Oscillatory spacer-mediated exchange ($E_{\mathrm{ex}}/A=-J(t){\hat{\mathbf{m}}}_1\cdot {\hat{\mathbf{m}}}_2$)

Electrons reflected by two magnetic interfaces can mediate an interaction across a nonmagnetic metal spacer. The phase accumulated on traversing the spacer varies with its thickness, producing oscillatory contributions to $J(t)$ from Fermi surface spanning vectors. In the sign convention in the title, positive $J$ favours parallel moments and negative $J$ antiparallel moments. Multiple periods, interface phase shifts, disorder and finite temperature prevent a universal sign rule based on thickness alone.