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 nonlinear 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 nonlinear sigma models, based on the one-to-one classification correspondence.

Figure 1

The $(\omega ,k)$ space with a Weyl point at the center. $M_{\pm }$ is the $k$ space of $G_{\pm }$ and $S^3$ is chosen to enclose the Weyl point.

Figure 2

The interpolation in the class AIII with $\lambda =0.2$.