One dimensional gapped quantum phases and enriched fusion categories

TL;DR

This paper validates that enriched fusion categories can describe all 1D gapped quantum phases, using Ising and Kitaev chains to explicitly demonstrate the categorical structures of these phases and their boundaries.

Contribution

It provides explicit categorical descriptions of 1D gapped quantum phases and their boundaries, confirming a unifying theoretical framework for various quantum phases.

Findings

Spacetime observables form a fusion category enriched in a braided fusion category.

Categorical descriptions of boundaries are explicitly constructed.

Classification of all 1D gapped phases with finite onsite symmetry is provided.

Abstract

In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and certain gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category. We also study the categorical descriptions of the boundaries of these models. In the end, we provide a classification of and the categorical descriptions of all 1-dimensional (the spatial dimension) gapped quantum phases with a finite onsite symmetry.