Pr_{2}Ir_{2}O_{7}: When Luttinger Semimetal Meets Melko-Hertog-Gingras Spin Ice State

In quantum materials with multiple degrees of freedom such as itinerant electrons and local moments, the interplay between them leads to intriguing phenomena and allows the mutual control of each other. Here, we study band topology and engineering from the interplay between local moments and itinerant electrons in pyrochlore iridates. For metallic Pr_{2}Ir_{2}O_{7}, 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 wave vector, dubbed the Melko-Hertog-Gingras state, by varying Ir and O contents. We point out that the Pr Ising order generates an internal field and reconstructs the Ir bands. Besides the broad existence of Weyl nodes, we predict that the magnetic translation of the Pr Melko-Hertog-Gingras state protects the Dirac-band touching at certain time-reversal invariant momenta for the Ir electrons. We propose the magnetic fields to control the Pr magnetism and thereby indirectly influence the Ir conduction electrons. Our prediction can be immediately tested in ordered Pr_{2}Ir_{2}O_{7} samples. Our theory should stimulate experiments on pyrochlore iridates, constitute a nontrivial and realistic example for the interplay between itinerant electrons and local moments in three dimensions, and shed light on hybrid quantum materials with multiple degrees of freedom