A theory of 2+1D bosonic topological orders

TL;DR

This paper reviews the mathematical framework for classifying 2+1D bosonic topological orders, highlighting their unique properties beyond traditional symmetry-breaking phases, based on long-range entanglement and topological invariants.

Contribution

It introduces a systematic classification scheme for 2+1D bosonic topological orders using measurable quantities (S, T, c), advancing understanding of topologically ordered phases.

Findings

Describes a framework based on measurable quantities (S, T, c) for topological orders.

Classifies 2+1D bosonic topological phases systematically.

Connects topological order with long-range entanglement and mathematical categories.

Abstract

In primary school, we were told that there are four phases of matter: solid, liquid, gas, and plasma. In college, we learned that there are much more than four phases of matter, such as hundreds of crystal phases, liquid crystal phases, ferromagnet, anti-ferromagnet, superfluid, etc. Those phases of matter are so rich, it is amazing that they can be understood systematically by the symmetry breaking theory of Landau. However, there are even more interesting phases of matter that are beyond Landau symmetry breaking theory. In this paper, we review new "topological" phenomena, such as topological degeneracy, that reveal the existence of those new zero-temperature phases -- topologically ordered phases. Microscopically, topologically orders are originated from the patterns of long-range entanglement in the ground states. As a truly new type of order and a truly new kind of phenomena, topological order and long-range entanglement require a new language and a new mathematical framework, such as unitary fusion category and modular tensor category to describe them. In this paper, we will describe a simple mathematical framework based on measurable quantities of topological orders $(S,T,c)$ proposed around 1989. The framework allows us to systematically describe/classify 2+1D topological orders (ie topological orders in local bosonic/spin/qubit systems)..