Surface states and anomalies in 3D topological semimetals from momentum-spin multipole moments

Topological semimetals exhibit a variety of anomalous responses determined by the geometry of their gapless nodal points and lines. For example, 3D Weyl semimetals show the anomalous Hall effect whose conductivity is determined by the chirality weighted momentum dipole moment. In this work, we study 3D Weyl semimetals with vanishing anomalous Hall coefficients, but nonzero mixed quadrupole moments of the momentum and spin. Such a system exhibits a 3D analog of the quantum spin Hall effect. On the 2D surface, gapless surface states are characterized by commuting spin-momentum locking terms in contrast to the anti-commuting spin-momentum locking of Dirac cones. These surface states induce anomalous charge and spin currents under external fields. We study the anomaly cancellation between bulk anfd surface by deriving a bulk topological action with electromagnetic and spin gauge fields and show that the response coefficient is proportional to the mixed momentum-spin quadrupole moment. Finally, we discuss the connection between our system and responses to higher-rank tensor field theory.