### Joseph Flannery with Harsh Mathur

*Dirac theory of topological insulator boundaries*

A topological insulator is a material that behaves as an insulator in its interior but whose surface contains bound conducting states. The bulk of a topological insulator may be described by a massive Dirac model while the gapless surface states follow the massless Dirac equation. Currently lattice models are needed to give a unified description of the bulk and surface states [1]. In this project we will construct a unified description of the bulk and surface states working with a continuum Dirac description by imposing appropriate boundary conditions on the massive Dirac equation at the insulator surface. These boundary conditions must be consistent with the requirement that the Dirac Hamiltonian operator be hermitian and symmetries such as time-reversal and charge conjugation. The continuum description of topological insulators will be used to analyze insulators with nontrivial topologies and geometries. Electrons on the surface of cylindrical topological insulators have been shown to behave as though the cylinder is threaded by an Aharonov-Bohm flux [2]. In this project we will explore analogous effects on spheres and tori.

[1] M.Z. Hasan and C.L. Kane, Reviews of Modern Physics **82,** 3045 (2010)

[2] Yi Zhang and Ashvin Vishwanath, Physical Review Letters **105**, 206601 (2010)