From entanglement renormalisation to the disentanglement of quantum double models

We describe how the entanglement renormalisation approach to topological lattice systems leads to a general procedure for treating the whole spectrum of these models in which the Hamiltonian is gradually simplified along a parallel simplification of the connectivity of the lattice. We consider the case of Kitaev’s quantum double models, both Abelian and non-Abelian, and we obtain a rederivation of the known map of the toric code to two Ising chains; we pay particular attention to the non-Abelian models and discuss their space of states on the torus. Ultimately, the construction is universal for such models and its essential feature, the lattice simplification, may point towards a renormalisation of the metric in continuum theories.

► The toric code is explicitly mapped to two Ising chains and their diagonalisation. ► The procedure uses tensor network ideas, notably entanglement renormalisation. ► The construction applies to all of Kitaev’s non-Abelian quantum double models. ► The algebraic structure of non-Abelian models is thoroughly discussed. ► The construction is universal and may work on the metric in the continuum limit.