Recent papers by Finkelstein, Galiautdinov, and co-workers [J. Math. Phys. 42, 1489 (2001); 42, 3299 (2001)] discuss a suggestion by Wilczek that non-Abelian projective representations of the permutation group can be used as a new type of particle statistics, valid in any dimension. Wilczek’s suggestion was based in part on an analysis by Nayak and Wilczek (NW) of the non-Abelian representation of the braid group in a quantum Hall system. We point out that projective permutation statistics is not possible in a local quantum field theory as it violates locality, and show that the NW braid group representation is not equivalent to a projective representation of the permutation group. The structure of the finite image of the braid group in a 2n/2−1-dimensional representation is obtained.
Recent papers by Finkelstein, Galiautdinov, and co-workers [J. Math. Phys. 42, 1489 (2001); 42, 3299 (2001)] discuss a suggestion by Wilczek that non-Abelian projective representations of the permutation group can be used as a new type of particle statistics, valid in any dimension. Wilczek’s suggestion was based in part on an analysis by Nayak and Wilczek (NW) of the non-Abelian representation of the braid group in a quantum Hall system. We point out that projective permutation statistics is not possible in a local quantum field theory as it violates locality, and show that the NW braid group representation is not equivalent to a projective representation of the permutation group. The structure of the finite image of the braid group in a 2n/2−1-dimensional representation is obtained.
In a recent paper, Teo and Kane [Phys. Rev. Lett. 104, 046401 (2010)] proposed a three-dimensional (3D) model in which the defects support Majorana fermion zero modes. They argued that exchanging and twisting these defects would implement a set R of unitary transformations on the zero-mode Hilbert space which is a ``ghostly'' recollection of the action of the braid group on Ising anyons in two dimensions. In this paper, we find the group T2n, which governs the statistics of these defects by analyzing the topology of the space K2n of configurations of 2n defects in a slowly spatially varying gapped free-fermion Hamiltonian: ${\mathcal{T}}_{2n}\ensuremath{\equiv}{\ensuremath{\pi}}_{1}({K}_{2n})$. We find that the group ${\mathcal{T}}_{2n}=\mathbb{Z}\ifmmode\times\else\texttimes\fi{}{\mathcal{T}}_{2n}^{r}$, where the ``ribbon permutation group'' T2nr is a mild enhancement of the permutation group S2n: ${\mathcal{T}}_{2n}^{r}\ensuremath{\equiv}{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}E\mathbf{(}({\mathbb{Z}}_{2}){}^{2n}\ensuremath{\rtimes}{S}_{2n}\mathbf{)}$. Here, $E\mathbf{(}({\mathbb{Z}}_{2}){}^{2n}\ensuremath{\rtimes}{S}_{2n}\mathbf{)}$ is the ``even part'' of $({\mathbb{Z}}_{2}){}^{2n}\ensuremath{\rtimes}{S}_{2n}$, namely, those elements for which the total parity of the element in (Z2)2n added to the parity of the permutation is even. Surprisingly, R is only a projective representation of T2n, a possibility proposed by Wilczek [e-print arXiv:hep-th/9806228]. Thus, Teo and Kane's defects realize projective ribbon permutation statistics,'' which we show to be consistent with locality. We extend this phenomenon to other dimensions, codimensions, and symmetry classes. We note that our analysis applies to 3D networks of quantum wires supporting Majorana fermions; thus, these networks are not required to be planar. Because it is an essential input for our calculation, we review the topological classification of gapped free-fermion systems and its relation to Bott periodicity.
A central question in the theory of topological order asks whether the braid group representation BN carried by N non-Abelian anyons uniquely determines the modular data — the S and T matrices — of the underlying topological quantum field theory. We reconcile three lines of analysis into a single assessment. Structurally, we formalize the question as a reduction map Φ from modular tensor category data to projective braid group representations, whose kernel is the locus of potential counterexamples. Two rigorous finiteness results constrain the problem: braid representations from twisted quantum doubles of finite groups factor through finite groups, and the SO(N)2/O(N)2 braid representations are Gaussian with finite image. We then perform an explicit computation on the Ising anyon model — the canonical non-Abelian order of Majorana zero modes — reconstructing the topological spins θσ=eiπ/8, θψ=−1, the total quantum dimension D=2, and the full S matrix from the braid eigenvalues, with all arithmetic shown. We exhibit a Galois-conjugate deformation Ising(7) with identical fusion rules but distinct S matrix, and verify that its braid representation is inequivalent to the Ising representation on B3 by an explicit trace computation, while remaining indistinguishable at the level of fusion rules. We conclude that braid representations constrain but do not provably uniquely determine modular data in general: uniqueness can hold only under additional hypotheses, and the Majorana fusion-rule classification is not closed by braid data alone. The affirmative claim that braiding is a complete invariant is restricted to a narrower regime than sometimes asserted, and we document this divergence explicitly. ## 1. Introduction Non-Abelian anyons in two spatial dimensions are characterized by two layers of algebraic structure. The first is combinatorial: fusion rules Nabc specifying how anyon c Full text and updates: papers.qnfo.org/papers/braid-group-representations-and-modular-data-non-uniqueness-finite-images-and-th/
