Hexadecapolar Kondo effect in URu$_2$Si$_2$?

Motivated by recent findings on the electronic structure of URu$_2$Si$_2\,$, we derive the coupling of a localized hexadecapolar mode to itinerant fermionic quasiparticles in tetragonal crystal field, and show how it maps onto the two-channel Kondo (2CK) model. Channel symmetry is a consequence of time-reversal symmetry, and a 2CK regime can be observed if the crystal field splitting is less than the Kondo temperature. Corollary to the derivation, for an $f^2$-configuration in tetragonal environment, a relevant crystal field splitting is always present in addition to the 2CK interaction--even if the local degrees of freedom are a $\Gamma_5$ doublet. Solving the coupling by the numerical renormalization group, we are able to fit the susceptibility and the specific heat of the dilute system, Th$_{1-x}$U$_x$Ru$_2$Si$_2$, in magnetic field and place the measurements on the verge of the local moment and the 2CK regimes.