Frustrated and Quantum Antiferromagnetism on the Diamond Sublattice of A-site Magnetic Spinels

Spinel crystals, with the chemical formula AB$_2$X$_4$, in which only the A atom is magnetic, realize antiferromagnetism on a diamond sublattice. We first discuss examples, such as CoAl$_2$O$_4$ and MnSc$_2$S$_4$, which exhibit ``bond frustration'' due to the competing effects of first and second neighbor interactions. This is well modeled by a classical Heisenberg Hamiltonian, which leads to a remarkable ground state degeneracy of coplanar spirals, in which the wavevector of the spiral can lie anywhere on a ``spiral surface'' in momentum space. We describe how thermal fluctuations lead to a broad spin liquid regime, with unique properties, and magnetic ordering at low temperatures. We next discuss the intriguing case of FeSc$_2$S$_4$, in which orbital degeneracy leads to a persistant ``spin orbital liquid'' down to the lowest temperatures. We argue that this material is in the vicinity of an unusual quantum critical point driven by a competition between exchange and spin-orbit interactions.