General procedure for determining braiding and statistics of anyons using entanglement interferometry

Recently, it was argued that the braiding and statistics of anyons in a two-dimensional topological phase can be extracted by studying the quantum entanglement of the degenerate ground states on the torus. This construction either required a lattice symmetry (such as $\pi /2$ rotation) or tacitly assumed that the minimum entanglement states (MESs) for two different bipartitions can be uniquely assigned quasiparticle labels. Here we describe a procedure to obtain the modular $\mathcal{S}$ matrix, which encodes the braiding statistics of anyons, which does not require making any of these assumptions. Our strategy is to coherently compare MESs of three independent entanglement bipartitions of the torus, which leads to a unique modular $\mathcal{S}$. This procedure also puts strong constraints on the modular $\mathcal{T}$ and $\mathcal{U}$ matrices without requiring any symmetries, and in certain special cases, completely determines it. Our method applies equally to Abelian and non-Abelian topological phases.

Figure 1

Left panel: Two nontrivial entanglement bipartitions shown as the dashed lines that separate the system into two cylindrical subsystems (periodic boundary condition is assumed through all boundaries). Right panel: Two sets of primitive vectors for the same torus; we use ${\vec{w}}_2$ to label the boundary direction of each entanglement bipartition.

Figure 2

Left panel: An additional and inequivalent entanglement bipartition scheme. Right panel: The corresponding primitive vectors, where ${\vec{w}}_2$ labeling the boundary direction of the entanglement bipartition is the sum of the previous two in Fig. 1.