Diagrammatics for Bose condensation in anyon theories
I. S. Eliëns, J. C. Romers, F. A. Bais
TL;DR
The authors introduce vertex lifting coefficients (VLCs) to establish a full diagrammatic mapping between an anyon theory and its Bose-condensed phase, enabling the complete reconstruction of topological data (S, T, F, R) in the condensed phase from the parent theory. They present a concrete lifting recipe, involving vacuum exchange lines and VLCs, to evaluate any condensed-phase diagram using only the parent theory’s data and a finite set of VLCs. The work covers two primary VLC classes (condensate-only and condensate-involving vertices) and demonstrates the method with explicit cases like SU(2)_4 → SU(3)_1 and SU(2)_10 → SO(5)_1, while also addressing noncondensable bosons. Overall, VLCs provide a powerful, general framework for topological symmetry breaking, with potential applications to Levin-Wen models, boundary phenomena, and topological quantum computation.
Abstract
Phase transitions in anyon models in (2+1)-dimensions can be driven by condensation of bosonic particle sectors. We study such condensates in a diagrammatic language and explicitly establish the relation between the states in the fusion spaces of the theory with the condensate, to the states in the parent theory using a new set of mathematical quantities called vertex lifting coefficients (VLCs). These allow one to calculate the full set of topological data ($S$-, $T$-, $R$- and $F$-matrices) in the condensed phase. We provide closed form expressions of the topological data in terms of the VLCs and provide a method by which one can calculate the VLCs for a wide class of bosonic condensates. We furthermore furnish a concrete recipe to lift arbitrary diagrams directly from the condensed phase to the original phase, such that they can be evaluated using the data of the original theory and a limited number of VLCs. Some representative examples are worked out in detail.