| Summary: | We demonstrate that ultracold symmetric top molecules loaded into an optical lattice can realize highly tunable and unconventional models of quantum magnetism, such as an XYZ Heisenberg spin model. We show that anisotropic dipole–dipole interactions between molecules can lead to effective spin–spin interactions which exchange spin and orbital angular momentum. This exchange produces effective spin models which do not conserve magnetization and feature tunable degrees of spatial and spin–coupling anisotropy. In addition to deriving pure spin models when molecules are pinned in a deep optical lattice, we show that models of itinerant magnetism are possible when molecules can tunnel through the lattice. Additionally, we demonstrate rich tunability of effective model parameters using only a single microwave frequency, in contrast to proposals with $^{1}\Sigma $ diatomic molecules, which often require many microwave frequencies. Our results are germane not only for experiments with polyatomic symmetric top molecules, such as methyl fluoride (CH _3 F), but also diatomic molecules with an effective symmetric top structure, such as the hydroxyl radical OH.
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