Two-stage multipolar ordering in Pr(TM)$_2$Al$_{20}$ Kondo materials

Among heavy fermion materials, there is a set of rare-earth intermetallics with non-Kramers Pr$^{3+}$ $4f^2$ moments which exhibit a rich phase diagram with intertwined quadrupolar orders, superconductivity, and non-Fermi liquid behavior. However, more subtle broken symmetries such as multipolar orders in these Kondo materials remain poorly studied. Here, we argue that multi-spin interactions between local moments beyond the conventional two-spin exchange must play an important role in Kondo materials near the ordered to heavy Fermi liquid transition. We show that this drives a plethora of phases with coexisting multipolar orders and multiple thermal phase transitions, providing a natural framework for interpreting experiments on the Pr(TM)$_2$Al$_{20}$ class of compounds.