Spontaneous symmetry breaking from anyon condensation

As a cornerstone for condensed matter physics, Landau theory of symmetry breaking dictates that the symmetry group G of a physical system can spontaneously breaks down to any subgroup H, characterized by a local order parameter valued in the quotient group G/H. However this paradigm breaks down when topological orders and anyons come into play. What is the relation between symmetry groups G and H of two symmetry enriched topological orders (SETOs), if they are connected to each other through a continuous quantum phase transition (QPT)? To address this issue, we develop a mathematical framework within the category theory of topological orders, which determines the compatibility of symmetry groups G and H for two SETOs connected by a QPT driven by anyon condensation. We identify two symmetry obstructions for such anyon-condensation transitions. which are demonstrated by many examples.