Indirect exchange interaction with the antipodes: RKKY interaction on the sphere

Abstract

The exact expression for the RKKY indirect exchange interaction is obtained for a two-dimensional free-electron gas on a sphere. The uniform space curvature significantly modifies the flat space two-dimensional form and a remarkable effect takes place between antipolar points on the sphere. Two local spins in a mutually antipodal position interact strongly with the anti-ferromagnetic sign. When the two spins are situated in the same hemisphere the interaction decreases with the (spherical) distance between them, and has the characteristic oscillations. The curvature introduces a distance-dependent phase-shift, which becomes noticeable as the distance becomes comparable to the radius of the sphere. As the equator, relative to the polar position of one of the spins, is crossed, the interaction strength starts to increase. The interaction between spins in opposite hemispheres is anti-ferromagnetic, except that it vanishes along regularly spaced latitude bands. At a sharply defined point, exactly antipolar to the first spin, the interaction rises to a strength comparable to what it is between spins situated in the same hemisphere.