On the structure of categorical duality operators

TL;DR

This paper explores the structure of categorical duality operators in spin and anyon chains, linking them to quantum cellular automata and fusion categories, revealing their behavior under symmetries and renormalization flows.

Contribution

It introduces a systematic framework for understanding duality operators via QCA and fusion categories, highlighting their classification and symmetry properties in quantum models.

Findings

Duality operators form a simplex with extreme points linked to simple objects.

External symmetries generated by duality operators lead to weakly integral fusion categories in the IR.

QCA define invertible bimodule categories that classify duality operators.

Abstract

We systematically study categorical duality operators on spin (and anyon) chains with respect to an internal fusion category symmetry C. We parameterize duality operators on the quasi-local algebra in terms of data dependent on the associated quantum cellular automata (QCA) on the symmetric subalgebra $B$. In particular, a QCA $\alpha$ on $B$ defines an invertible C-C bimodule category $M_{\alpha}$, and the duality operators extending $\alpha$ form a simplex, with extreme points in bijective correspondence with the simple object of $M_{\alpha}$. Then we consider the structure of external symmetries generated by a family of duality operators, and show that if the UV models are all defined on tensor product Hilbert spaces, these categories necessarily flow to weakly integral fusion categories in the IR.