A theory of 2+1D bosonic topological orders

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 all 2+1D bosonic topological orders (i.e. topological orders in local bosonic/spin/qubit systems).

Hierarchy Construction and Non-Abelian Families of Generic Topological Orders

“…A classic example of such a classification result is the "periodic table" for chemical elements. As for the topological ordered [1,2] phases of matter, which has drawn more and more research interest recently, we are already able to create some tables for them [3][4][5][6][7][8], via the theory of (pre-)modular categories. However, efforts are needed to further understand the tables, for example, to reveal some "periodic" structures in the table.…”

“…Such a theory is the so-called "unitary modular tensor category (UMTC) theory" (for a review and much more details on the UMTC, see Ref. [5]). …”

“…where N ij k is the fusion coefficient in C. 5,6]. It is important to fix a "gauge" for t i , say by choosing t i ∈ ½0; 1Þ.…”

“…As an application, let us explain the "eightfold way" observed in the table of topological orders [5,6]: whenever there is a fermionic quasiparticle, the topological order has eight companions with the same rank and quantum dimensions but different spins and central charges. If we apply the one-step condensation with a c being a fermion, and M c ¼ AE1, a new topological order of the same rank is obtained [22].…”

“…A more interesting non-Abelian state (which can perform universal topological quantum computation [26]) is Ψ ν¼ 3 2 ðfz i gÞ ¼ ½χ 3 ðfz i gÞ 2 , whose edge has c ¼ 21 5 . The bulk effective theory is the SUð2Þ f −3 CS theory with four types of anyons [17,19].…”