Relativistic studies of anisotropic properties in magnetic lanthanide erbium and thulium dimers

The recent realization of quantum degenerate gases of magnetic atoms, cooled and trapped to sub-$\mu$K temperatures, is expected to stimulate studies of magnetic superfluid matter. Atom-atom interactions play a central role in the behavior of such matter. Here, we study the anisotropic interactions for two high-spin magnetic species: ${j=6}$ erbium and ${j=7/2}$ thulium. Non-relativistic coupled-cluster calculations are combined with fully-relativistic configuration-interaction calculations to determine the electronic potentials for homonuclear Er$_2$ and Tm$_2$ dissociating to ground-state atoms with their partially-filled f-electron shells. State-of-the-art non-relativistic coupled-cluster calculations determine the potential for the ``stretched'' electronic state with the largest electron spin and orbital angular-momentum projection quantum numbers, while relativistic configuration-interaction calculations give the splittings among the potentials. We find a complicated picture of 91 and 36 potentials characterizing the Er$_2$ and Tm$_2$ dimers, respectively. An analysis in terms of spin-spin Hamiltonians shows that the splittings are well described by an anisotropic dipolar coupling between the atomic angular momentum with the mechanical rotation of the atom pair.