Effective Field Theory of the Disordered Weyl Semimetal

In disordered Weyl semimetals, mechanisms of topological origin lead to the protection against Anderson localization, and at the same time to different types of transverse electromagnetic response—the anomalous Hall and the chiral magnetic effect. We here apply field theory methods to discuss the manifestation of these phenomena at length scales that are beyond the scope of diagrammatic perturbation theory. Specifically, we show how an interplay of symmetry breaking and the chiral anomaly leads to a field theory containing two types of topological terms. Generating the unconventional response coefficients of the system, these terms remain largely unaffected by disorder, i.e., information on the chirality of the system remains visible even at large length scales.

Figure 1

Schematic of two Weyl nodes split in energy and momentum by $2\mu$ and $2\mathbf{b}$, respectively.

Figure 2

Building blocks of the field theory. Upper panel: decoupling the impurity scattering by a Hubbard-Stratonovich field which describes the particle-hole interference; the mean field value of the field is determined by the self-consistent Born approximation impurity self-energy (right). Center panel: soft modes (generated by the fields $A$ of the text) describe diffusive propagation (left) and nonlinear “interaction” due to higher order scattering vertices (right). Bottom panel: two “triangle diagrams” generating an $A^3$ (left) and $A\partial A$ (right) vertex, with the wavy line representing a derivative vertex.