Interacting Anyons in One and Two Dimensions: Strong Zero Modes in Anyon Chains and Non-Abelian Anyons on a Torus

The work in this thesis is split into two distinct parts. The first focuses on the identification and construction of strong zero modes in non-abelian anyon chains. We show a Tambara-Yamagami chain is equivalent to a Zn chiral clock model. We then present numerical evidence for a strong zero mode in an SU(2)4 anyon chain. By rewriting the chain as an XXZ spin chain, we construct the zero mode explicitly in terms of spin operators. Finally, we write the zero mode in the diagrammatic formalism of anyons. In the second part, we construct a hopping model of non-abelian anyons on a torus. We demonstrate that that the model possesses a translational symmetry around each non-trivial torus loop. By calculating the level spacing statistics of the model, we show that the model is non-integrable for Fibonacci anyons, Ising anyons and abelian anyons. Lastly, we carry out the groundwork for a future project. We add local interactions between the anyons to the Hamiltonian. We then calculate the entanglement spectrum of a ground state of the system after cutting the torus into two cylindrical pieces. The low lying states of this spectrum have a linear dispersion relation for several systems we examined, suggesting the entanglement spectrum is described by a conformal field theory spectrum.

Name dc:type.qualificationname

phd

Level dc:type.qualificationlevel

doctoral

Grantor dc:publisher.institution

National University of Ireland, Maynooth

Year dc:date.issued

2021

Author dc:creator

• Millar, Darragh

Language dc:language

en