Topological quantum color code model on infinite lattice

TL;DR

This paper rigorously analyzes the topological quantum color code model on an infinite lattice, classifying its anyonic excitations and confirming its topological order as a double layer of the toric code using operator algebra methods.

Contribution

It provides a mathematical classification of the color code's excitations and topological order in the thermodynamic limit, extending finite lattice results to infinite systems.

Findings

Classified all irreducible anyon superselection sectors.

Constructed explicit string operators for excitations.

Identified the topological order as a double layer of the toric code.

Abstract

The color code model is a crucial instance of a Calderbank--Shor--Steane (CSS)-type topological quantum error-correcting code, which notably supports transversal implementation of the full Clifford group. Its robustness against local noise is rooted in the structure of its topological excitations. From the perspective of quantum phases of matter, it is essential to understand these excitations in the thermodynamic limit. In this work, we analyze the color code model on an infinite lattice within the quasi-local $C^{*}$-algebra framework, using a cone-localized Doplicher-Haag-Roberts (DHR) analysis. We classify its irreducible anyon superselection sectors and construct explicit string operators that generate anyonic excitations from the ground state. We further examine the fusion and braiding properties of these excitations. Our results show that the topological order of the color code is described by $\mathsf{Rep}(D(\mathbb{Z}_2 \times \mathbb{Z}_2)) \simeq \mathsf{Rep}(D(\mathbb{Z}_2)) \boxtimes \mathsf{Rep}(D(\mathbb{Z}_2))$, which is equivalent to a double layer of the toric code and consistent with established analyses on finite lattices.