Anyons are not energy eigenspaces of quantum double Hamiltonians

Abstract

Kitaev's quantum double models, including the toric code, are canonical examples of quantum topological models on a two-dimensional spin lattice. Their Hamiltonian defines the ground space by imposing an energy penalty to any nontrivial flux or charge, but does not distinguish among those. We generalize this construction by introducing a family of Hamiltonians made of commuting four-body projectors that provide an intricate splitting of the Hilbert space by discriminating among nontrivial charges and fluxes. Our construction highlights that anyons are not in one-to-one correspondence with energy eigenspaces, a feature already present in Kitaev's construction. This discrepancy is due to the presence of local degrees of freedom in addition to topological ones on a lattice.

Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview

“…As mentioned previously, the magnetic excitations in this model are analogous to the magnetic excitations from Kitaev's Quantum Double model [13], and the potential start-point excitation is also present for the magnetic ribbon operators in that model (see for example Ref. [79]). We can interpret the start-point of the magnetic membrane operators in this model, as well as the start-points of the magnetic ribbon operators in Kitaev's Quantum Double model, in terms of gauge theory.…”

Excitations in the Higher Lattice Gauge Theory Model for Topological Phases III: the 3+1d Case

“…This means that, as we stated earlier, even if the coefficients α e are such that the blob ribbon operator excites the start-point vertex, this is not reflected in the charge of the excitation. The idea that the extra vertex excitation on an object may not correspond to an additional charge is something that is familiar from Kitaev's Quantum Double model in 2+1d [11,24].…”