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We study the quantum critical phenomena emerging at the transition fromtriple-Weyl semimetal to band insulator, which is a topological phasetransition described by the change of topological invariant. The critical pointrealizes a new type of semimetal state in which the fermion dispersion is cubicalong two directions and quadratic along the third. Our renormalization groupanalysis reveals that, the Coulomb interaction is marginal at low energies andeven arbitrarily weak Coulomb interaction suffices to induce an infrared fixedpoint. We compute a number of observable quantities, and show that they allexhibit non-Fermi liquid behaviors at the fixed point. When the interplaybetween the Coulomb and short-range four-fermion interactions is considered,the system becomes unstable below a finite energy scale. The system undergoes afirst-order topological transition when the fermion flavor $N$ is small, andenters into a nematic phase if $N$ is large enough. Non-Fermi liquid behaviorsare hidden by the instability at low temperatures, but can still be observed athigher temperatures. Experimental detection of the predicted phenomena isdiscussed.

Topological quantum critical point in a triple-Weyl semimetal: non-Fermi-liquid behaviors and instabilities