Topological theory of gapless and gapped fermionic systems

Abstract

Recently the discoveries of graphene, Weyl semi-metal and Dirac semi-metal are drawing more and more people’s attentions back to the topological characteristics of Fermi surfaces, which may be tracked back to the pair of Weyl points observed experimentally in the phase A of Helium three in 80s in the last century. Based on the pioneer works by Volovik and Hořava, we classify all kinds of Fermi surfaces with respect to anti-unitary symmetries and codimensions of Fermi surfaces. The first chapter of this thesis is attributed to develop symmetry-dependent topological invariants to characterize topological properties of Fermi surfaces, and map out the periodic classification tables of Fermi surfaces. Compared with the existing classification of topological insulators (TIs) and superconductors (TSCs), it is observed that there exists a two-step dimension shift from our classification of Fermi surfaces. Actually the two classifications can both be derived rigorously in the framework of K-theory, a mathematical algebraic topology theory for stable fiber bundles, where the dimension shift can also be derived rigorously by constructing maps between Fermi surfaces and TIs/TSCs. This unified treatment of the two classifications is of mathematical elegance, even providing us deeper understandings of these topological phenomena, and is the subject of chapter II of this thesis.

In the beginning of chapter III, when applying our theory of topological Fermi surfaces on the boundary of TIs/TSCs, a general index theory is conjectured describing a faithful boundary-bulk correspondence of TIs/TSCs, which is motivated by the dimension shift in the two classifications. Then we construct all kinds of TIs/TSCs and Fermi surfaces by Dirac matrices, which is actually a physical interpretation of the Atiyah-Bott-Shapiro construction as a mathematical theory, and provides us a rigorous proof of our general index theorem. We also provide applications of our theory and its connections to nonlinear sigma models of disordered systems.

The last chapter of this thesis may be regarded as a collection of applications of the boundary-bulk correspondence described by the general index theorem for spatially one-dimensional systems. Specifically one-dimensional superconductor models in the other three nontrivial cases are constructed as generalizations of the Kitaev’s model that is one of four nontrivial cases, and every model is solved in detail by methods similar to that provided by Kitaev. Then we analyze each model in the framework of the general index theorem, focusing on the topological properties of Majorana zero-modes with codimension zero at the ends of these models under the open boundary condition. The possible applications of these models to universal quantum manipulations are also discussed.