Disordered Weyl Semimetals and Their Topological Family

We develop a topological theory for disordered Weyl semimetals in the framework of the gauge invariance of the replica formalism and boundary-bulk correspondence of Chern insulators. An anisotropic topological θ term is analytically derived for the effective nonlinear σ model. It is this nontrivial topological term that ensures that the bulk transverse transport of Weyl semimetals is robust against disorders. Moreover, we establish a general diagram that reveals the intrinsic relations among topological terms in the nonlinear σ models and gauge response theories, respectively, for (2n+2)-dimensional topological insulators, (2n+1)-dimensional chiral fermions, (2n+1)-dimensional chiral semimetals, and (2n)-dimensional topological insulators with n being a positive integer.