Geometrically Frustrated Quadrupoles on the Pyrochlore Lattice and Generalized Spin Liquids

In this paper we explore the physics of geometrically frustrated quadrupoles on the pyrochlore lattice. This may be viewed as a generalization of the dipoles, \(L=1\) angular momenta, to \(L=2\) angular momenta degrees of freedom. Whereas dipoles have three degrees of freedom, quadrupoles have five. Using the same methods as in my work on the dipole model phase diagram, we construct the most general symmetry-allowed nearest-neighbor Hamiltonian, which has nine allowed couplings, and map out in broad strokes the structure of the phase diagram. Importantly, it looks similar to the the dipolar phase diagram, but all of the easy-plane degree of freedom and phases are doubled. It turns out that quadrupoles behave in surprising ways which depend strongly on the spin quantum number, because the quadrupolar \(3z^2 - r^2\) operators is not unitarily related to the other four, which we discuss at length. We explore order-by-disorder in this model and construct some examples of quadrupolar spin liquids. Most interestingly, we construct a rank-3 tensor spin liquid which exhibits 6-fold pinch points, by exploiting intuition gained from a multipole decomposition. We discuss the relevance of this model to frustrated magnetic materials, especially those with non-Kramers crystal field doublets or low-lying excited doublets like Tb\(_2\)Ti\(_2\)O\(_7\).