Local disorder, topological ground state degeneracy and entanglement entropy, and discrete anyons

TL;DR

This paper rigorously analyzes Kitaev's abelian models on surfaces, establishing topological ground state degeneracy, entanglement entropy, and anyonic excitations through homology and cohomology groups, advancing understanding of topological order.

Contribution

Provides complete proofs of topological degeneracy, entanglement entropy, and anyonic excitations in Kitaev's models using homology and cohomology, offering a rigorous mathematical framework.

Findings

Confirmed topological ground state degeneracy.

Computed entanglement entropy exactly.

Characterized elementary anyonic excitations.

Abstract

In this comprehensive study of Kitaev's abelian models defined on a graph embedded on a closed orientable surface, we provide complete proofs of the topological ground state degeneracy, the absence of local order parameters, compute the entanglement entropy exactly and characterise the elementary anyonic excitations. The homology and cohomolgy groups of the cell complex play a central role and allow for a rigorous understanding of the relations between the above characterisations of topological order.