The algebraic tools used to study topological phases of matter are not clearly suited to studying processes in which the bulk energy gap closes, such as phase transitions. We describe an elementary two edge thought experiment, which reveals the effect of an anyon condensation phase transition on the robust edge properties of a sample, bypassing a limitation of the algebraic description. In particular, the two edge construction allows some edge degrees of freedom to be tracked through the transition, despite the bulk gap closing. The two edge model demonstrates that bulk anyon condensation induces symmetry breaking in the edge model. Further, the construction recovers the expected result that the number of chiral current carrying modes at the edge cannot change through anyon condensation. We illustrate the construction through detailed analysis of anyon condensation transitions in an achiral phase, the toric code, and in chiral phases, the Kitaev spin liquids.
A topological order is a new quantum phase that is beyond Landau's symmetry-breaking paradigm. Its defining features include robust degenerate ground states, long-range entanglement and anyons. It was known that
R- and
F-matrices, which characterize the fusion-braiding properties of anyons, can be used to uniquely identify topological order. In this article, we explore an essential question: how can the
R- and
F-matrices be experimentally measured? By using quantum simulations based on a toric code model with boundaries and state-of-the-art technology, we show that the braidings, i.e. the
R-matrices, can be completely determined by the half braidings of boundary excitations due to the boundary-bulk duality and the anyon condensation. The
F-matrices can also be measured in a scattering quantum circuit involving the fusion of three anyons in two different orders. Thus we provide an experimental protocol for measuring the unique identifiers of topological order.