  The multipolar Kondo problem, wherein the quantum impurity carries
higher-rank multipolar moments, has seen recent theoretical and experimental
interest due to proposals of novel non-Fermi liquid states and the availability
of a variety of material platforms. The multipolar nature of local moments, in
conjunction with constraining crystal field symmetries, leads to a vast array
of possible interactions and resulting non-Fermi liquid ground states. Previous
works on Kondo physics have typically focussed on impurities that have two
degenerate internal states. In this work, inspired by recent experiments on the
tetragonal material YbRu$_{2}$Ge$_{2}$, which has been shown to exhibit a local
moment with a quasi-fourfold degenerate ground state, we consider the Kondo
effect for such a quasi-quartet multipolar impurity. In the tetragonal crystal
field environment, the local moment supports dipolar, quadrupolar, and
octupolar moments, which interact with conduction electrons in entangled spin
and orbital states. Using renormalization group analysis, we uncover a number
of emergent quantum ground states characterized by non-trivial fixed points. It
is shown that these previously unidentified fixed points are described by
truncated SU(4) Kondo models, where only some of the SU(4) generators
(representing the impurity degrees of freedom) are coupled to conduction
electrons. Such novel non-trivial fixed points are unique to the quasi-quartet
multipolar impurity, reinforcing the idea that an unexplored rich diversity of
phenomena may be produced by multipolar quantum impurity systems.
