Categories of quantum liquids I

TL;DR

This paper develops a comprehensive mathematical framework for quantum liquids, unifying various phases like topological orders, symmetry-breaking, and gapless states through higher category theory and condensation completion.

Contribution

It introduces a new mathematical theory of higher categories for quantum liquids, including results on $E_m$-multi-fusion categories and a unified approach to diverse quantum phases.

Findings

Unified mathematical description of all quantum liquids.

Explicit computation of equivalence types of quantum liquids.

Foundation for classifying quantum phases via higher categories.

Abstract

We develop a mathematical theory of separable higher categories based on Gaiotto and Johnson-Freyd's work on condensation completion. Based on this theory, we prove some fundamental results on $E_m$-multi-fusion higher categories and their higher centers. We also outline a theory of unitary higher categories based on a $*$-version of condensation completion. After these mathematical preparations, based on the idea of topological Wick rotation, we develop a unified mathematical theory of all quantum liquids, which include topological orders, SPT/SET orders, symmetry-breaking orders and CFT-like gapless phases. We explain that a quantum liquid consists of two parts, the topological skeleton and the local quantum symmetry, and show that all $n$D quantum liquids form a $*$-condensation complete higher category whose equivalence type can be computed explicitly from a simple coslice 1-category.