Viscosity as a probe of nodal topology

Electronic materials can sustain a variety of unusual, but symmetry protected touchings of valence and conduction bands, each of which is identified by a distinct topological invariant. Well known examples include linearly dispersing pseudo-relativistic fermions in monolayer graphene, Weyl and nodal-loop semimetals, bi-quadratic (-cubic) band touching in bilayer (trilayer) graphene, as well as mixed dispersions in multi-Weyl systems. We here show that depending on the underlying band curvature, the shear viscosity in the collisionless regime displays unique power-law scaling with frequency at low temperatures, bearing the signatures of the band topology, which are distinct from the ones when the system resides at the brink of a topological phase transition into a band insulator. Therefore, besides density of states (governing specific heat, compressibility) and dynamic conductivity, shear viscosity can be instrumental to pin nodal topology in electronic materials.