Bogoliubov excitations in a Bose-Hubbard model on a hyperhoneycomb lattice

We study the topological properties of Bogoliubov excitation modes in a Bose-Hubbard model of three-dimensional (3D) hyperhoneycomb lattices. For the non-interacting case, there exist nodal loop excitations in the Bloch bands. As the on-site repulsive interaction increases, the system is first driven into the superfluid phase and then into the Mott-insulator phase. In both phases, the excitation bands exhibit robust nodal-loop structures and bosonic surface states. From a topology point of view, these nodal-loop excitation modes may be viewed as a permanent fingerprint left in the Bloch bands.