A theory of 2+1D bosonic topological orders

TL;DR

The paper develops a program to characterize 2+1D bosonic topological orders beyond Landau theory by using universal invariants. It presents two equivalent data frameworks, $(S,T,c)$ and $(N^{ij}_k,s_i,c)$, from which fusion, spin, and braiding data, as well as the chiral central charge, are derived via Verlinde relations and modular transformations. A substantial portion is devoted to a numerical classification of low-rank bosonic orders, including their Abelian and non-Abelian content, stacking behavior, and connections to parafermion theories and $SO(k)_2$ structures. The work also discusses physical realizations, time-reversal properties, and the interpretation of 1+1D gravitational anomalies as boundary manifestations of 2+1D topological orders, outlining a path toward a complete modular tensor category-based theory of topological order.

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)..

Figures (20)

• Figure 1: The energy levels of a topologically ordered state with a finite size. The splitting $\eps$ of the nearly degenerate ground states approaches to zero in the large-system limit.
• Figure 2: Two types of phase transitions between two gapped states. (a) Energy levels of a Hamiltonian $H_a(g)$ as functions of the coupling constant $g$. The topologically order state for $g<g_c$ changes into a trivial state for $g>g_c$ via a first order phase transition. (b) Energy levels of a Hamiltonian $H_b(g)$ as functions of the coupling constant $g$. The topologically order state for $g<g_c$ changes into a trivial state for $g>g_c$ via a continuous phase transition. In this case, the ground state of $H_b(g_c)$ is a quantum critical state.
• Figure 3: An impossible $g$ dependence of energy levels.
• Figure 4: The energy density distribution of a quasiparticle.
• Figure 5: The braiding procedure to derive vafaT.