Geometrically Frustrated Quadrupoles on the Pyrochlore Lattice and Generalized Spin Liquids

As an instance of geometrical frustration with non-magnetic degrees of freedom, we explore the physics of local spin $S\geq 1$ moments on the pyrochlore lattice that interact via pure quadrupolar couplings. In the presence of spin-orbit coupling, there are nine allowed couplings between nearest neighbor quadrupoles. We determine the semi-classical phases and survey the phase diagram of the model. One may view the Hamiltonian as being composed of two copies of the well-studied dipolar model with couplings between the copies, and we find that each easy-plane dipolar phase has two quadrupolar counterparts. As geometrical frustration is important over broad swathes of the parameter space, there are many classical quadrupolar liquids and regions with order-by-disorder selection of discrete states. Order-by-disorder with quadrupoles admits cubic terms in the Landau theory whose effects appear in Monte Carlo simulations and flavor wave calculations for fixed spin $S$. We showcase a number of examples of quadrupolar spin liquids, including one realizing a rank-3 symmetric tensor gauge theory exhibiting 6-fold pinch point singularities. We discuss remarkable differences between the quadrupolar physics of different spin quantum number. We also discuss connections to the non-Kramers rare earth pyrochlore materials.