface between identical topological states according to symmetries. This classification also applies to gapped edges of a wide range of fractional quantum spin Hall (QSH) states. We show that the edge states of the ADEADEADE QSH systems can be gapped even in the presence of time-reversal and charge-conservation symmetries. We distinguish topological point defects according to anyonic symmetries and bound quasiparticles. Although in an Abelian system, they surprisingly exhibit non-Abelian fractional Majorana-like characteristics from their fusion behavior.
We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions of unitary modular tensor categories. The algebraic data of these categories afford projective unitary representations of the braid group. Elements in the image of such representations correspond to quantum gates arising from exchanging anyons and symmetry defects in topological phases of matter with symmetry. We provide some small examples that highlight features of practical interest for quantum computing. In particular, symmetry defects show the potential to generate non-abelian statistics from abelian topological phases and to be used as as a tool to enlarge the set of quantum gates accessible to an anyonic device.
Defects between gapped boundaries provide a possible physical realization of projective non-Abelian braid statistics. A notable example is the projective Majorana/parafermion braid statistics of boundary defects in fractional quantum Hall/topological insulator and superconductor heterostructures. In this paper, we develop general theories to analyze the topological properties and projective braiding of boundary defects of topological phases of matter in two spatial dimensions. We present commuting Hamiltonians to realize defects between gapped boundaries in any (2+1)D(2+1)D(2+1)D untwisted Dijkgraaf-Witten theory, and use these to describe their topological properties such as their quantum dimension. By modeling the algebraic structure of boundary defects through multifusion categories, we establish a bulk-edge correspondence between certain boundary defects and symmetry defects in the bulk. Even though it is not clear how to physically braid the defects, this correspondence elucidates the projective braid statistics for many classes of boundary defects, both amongst themselves and with bulk anyons. Specifically, three such classes of importance to condensed matter physics/topological quantum computation are studied in detail: (1) a boundary defect version of Majorana and parafermion zero modes, (2) a similar version of genons in bilayer theories, and (3) boundary defects in D(S3)\mathfrak{D}({S}_{3})D(S3​).
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It has recently been realized that a general class of non-Abelian defects can be created in conventional topological states by introducing extrinsic defects, such as lattice dislocations or superconductor-ferromagnet domain walls in conventional quantum Hall states or topological insulators. In this paper, we begin by placing these defects within the broader conceptual scheme of extrinsic twist defects associated with symmetries of the topological state. We explicitly study several classes of examples, including Z2{Z}_{2}Z2​ and Z3{Z}_{3}Z3​ twist defects, where the topological state with NNN twist defects can be mapped to a topological state without twist defects on a genus $g\ensuremath{\propto}N$ surface. To emphasize this connection we refer to the twist defects as genons. We develop methods to compute the projective non-Abelian braiding statistics of the genons, and we find the braiding is given by adiabatic modular transformations, or Dehn twists, of the topological state on the effective genus ggg surface. We study the relation between this projective braiding statistics and the ordinary non-Abelian braiding statistics obtained when the genons become deconfined, finite-energy excitations. We find that the braiding is generally different, in contrast to the Majorana case, which opens the possibility for fundamentally novel behavior. We find situations where the genons have quantum dimension 2 and can be used for universal topological quantum computing (TQC), while the host topological state is by itself nonuniversal for TQC.