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We demonstrate that ultracold symmetric top molecules loaded into an opticallattice can realize highly tunable and unconventional models of quantummagnetism, such as an XYZ Heisenberg spin model. We show that anisotropicdipole-dipole interactions between molecules can lead to effective spin-spininteractions which exchange spin and orbital angular momentum. This exchangeproduces effective spin models which do not conserve magnetization and featuretunable degrees of spatial and spin-coupling anisotropy. In addition toderiving pure spin models when molecules are pinned in a deep optical lattice,we show that models of itinerant magnetism are possible when molecules cantunnel through the lattice. Additionally, we demonstrate rich tunability of theeffective models' parameters using only a single microwave frequency, incontrast to proposals with $^1\Sigma$ diatomic molecules, which often requiremany microwave frequencies. Our results are germane not only for experimentswith polyatomic symmetric top molecules, such as methyl fluoride (CH$_3$F), butalso diatomic molecules with an effective symmetric top structure, such as thehydroxyl radical OH.

Realizing unconventional quantum magnetism with symmetric top molecules