A Bootstrap Approach to Fermion Condensation

Anyon condensation provides a powerful framework for understanding phase transitions between topological orders. While a systematic bootstrap analysis for bosonic anyon condensation has been well established (Kong 2014), a rigorous mathematical treatment of fermion condensation remains underdeveloped. In this talk, we apply a bootstrap  approach to $2d$ condensation from a bosonic topological order (described by $\mathcal{C}$, a unitary modular tensor category (UMTC)) to a fermionic topological order (described by $\mathcal{D}$, a UMTC over $\text{sRep}(\mathbb{Z}_2^f)$).

We show that the local excitations in $\mathcal{D}$ — a local boson and a local fermion — correspond to a direct sum $A = A_0 \oplus A_1$ in $\mathcal{C}$, which forms a connected supercommutative symmetric normalized-special super $*$-Frobenius algebra with $\dim A_0 = \dim A_1$. We call this a fermionic $2d$-condensable algebra. Then we use bootstrap method to analyze the properties of the condensed phase $\CD$ and the domain wall $\CE$ between these two topological phases.  We also discuss the bulk-to-wall maps and compare with the familiar bosonic condensation scenario.