  Notable non-Fermi liquid and quantum critical behaviors are observed in
rare-earth metallic systems with non-Kramers local moments supporting a number
of different multipolar moments. A prominent example is
$\text{Pr(Ti,V)}_{2}\text{Al}_{20}$, where the non-Kramers doublet of the
$\text{Pr}^{3+}$ ion allows quadrupolar and octupolar moments, but lacks a
dipolar moment. Previous theoretical studies show that a single impurity Kondo
problem with such an unusual local moment leads to novel non-Fermi liquid
states. In this work, we investigate possible quantum critical behaviors
arising from the competition between non-Fermi liquid states and
multipolar-ordered phases induced by the RKKY interaction. We consider a local
version of the corresponding Kondo lattice model, namely the Bose-Fermi Kondo
model. Here, the multipolar local moments are coupled to fermionic and bosonic
bath degrees of freedom representing the multipolar Kondo effect and RKKY
interactions. Using a perturbative renormalization group (RG) study up to two
loop order, we find critical points between non-Fermi liquid Kondo fixed points
and a quadrupolar ordered fixed point. The critical points describe quantum
critical behaviors at the corresponding phase transitions and can be
distinguished by higher order corrections in the octupolar susceptibility that
can be measured by ultrasound experiments. Our results imply the existence of a
rich expansion of the phases and quantum critical behaviors in multipolar heavy
fermion systems.
