Crossed module and higher order versions of Kitaev-Model

Joao Faria Martins, University of Leeds. Part of the applied mathematics integrable systems seminars series.

The Kitaev Model is a finite dimensional totally solvable quantum field theory model for discrete gauge connections on  a surface, modelling topological phases of matter. Point-like topological excitations of Kitaev model possess anyonic behaviour and models for topological quantum computing can be derived from them.

In this talk, I will review the notion of crossed modules, higher gauge fields, discrete higher gauge fields, and their 2-dimensional holonomy. With those tools in hand I will show a higher gauge field notion of Kitaev model. I will lightly discuss loop-like topological excitations of the model, and how they can be used to construct isotopy invariants of loop braids in 3+1D space.

References:

--  A. Yu. Kitaev:  Fault-tolerant quantum computation by anyons.  Annals Phys. 303 (2003) 2-30

-- Bullivant A, Calçada M, Kádár Z; Martin P, Faria Martins J: Topological phases from higher gauge symmetry in 3+1 dimensions. PHYSICAL REVIEW B 95, 155118 (2017)

-- Bullivant A, Calçada M, Kádár Z, Martin P, Faria Martins J:   Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry.  arXiv:1702.00868 [math-ph]