  Understanding correlation effects in topological phases and their transitions
is a cutting-edge area of research in recent condensed matter physics. We study
topological quantum phase transitions (TQPTs) between double-Weyl semimetals
(DWSMs) and insulators, and argue that a novel class of quantum criticality
appears at the TQPT characterized by emergent anisotropic non-Fermi liquid
behaviors, in which the interplay between the Coulomb interaction and
electronic critical modes induces not only anisotropic renormalization of the
Coulomb interaction but also strongly correlated electronic excitation in three
spatial dimensions. Using the standard renormalization group methods, large
$N_f$ theory and the $\epsilon= 4-d$ method with fermion flavor number $N_f$
and spatial dimension $d$, we obtain the anomalous dimensions of electrons
($\eta_f=0.366/N_f $) in large $N_f$ theory and the associated anisotropic
scaling relations of various physical observables. Our results may be observed
in candidate materials for DWSMs such as HgCr$_2$Se$_4$ or SrSi$_2$ when the
system undergoes a TQPT.
