Fermion condensation and super pivotal categories

TL;DR

This work develops a comprehensive framework for fermionic topological phases via fermion condensation, connecting bosonic parent theories to fermionic condensed theories using back-wall spin-structure constructions and a super pivotal categorical formalism. It introduces the distinction between $m$-type and $q$-type objects and develops a fermionic tube category, establishing key correspondences like Tube($\mathcal{C}/\psi$) ≅ $\mathcal{C} \times \overline{\mathcal{C}/\psi}$, which underpins modular data, Verlinde-type dimension formulas, and state-sum constructions. The authors build a fermionic Levin-Wen-like lattice Hamiltonian and a fermionic TVBW state sum, and devote substantial effort to detailed condensation examples including Ising, $SO(3)_6$, and $12E_6/\psi$, illustrating how quasiparticle spectra and modular data arise in fermionic settings. The framework provides explicit computational tools for tube categories, spin structures, and their interplay with condensation, offering a solid foundation for classifying and realizing fermionic topological orders with potential links to quantum computation through Majorana-type excitations and Kitaev-chain dynamics.

Abstract

We study fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases which contain a fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions, and condensing pairs of physical and emergent fermions. There are two distinct types of objects in fermionic theories, which we call "m-type" and "q-type" particles. The endomorphism algebras of q-type particles are complex Clifford algebras, and they have no analogues in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations in fermionic topological phases. We then prove a series of results relating data in condensed theories to data in their parent theories; for example, if $\mathcal{C}$ is a modular tensor category containing a fermion, then the tube category of the condensed theory satisfies $\textbf{Tube}(\mathcal{C}/ψ) \cong \mathcal{C} \times (\mathcal{C}/ψ)$. We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted to three detailed examples of performing fermion condensation to produce fermionic topological phases: we condense fermions in the Ising theory, the $SO(3)_6$ theory, and the $\frac{1}{2}\text{E}_6$ theory, and compute the quasiparticle excitation spectrum in each of these examples.