We determine and study the statistical interparticle potential of an ideal system of non-Abelian Chern-Simons (NACS) particles, comparing our results with the corresponding results of an ideal gas of Abelian anyons. In the Abelian case, the statistical potential depends on the statistical parameter and it has a "quasi-bosonic" behaviour for statistical parameter in the range (0,1/2) (non-monotonic with a minimum) and a "quasi-fermionic" behaviour for statistical parameter in the range (1/2,1) (monotonically decreasing without a minimum). In the non-Abelian case the behavior of the statistical potential depends on the Chern- Simons coupling and the isospin quantum number: as a function of these two parameters, a phase diagram with quasi-bosonic, quasi-fermionic and bosonic-like regions is obtained and investigated. Finally, using the obtained expression for the statistical potential, we compute the second virial coefficient of the NACS gas, which correctly reproduces the results available in literature.
Abstract We review and develop the many-body spectral theory of ideal anyons, i.e. identical quantum particles in the plane whose exchange rules are governed by unitary representations of the braid group on N strands. Allowing for arbitrary rank (dependent on N ) and non-abelian representations, and letting
N→∞ N → ∞ , this defines the ideal non-abelian many-anyon gas. We compute exchange operators and phases for a common and wide class of representations defined by fusion algebras, including the Fibonacci and Ising anyon models. Furthermore, we extend methods of statistical repulsion (Poincaré and Hardy inequalities) and a local exclusion principle (also implying a Lieb–Thirring inequality) developed for abelian anyons to arbitrary geometric anyon models, i.e. arbitrary sequences of unitary representations of the braid group, for which two-anyon exchange is nontrivial.
Topological quantum computation relies on braiding non-Abelian anyons, but requires the underlying topological order to survive imperfect state preparation and environmental noise. We show that the instability of topological order to wavefunction deformations and to decoherence, with the latter probed by syndrome distributions, is generically captured by stat-mech models whose
G×Rep(G) symmetry naturally exposes the corrupting anyonic excitations. As an example, we combine this framework with Monte-Carlo simulations to resolve the stability of
D4 topological order under deformations and quantum channels that proliferate multiple non-Abelian anyon species that are individually unable to condense. We show that beyond a finite threshold, proliferation of two non-Abelian anyon species parasitically condenses a shared Abelian-anyon fusion outcome---destroying the topological order. Our symmetry-based approach suggests the resulting trivial phase is distinct from that obtained by condensing all Abelian anyons; in other words, the trivial phase "remembers" which anyons condensed. This framework provides a first step into identifying the relevant symmetry for optimal decoders, conditioned on syndrome measurements, of non-Abelian topological order.
As opposed to classical mechanics, quantum mechanical particles can be truly identical and lead to new and interesting phenomena. Identical particles can be of different types, determined by their exchange symmetry, which in turn gives rise to statistical repulsion. The exchange symmetry is given by a representation of the exchange group; the fundamental group of the configuration space of identical particles. In three dimensions the exchange group is the permutation group and there are only two types of identical particles; bosons and fermions. While any number of bosons can be at the same place or in the same state, fermions repel each other. In two dimensions the exchange group is the braid group and essentially any exchange symmetry is allowed, such particles are called anyons. Abelian anyons are described by abelian representations of the exchange group and can be seen as giving a continuous interpolation between bosons and fermions. Non-abelian anyons are much more complex and their statistical repulsion is yet largely unexplored. We use the framework of modular tensor categories to show how the statistical repulsion of non-abelian anyons depends on the exchange symmetry. The Fibonacci anyon model is studied, for which explicit results are obtained. We also show how Fibonacci anyons can be used to implement topological quantum computation, providing topologically stable quantum information encoded in the state of non-abelian anyons that can be manipulated via the non-abelian exchange symmetry.