Axionic quantum criticality of generalized Weyl semimetals

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 $\ensuremath{\varepsilon}=2\ensuremath{-}{d}_{M}$ 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.