Mixed Charge-Momentum Responses in Higher-Order Topological Semimetals

The theory of topological responses in insulators and semimetals has recently been extended to include mixed translation-electromagnetic responses. These responses encode the charge response to lattice strain and the momentum response to electromagnetic fields. In this work we identify a higher-order topological semimetal that possesses mixed translation-electromagnetic responses in both its bulk and its surface. These responses originate from higher-order Weyl nodes in the 3D bulk and conventional Dirac nodes on 2D surfaces. In addition, when projected to the reduced Brillouin zone on a hinge, the interval between the bulk Weyl and surface Dirac nodes is spanned by flat-band hinge states. These hinge bands realize an equilibrium quadrupole moment of crystal momentum that has not previously been considered.