Ground state degeneracy and module category

TL;DR

This paper introduces a systematic classification method for connected étale algebras in pre-modular categories, linking mathematical structures to physical ground state degeneracies and symmetry breaking in gapped phases.

Contribution

It develops a new classification framework for étale algebras in pre-modular categories, with applications to physics, especially in understanding ground state degeneracies and symmetry breaking.

Findings

Bound on ranks of module categories by FPdim

Classification of étale algebras in categories from physics

Most examples show spontaneous symmetry breaking

Abstract

We develop a systematic method to classify connected \'etale algebras $A$'s in (possibly degenerate) pre-modular category $\mathcal B$. In particular, we find the category of $A$-modules, $\mathcal B_A$, have ranks bounded from above by $\lfloor\text{FPdim}(\mathcal B)\rfloor$. For demonstration, we classify connected \'etale algebras in some $\mathcal B$'s, which appear in physics. Physically, the results constrain (or fix) ground state degeneracies of (certain) $\mathcal B$-symmetric gapped phases. We study massive deformations of rational conformal field theories such as minimal models and Wess-Zumino-Witten models. In most of our examples, the classification suggests the symmetries $\mathcal B$'s are spontaneously broken.