Title: Pr$_2$Ir$_2$O$_7$: when Luttinger semimetal meets Melko-Hertog-Gingras spin ice state,
Abstract: We study the band structure topology and engineering from the interplay
between local moments and itinerant electrons in the context of pyrochlore
iridates. For the metallic iridate 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 wavevector, dubbed
"Melko-Hertog-Gingras" state, by varying Ir and O contents. We point out that
the ordered spin ice of the Pr local moments generates an internal magnetic
field that reconstructs the band structure of the Luttinger semimetal. Besides
the broad existence of Weyl nodes, we predict that the magnetic translation of
the "Melko-Hertog-Gingras" state for the Pr moments protects the Dirac band
touching at certain time reversal invariant momenta for the Ir conduction
electrons. We propose the magnetic fields to control the Pr magnetic structure
and thereby indirectly influence the topological and other properties of the Ir
electrons. Our prediction may be immediately tested in the ordered
Pr$_2$Ir$_2$O$_7$ samples. We expect our work to stimulate a detailed
examination of the band structure, magneto-transport, and other properties of
Pr$_2$Ir$_2$O$_7$.