Hyperbolic dispersion enables unprecedented abilities for wave-field engineering which so far chiefly has been realized by man-made metamaterials. Recent classical explorations of topological media and semimetals suggest that these exotic structures may enable a novel route toward hyperbolic sound control. Here, we demonstrate that a three-dimensional acoustic semimetal implementation exhibits nodal lines of hyperbolic shape that are topologically protected due to translational symmetry. The structure comprises a cubic arrangement of crossed hollow channels whose hopping strengths are determined by their cross sections, which facilitate analytically exact prediction of tunable spin-Hall textures. Interestingly, thanks to these nodal signatures of hyperbolic shape, we are able to acquire an array of unusual emission features, spanning from directional collimation to either horizontally or vertically split radiation. We foresee that our findings will provide remarkable opportunities for advanced wave control with hyperbolic and topological attributes.

• Received 30 September 2020
• Revised 3 February 2021
• Accepted 4 February 2021

DOI:https://doi.org/10.1103/PhysRevB.103.064307