Citation: Bonderson, Parsa. "Splitting the topological degeneracy of non-abelian anyons." Physical review letters 103.11 (2009): 110403.
Web: https://arxiv.org/abs/0905.2726
Tags: Physical, Modular-tensor-categories
In this paper, Parsa Bonderson studies the implications of finite-size effects in topological order on the level of modular categories. Or, in his words, "We examine tunneling of topological charge between non-Abelian anyons as a perturbation of the long-range effective theory of a topologically ordered system".
The idea is that all errors in topological quantum systems at low temperature are essentially equivalent to uncontrolled trajectories of anyons. This means that one should introduce a physically meaningful Hamiltonian which is a sum over all possible anyon trajectories, and add amplitudes corresponding to the likelihood of these events happening.
Parsa Bonderson argues that the main terms to analyze should be nearest-neighbor tunneling terms, which are mathematically the same as projections onto overall topological charge. This is because other terms (like a third anyon tunneling around two neighbors) have mathematically identical structure, and more exotic terms (like three-anyon interactions) are much more unlikely so the amplitudes corresponding to them should be negligible.
After this general philosophy and modeling, Bonderson demonstrates that one should generically expect these tunneling terms to completely split the degeneracy between the different fusion channels.