From String Nets to Nonabelions

TL;DR

This paper explores the theoretical framework of string net Hilbert spaces and their relation to topological phases, proposing conditions for Hamiltonians to realize non-abelian anyons in quantum spin models.

Contribution

It introduces a method to realize non-abelian topological phases using string net models and details conditions for Hamiltonians to support such phases.

Findings

String net Hilbert spaces can model non-abelian topological phases.

Conditions for Hamiltonians to realize the DFib phase are identified.

Connections between string nets, chromatic polynomial, and Potts model are established.

Abstract

We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for its ground state and low-lying quasiparticle excitations to be in the DFib topological phase. Using the string net wavefunction, we describe the properties of this phase. Our discussion is informed by mappings of string net wavefunctions to the chromatic polynomial and the Potts model.