An algebraic quantum field theoretic approach to toric code with gapped boundary

TL;DR

This paper extends the algebraic quantum field theoretic framework to analyze the toric code with gapped boundaries, revealing the boundary's structure as a module tensor category over the bulk, which is important for understanding topological quantum computation.

Contribution

It adapts Naaijkens' operator algebraic approach to the toric code with gapped boundary, providing a rigorous mathematical description of boundary condensation phenomena.

Findings

Boundary theory forms a module tensor category over the bulk

Reproduces known condensation results in a rigorous framework

Extends algebraic methods to systems with boundaries

Abstract

Topologically ordered quantum spin systems have become an area of great interest, as they may provide a fault-tolerant means of quantum computation. One of the simplest examples of such a spin system is Kitaev's toric code. Naaijkens made mathematically rigorous the treatment of toric code on an infinite planar lattice (the thermodynamic limit), using an operator algebraic approach via algebraic quantum field theory. We adapt his methods to study the case of toric code with gapped boundary. In particular, we recover the condensation results described in Kitaev and Kong and show that the boundary theory is a module tensor category over the bulk, as expected.