Ordering of the pyrochlore Ising model with the long-range RKKY interaction

Recently, the ordering of pyrochlore magnets consisting of corner-sharing network of tetrahedra has attracted much attention due to an intrinsic effect of geometrical frustration. Intensive studies were performed in particular on “spin-ice” magnets, e.g., Ho2Ti2O7 and Dy2Ti2O7, which possesses a strong Ising-like < 111 > magnetic anisotropy and exhibits an intriguing glassy behavior analogous to the one associated with the proton ordering of water ice.1–3) In spin-ice magnets, the local spin configuration at each tetrahedron is known to be the socalled “2-in 2-out” configuration, which is stabilized by an effective ferromagnetic interaction. Indeed, this “2-in 2-out” condition is reminiscent of the “ice-rule” condition familiar in the proton ordering of water ice. While an earlier analysis on spin ice was made based on the nearest-neighbor ferromagnetic exchange model,1) the dominant interaction in real spin-ice magnets is the long-range dipolar interaction, which falls off as 1/r3 with distance r.3–6) Thanks to extensive experimental and theoretical studies, the experimental results on spin ice magnets now seem to be relatively well understood on the basis of the dipolar spin-ice model, i.e., the pyrochlore Ising model with the long-range dipolar interaction. Typical spin-ice magnets Ho2Ti2O7 and Dy2Ti2O7 are insulators. Ordering properties of the corresponding metallic pyrochlore magnets are then of special interest.3) Recently, Nakatsuji and collaborators reported that a metallic Ising-like pyrochlore magnet Pr2Ir2O7 exhibits an interesting behavior at low temperatures.7) The magnetism of this compound is borne by the localized moment of Pr3+ which possesses a strong < 111 > Ising-like magnetic anisotropy. Reflecting the metallic nature of the compound, the dominant magnetic interaction is expected to be the long-range RKKY interaction mediated by the Ir4+ conduction electrons, rather than the dipolar interaction. Although the Curie-Weiss constant of Pr2Ir2O7 determined from the