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The low-energy behaviors of gapless double- and triple-Weyl fermions causedby the interplay of long-range Coulomb interaction and quenched disorder arestudied by performing a renormalization group analysis. It is found that anarbitrarily weak disorder drives the double-Weyl semimetal to undergo a quantumphase transition into a compressible diffusive metal, independent of thedisorder type and the Coulomb interaction strength. In contrast, the nature ofthe ground state of triple-Weyl fermion system relies sensitively on thespecific disorder type in the noninteracting limit: The system is turned into acompressible diffusive metal state by an arbitrarily weak random scalarpotential or $z$ component of random vector potential but exhibits stablecritical behavior when there is only $x$ or $y$ component of random vectorpotential. In case the triple-Weyl fermions couple to random scalar potential,the system becomes a diffusive metal in the weak interaction regime but remainsa semimetal if Coulomb interaction is sufficiently strong. Interplay of Coulombinteraction and $x$, or $y$, component of random vector potential leads to astable infrared fixed point that is likely to be characterized by criticalbehavior. When Coulomb interaction coexists with the $z$ component of randomvector potential, the system flows to the interaction-dominated strong couplingregime, which might drive a Mott insulating transition. It is thus clear thatdouble- and triple-Weyl fermions exhibit distinct low-energy behavior inresponse to interaction and disorder. The physical explanation of suchdistinction is discussed in detail. The role played by long-range Coulombimpurity in triple-Weyl semimetal is also considered.

Quantum phase transition and unusual critical behavior in multi-Weyl semimetals