Theory of topological insulators
Abstract
An important goal of condensed matter physics is to search for new phases of matter. This thesis is about a new insulating phase. Traditionally an insulator is defined as a material that does not conduct electricity. In most insulators the absence of electrical conduction is explained by the band theory of solids--a triumph of quantum mechanics in the twentieth century. According to band theory, an insulator has an energy gap separating the conduction and valence bands. As a result, there is no low energy electronic states inside an insulator to accommodate a charge flow. In the past few years, a new kind of insulators has been theoretically predicted, which has a band structure that is topologically different from an ordinary insulator. For this reason, this new state is called a topological insulator. Despite having an energy gap in the bulk, a topological insulator has unique gapless states bound to the sample surface as a consequence of the topological order in the bulk. Recent experimental observations of these unique surface electron states have provided direct evidence of the topological insulator phase in a number of materials. In this thesis we present the theory of topological insulators. Specifically, we describe the mathematical formulation of the topological order of insulating band structures, which leads to the theoretical discovery of three-dimensional topological insulator phases. We also give the physical characterization of the topological order, thereby establishing that the hallmark signatures of topological insulators are the distinctive surface states. Unlike any other two dimensional metal, the surface of a topological insulator has unique properties giving rise to unusual phases. In particular, we show that depositing a superconductor on the surface leads, via proximity effect, to a novel superconducting state which hosts zero energy Majorana fermions. A Majorana fermion is theoretically defined as a particle that is its own anti-particle but has never been found in nature. Zero-energy Majorana fermions are predicted to have non-Abelian statistics which, if observed, will greatly advance our understanding of the fundamental principles of quantum statistics and open the door to potential quantum computation applications. We propose that the superconductor-topological insulator interface provides a new venue for observing Majorana fermions and their non-Abelian statistics.
Subject Area
Condensed matter physics
Recommended Citation
Liang Fu,
"Theory of topological insulators"
(January 1, 2009).
Dissertations available from ProQuest.
Paper AAI3363356.
http://repository.upenn.edu/dissertations/AAI3363356
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