A Wilson line picture of the Levin–Wen partition functions

Abstract

Lattice Hamiltonians (e.g. Levin and Wen 2005 Phys. Rev. B 71 045110) can be constructed that have a low energy description, that is a doubled Chern–Simons (CS) theory—two independent opposite chirality topological sectors. We show that the partition function of these theories is an expectation of Wilson loops that form a link in 2+1 dimensional spacetime known in the mathematical literature as chain–mail. This geometric construction establishes a concrete connection between the lattice models and continuum Chern–Simons theories, allowing us to use well-established results on the latter to obtain a physical interpretation of the lattice model Hilbert space and Hamiltonian, its topological invariance, exactness under coarse-graining and how two opposite chirality sectors of the doubled theory arise. These features of the lattice models can thus be situated in the broader context of topological invariants obtained from CS theories.