General response theory of topologically stable Fermi points and its implications for disordered cases

We develop a general response theory of gapless Fermi points with nontrivial topological charges for gauge and nonlinear sigma fields, which asserts that the topological character of the Fermi points is embodied as the terms with discrete coefficients proportional to the corresponding topological charges. Applying the theory to the effective non-linear sigma models for topological Fermi points with disorders in the framework of replica approach, we derive rigorously the Wess-Zumino terms with the topological charges being their levels in the two complex symmetry classes of A and AIII. Intriguingly, two nontrivial examples of quadratic Fermi points with the topological charge `2' are respectively illustrated for the classes A and AIII. We also address a qualitative connection of topological charges of Fermi points in the real symmetry classes to the topological terms in the non-linear sigma models, based on the one-to-one classification correspondence.Comment: 8 pages and 2 figures, revised version with appendi

Topological Classification and Stability of Fermi Surfaces

In the framework of the Cartan classification of Hamiltonians, a kind of topological classification of Fermi surfaces is established in terms of topological charges. The topological charge of a Fermi surface depends on its codimension and the class to which its Hamiltonian belongs. It is revealed that six types of topological charges exist, and they form two groups with respect to the chiral symmetry, with each group consisting of one original charge and two descendants. It is these nontrivial topological charges which lead to the robust topological protection of the corresponding Fermi surfaces against perturbations that preserve discrete symmetries.Comment: 5 pages, published version in PR

Revealing a topological connection between stabilities of Fermi surfaces and topological insulators/superconductors

A topology-intrinsic connection between the stabilities of Fermi surfaces (FSs) and topological insulators/superconductors (TIs/TSCs) is revealed. In particular, a one-to-one relation between the topological types of FSs and TIs/TSCs is rigorously derived; combining it with a well-established topological theory of FSs, we produce a complete table illustrating precisely topological types of all TIs/TSCs, while a valid part of it was postulated before. Moreover, we propose and prove a general index theorem that relates the topological charge of FSs on the natural boundary of a strong TI/TSC to its bulk topological number. Implications of the general index theorem on the boundary quasi-particles are also addressed.Comment: 5 pages with Supplemental Material, more content is adde