We construct an explicit projective representation of the braid group from the microscopic Hilbert space of a Kitaev chain extended with algebraically decaying long-range hopping and pairing. The construction proceeds in three steps. First, we identify the topological phase of the long-range Kitaev chain (power-law exponent α) and show that unpaired Majorana zero modes (MZMs) localize at domain walls, with localization length controlled by α. Second, we write the braid unitary U_ij = exp((π/4)γ_iγ_j) acting on the zero-mode Majorana operators γ_i and verify, by explicit matrix computation in a fixed-parity sector, that the braid relation σ₁σ₂σ₁ = σ₂σ₁σ₂ holds exactly while the relation σ⁴ = 1 is lifted: U⁴ = −𝟙 rather than +𝟙. This defines a genuine projective (2-cocycle-twisted) representation whose class is the Ising one, with per-exchange projective phase anchored at π/8. Third, we give quantitative estimates of the physical requirements: zero-mode splitting, gap, and cumulative phase drift under long-range perturbations. Our central quantitative result is that the Ising projective class is stable against long-range hopping with exponent α > 2 and open gap: our projection at α = 3 with representative parameters — based on the assumed, undischarged scaling ε ~ λ^s · s^{−α} rather than a derived bound — gives a per-braid drift of ≤ 1.5 × 10⁻¹⁷ rad and a cumulative drift over 10⁶ braids of ≤ 9.2 × 10⁻¹¹ rad. We connect the construction to projective representations in Chern–Simons theory, to projective ribbon permutation statistics in three dimensions, and to recent results on the non-uniqueness of modular data from braid representations. Falsification criteria and open questions are discussed. ## 1. Introduction Topological phases of matter support exotic quantum statistics: when quasiparticles are exchanged adiabatically, the wave function transforms by a unitary matrix, and for non-Abelian anyons these matrices form a representation of the braid group B_N. A Full text and updates: papers.qnfo.org/papers/projective-braid-group-representations-from-a-microscopic-long-range-kitaev-chai/
The general formulation of quantum statistical mechanics hints at interesting generalizations of the usual Bose/Fermi framework in two spatial dimensions. Anyon statistics, which is essentially a continuous interpolation between Bose and Fermi statistics, is relevant to the Fractional Quantum Hall Effect in two-dimensional (i.e., thin layer) condensed matter systems. In addition, the possibility of non-abelian statistics, in which the multi-particle wavefunction transforms as a representation of a non-abelian group under the exchange of indistinguishable particles, has been explored. Spontaneously broken non-abelian gauge theories in (2 + 1) dimensions often have stable topological defects, called non-abelian vortices, that experience non-abelian statistics. In addition, it has been suggested that degenerate quasihole multiplets in Quantum Hall systems also transform as non-abelian representations of the braid group under particle exchange. In this thesis, I explore the braiding properties of systems of two-cycle flux vortices in a residual S₃ discrete gauge group. The individual vortices are uncharged, but multi-vortex states can have Cheshire charge. The uncharged sectors all have non-vanishing bosonic subspaces, as do the non-abelian charged trivial flux sectors. A kinetic Hamiltonian for vortices on a periodic lattice is constructed. There is a modification to the translational symmetry in the periodically identified direction for non-trivial Z₂ charged sectors. The ground state energies for various three and four vortex sectors is numerically determined. Typically, the ground state is bosonic, with a gap separating it from a non-abelian subspace.
Braiding statistics of p- and q-dimensional topological excitations is conventionally defined in p+q+2 spatial dimensions. We find a novel statistical process WN(X,Y)=(Y−1X−1)N(YX)N for two order-N excitations in p+q+1 dimensions, detecting the Bockstein response A⌣β(B). This new statistics and fermionic loop statistics exhaust all loop statistics in three dimensions whose fusion rules form an Abelian group G, classified by H5(B2G,U(1)). Surprisingly, conventional three-loop braiding goes beyond this classification, so it must have non-Abelian fusion rules. We suggest viewing three-loop braiding as particle-loop braiding together with exotic fusion rules between loops and point-like defects. We also try to clarify the relationship between statistics and symmetry anomaly.
When leaving classical physics and entering the realm of quantum physics, there are many new concepts being introduced. One of the most fundamental ideas in quantum mechanics is that particles no longer have exact known positions, but instead expected values and prob- abilities. This leads to the phenomena of truly identical particles, since they no longer can be distinguished simply by their positions. An important property differentiating different kinds of particles is how a system behaves when two such identical particles are exchanged. Historically, this divided particles into bosons and fermions, corresponding to symmetry and antisymmetry under an exchange. However, in two dimensions a new type of particle appears. These particles are called anyons, and behave differently when particles are exchanged. Anyons can be further divided into abelian and non-abelian anyons, of which this thesis will focus on the latter. The ex- changes can then be represented by the fundamental group of the configuration space of the particles, and in two dimensions this fundamental group is the braid group. Using rotors from a Clifford algebra and studying excitations of Majorana fermions, this thesis will show a way to calculate the exchange matrices of non-abelian anyons, and their corresponding eigenvalues. Furthermore, suggestions on a generalization of this framework along with areas where it can be applied are given.