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 $\theta$ term is analytically derived for the effective nonlinear $\sigma$ 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 $\sigma$ 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.

Figure 1

Relationship diagram of topological terms and materials for the $n$th unit of a whole topological family of class $A$ [13, 24]. For each node, the upper, below-left, and below-right boxes denote the model, the topological term in $\mathrm{NL}\sigma \mathrm{M}$, and gauge theory, respectively. The dashed arrow is effectively composed of the three successive solid arrows. Nodes: TI denotes the topological insulator, $\mathrm{\Theta }$ represents the theta term, CS the Chern-Simons term, CF the chiral fermion, WZW the WZW term, CC the Chern character, CSM the chiral semimetal, $A\text{−}\mathrm{\Theta }$ the anisotropic $\mathrm{\Theta }$, and $A\text{−}\mathrm{CS}$ the anisotropic CS. Arrows: BBC denotes boundary-bulk correspondence, OC opposite coupling, and DR dimension reduction. Notably, the thick blue arrow represents a key step that deduces the central result of this work.

Figure 2

Three-vertex diagrams.