Pr₂Ir₂O₇: when Luttinger semimetal meets Melko-Hertog-Gingras spin ice state

In quantum materials with multiple flavors of degrees of freedom (DOF), the interplay between them leads to intriguing phenomena and allows the mutual control. Here we study band topology and engineering from the interplay between local moments and itinerant electrons in pyrochlore iridates. For metallic Pr₂Ir₂O₇, the Ir 5d conduction electrons interact with the Pr 4f local moments via the f-d exchange. While the Ir electrons form a Luttinger semimetal, the Pr moments can be tuned into an ordered spin ice with a finite ordering wavevector, dubbed Melko-Hertog-Gingras (MHG) state, by varying Ir and O contents. We point out that the Pr Ising order generates an internal field that reconstructs the Ir bands. Besides the broad existence of Weyl nodes, we predict that the magnetic translation of the Pr MHG state protects the Dirac band touching at certain time reversal invariant momenta for the Ir bands. We propose the magnetic fields to control the Pr magnetism and indirectly influence the Ir electrons. Our prediction can be immediately tested in ordered Pr₂Ir₂O₇ samples. Our theory constitutes a nontrivial and realistic example for the interplay between itinerant electrons and local moments in three dimensions, and shed lights on hybrid quantum materials with multiple flavors of DOF.