Space–time geometry of topological phases

The 2 + 1 dimensional lattice models of Levin and Wen (2005) [1] provide the most general known microscopic construction of topological phases of matter. Based heavily on the mathematical structure of category theory, many of the special properties of these models are not obvious. In the current paper, we present a geometrical space–time picture of the partition function of the Levin–Wen models which can be described as doubles (two copies with opposite chiralities) of underlying anyon theories. Our space–time picture describes the partition function as a knot invariant of a complicated link, where both the lattice variables of the microscopic Levin–Wen model and the terms of the Hamiltonian are represented as labeled strings of this link. This complicated link, previously studied in the mathematical literature, and known as Chain–Mail, can be related directly to known topological invariants of 3-manifolds such as the so-called Turaev–Viro invariant and the Witten–Reshitikhin–Turaev invariant. We further consider quasi-particle excitations of the Levin–Wen models and we see how they can be understood by adding additional strings to the Chain–Mail link representing quasi-particle world-lines. Our construction gives particularly important new insight into how a doubled theory arises from these microscopic models.