Crystalline electromagnetic responses of higher-order topological semimetals

Previous work has shown that time-reversal symmetric Weyl semimetals with a quadrupolar arrangement of first-order Weyl nodes exhibit a mixed crystalline-electromagnetic response. For systems with higher-order Weyl nodes, which are attached to both surface and hinge Fermi arcs, additional phenomena appear on surfaces of codimension $n>1$, such as electromagnetic responses of the hinges. Here we construct a model possessing a quadrupole of higher-order Weyl nodes to study the interplay between higher-order topology and mixed crystalline-electromagnetic responses. We show that the higher-order nature of the Weyl nodes yields a dipole of Dirac nodes on certain surfaces, leading to a mixed crystalline-electromagnetic surface response that binds charge to dislocations and momentum density to magnetic fields. In addition, we show that the model possesses a bulk quadrupole moment of crystal momentum that provides a link between the bulk and surface responses of the system.