Defects, topology, and the geometric phase in condensed matter physics

This thesis presents work on some topological applications in condensed matter physics, particularly geometric phases and defects. The first chapter deals with pentagonal disclinations in graphene and their associated bound states. This problem had been attacked previously, using index theorems, as well as the solutions of the continuum Dirac equation. We demonstrated \cite{roygraph} that these two approaches as well as bare numerical computation could be made consistent once one took into account boundary conditions at the defect site. For example, the continuum model considers two pentagons and a square to be alike. However there is a physical distinction between the two, as the sublattice symmetry is locally broken in the former case. The next chapter treats Berry phases using the Majorana representation of spin states as points on a sphere \cite{roymajorana}. The advantages of this approach are that one has a visual representation of the evolution of a state, which automatically absorbs the gauge freedom that drops out of a geometric phase. I show how non-abelian phases can be treated in this framework. The third chapter discusses the Kitaev toric code model and its generalizations, both to higher dimensions and richer braiding symmetries. The toric code is naturally associated to the mathematical structure of a chain complex. This leads to a unified treatment of braiding, degeneracy and effective field theory in higher dimensions \cite{roytoric}. The last part of this thesis is about twist defects in anyonic models. I discuss a general notion of a group defect that permutes anyons and use the toric code as well as a new honeycomb model \cite{roytwist}, as examples. I discuss fusion of these defects as well as ground state degeneracy. The latter is treated geometrically using covering spaces. A common theme running through this work is that topological phenomena can be grounded in a lattice model. This makes them more approachable and often clarifies physical details which might otherwise be missed.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Roy, Abhishek

Contributors dc:contributor

• Stone, Michael
• Ryu, Shinsei
• Hughes, Taylor L.
• Mason, Nadya

• condensed matter
• topology
• geometric phase
• defects
• anyons

Statement dc:rights

• Copyright 2013 Abhishek Roy

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/46645

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/46645