Magnetic-field control of interactions in alkaline-earth Rydberg atoms and applications to {\it XXZ} models

We study the magnetic-field dependence of the interactions between two alkaline-earth(-like) Rydberg atoms, ${}^{88}$Sr and ${}^{174}$Yb. Considering the pair of Rydberg states $|ns,{}^3S_1,m_J\rangle$ and $|(n+1)s,{}^3S_1,m_J\rangle$, we show that the effective Hamiltonian takes the form of an {\it XXZ}-type quantum spin model, as in the alkali-atom case [M. Kunimi and T. Tomita, Phys. Rev. A {\bf 112}, L051301 (2025)]. We find that the behavior of the anisotropy parameter for ${}^{174}$Yb at zero magnetic field is significantly different from that for other atomic species. This behavior originates from the strong spin-orbit coupling in ${}^{174}$Yb. We systematically calculate the interaction parameters of the {\it XXZ} model in the presence of a magnetic field and show that they can be tuned by the field. As applications to quantum many-body problems, we investigate one-dimensional systems in the large-anisotropy regime and show that the folded {\it XXZ} model can be realized in ${}^{174}$Yb systems without fine-tuning of the field. We also investigate two-dimensional square-lattice systems and show that a supersolid phase can emerge in the ground state at the mean-field level.