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The fluctuations-driven continuous quantum criticality has sparked tremendousinterest in condensed matter physics. It has been verified that the gaplessfermions fluctuations can change the nature of phase transition at criticality.In this paper, we study the fermionic quantum criticality with enlargedIsing$\times$Ising fluctuations in honeycomb lattice materials. TheGross-Neveu-Yukawa theory for the multicriticality between the semimetallicphase and two ordered phases that break Ising symmetry is investigated byemploying perturbative renormalization group approach. We first determine thecritical range in which the quantum fluctuations may render the phasetransition continuous. We find that the Ising criticality is continuous onlywhen the flavor numbers of four-component Dirac fermions $N_f\geq1/4$. Usingthe $\epsilon$ expansion in four space-time dimensions, we then study theIsing$\times$Ising multicriticality stemming from the symmetry-breakingelectronic instabilities. We analyze the underlying fixed-point structure andcompute the critical exponents for the Ising$\times$Ising Gross-Neveu-Yukawauniversality class. Further, the correlation scaling behavior for the fermionbilinear on the honeycomb lattice at the multicritical point are also brieflydiscussed.

Fermionic criticality with enlarged fluctuations in Dirac semimetals