Quantum cellular automata and categorical dualities of spin chains

TL;DR

This paper explores categorical dualities in quantum spin chains, providing criteria for extending these dualities to quantum cellular automata and classifying such extensions using advanced algebraic tools.

Contribution

It introduces a categorical criterion for extending dualities to quantum cellular automata in spin chains, utilizing Doplicher-Haag-Roberts bimodules for classification.

Findings

A criterion for duality extension exists based on categorical structures.

Extensions form a torsor over invertible objects in the symmetry category.

Classification results for dualities in the case of finite group symmetries.

Abstract

Dualities play a central role in the study of quantum spin chains, providing insight into the structure of quantum phase diagrams and phase transitions. In this work we study categorical dualities, which are defined as bounded-spread isomorphisms between algebras of symmetry-respecting local operators on a spin chain. We consider generalized global symmetries that correspond to unitary fusion categories, which are represented by matrix-product operator algebras. A fundamental question about dualities is whether they can be extended to quantum cellular automata on the larger algebra generated by all local operators that respect the unit matrix-product operator. For conventional global symmetries, which are on-site representations of finite groups, this larger algebra is simply the tensor product of algebras associated to individual spins in the chain. We present a solution to the extension problem using the machinery of Doplicher-Haag-Roberts bimodules. Our solution provides a crisp categorical criterion for when an extension of a duality exists. We show that the set of possible extensions form a torsor over the invertible objects in the relevant symmetry category. As a corollary, we obtain a classification result concerning dualities in the group case.