Diagrammatics for Bose condensation in anyon theories

We reformulate the topological symmetry-breaking scheme for phase transitions in systems with anyons in a graphical manner. A set of quantities called vertex lifiting coefficients (VLCs) is introduced and used to specify the full operator content of the broken phase. First, it is shown how the assumption that a set of charges behaves like the vacuum of a new theory naturally leads to diagrammatic consistency conditions for a condensate. This recovers the notion of a condensate used in earlier approaches and uncovers the connection to pure mathematics. The VLCs are needed to solve the consistency conditions and establish the mapping of the fusion and splitting spaces of the broken theory into the parent phase. This enables one to calculate the full set of topological data $(S,T,R$, and $F$ matrices) for the condensed phase and closed-form expressions in terms of the VLCs are provided. We furthermore furnish a concrete recipe to lift arbitrary diagrams directly from the condensed phase to the original phase using only a limited number of VLCs and we describe a method for the explicit calculation of VLCs for a large class of bosonic condensates. This allows for the explicit calculation of condensed-phase diagrams in many physically relevant cases and representative examples are worked out in detail.

Figure 1

Suspenders diagram: pictorial representation of the monodromy equation (A32) (taken from Ref. [18]).