Anyon and Loop Braiding Statistics in Field Theories with a Topological $\Theta-$term

TL;DR

This paper shows how semiclassical nonlinear sigma model field theories with a topological $ heta$-term can be used to derive the braiding statistics of anyons and loops in 2D and 3D topological phases, providing a geometric understanding.

Contribution

It introduces a formalism linking braiding statistics in topological phases to topological $ heta$-terms in nonlinear sigma models, offering a geometric interpretation.

Findings

Derivation of anyon and loop braiding statistics using $ heta$-terms.

Geometric interpretation of braiding processes as nontrivial field configurations.

Applicable to various 2D and 3D topological phases.

Abstract

We demonstrate that the anyon statistics and three-loop statistics of various 2d and 3d topological phases can be derived using semiclassical nonlinear Sigma model field theories with a topological $\Theta$-term. In our formalism, the braiding statistics has a natural geometric meaning: The braiding process of anyons or loops leads to a nontrivial field configuration in the space-time, which will contribute a braiding phase factor due to the $\Theta$-term.