We formulate a field-theoretic description for $d$-dimensional interacting nodal semimetals, featuring dispersion that scales with the linear and $n\mathrm{th}$ power of momentum along $d_L$ and $d_M$ mutually orthogonal directions around a few isolated points in the reciprocal space, respectively, with $d_L+d_M=d$, and residing at the brink of isotropic insulation, described by $N_b$-component bosonic order parameter fields. The resulting renormalization group (RG) procedure, tailored to capture the associated quantum critical phenomena, is controlled by a “small” parameter $\varepsilon =2-d_M$ and $1/N_f$, where $N_f$ is the number of identical fermion copies (flavor number) when in conjunction $d_L=1$. When applied to three-dimensional interacting general Weyl semimetals ($d_L=1$ and $d_M=2$), characterized by the Abelian monopole charge $n>1$, living at the shore of the axionic insulation ($N_b=2$), a leading-order RG analysis suggests the Gaussian nature of the underlying quantum phase transition, around which the critical exponents assume mean-field values. A traditional field-theoretic RG analysis yields the same outcomes for simple Weyl semimetals ($n=1, d_L=3$, and $d_M=0$). Consequently, emergent marginal Fermi liquids showcase only logarithmic corrections to physical observables at intermediate scales of measurements.

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• Effective field theory
• Quantum phase transitions
• Renormalization group
• Weyl semimetal
• Critical phenomena
• Non-Fermi-liquid theory