Itinerant spin ice order, Weyl metal, and anomalous Hall effect in Pr$_2$Ir$_2$O$_7$
In the global phase diagram of 227 pyrochlore iridates, Pr$_2$Ir$_2$O$_7$ resides in the proximity of a metal insulator transition, supporting an anomalous Hall effect below 1.5 K, and the normal state of this material is described by a parabolic semimetal with quadratic band touching at the \Gamma$ point. For such a parabolic semimetal which is believed to be the parent state of many 227 pyrochlore iridate compounds, we demonstrate a strong competition between the "spin ice" and "all in all out" orders, and based on a Landau theory we construct possible global phase diagrams for this class of materials. When "spin ice" order has the dominant susceptibility, we show that the low temperature phase displays an itinerant "3 in, 1 out" order. For a generic model of particle hole anisotropic parabolic semimetal, the "3 in, 1 out" order is an admixture of both "spin ice" and "all in all out" spin configurations on Ir tetrahedron. The "3 in, 1 out" order gives rise to a Weyl metal with two Weyl nodes along one of the eight possible [1,1,1] directions supporting a large anomalous Hall effect without any appreciable magnetic moment for Ir${4+ and Pr${3+ ions, as observed in Pr$_2$Ir$_2$O$_7$ at low temperatures. The anomalous Hall conductivity inside the "3 in, 1 out" ordered phase varies as square root of the magnetization. Inside the metallic "3 in, 1 out" phase, the "spin ice" and "all in all out" components couple to the fluctuating $T_{2g} quadrupolar order parameter, leading to a small nematicity. Since an externally applied strain along [1,1,1] direction induces a static $T_{2g} quadrupolar order, we show how it can amplify the "3 in, 1 out" order, raising the possibility of observing anomalous Hall effect at higher temperatures.