  We establish a scenario where fluctuations of new degrees of freedom at a
quantum phase transition change the nature of a transition beyond the standard
Landau-Ginzburg paradigm. To this end we study the quantum phase transition of
gapless Dirac fermions coupled to a $\mathbb{Z}_3$ symmetric order parameter
within a Gross-Neveu-Yukawa model in 2+1 dimensions, appropriate for the
Kekul\'e transition in honeycomb lattice materials. For this model the standard
Landau-Ginzburg approach suggests a first order transition due to the
symmetry-allowed cubic terms in the action. At zero temperature, however,
quantum fluctuations of the massless Dirac fermions have to be included. We
show that they reduce the putative first-order character of the transition and
can even render it continuous, depending on the number of Dirac fermions $N_f$.
A non-perturbative functional renormalization group approach is employed to
investigate the phase transition for a wide range of fermion numbers. For the
first time we obtain the critical $N_f$, where the nature of the transition
changes. Furthermore, it is shown that for large $N_f$ the change from the
first to second order of the transition as a function of dimension occurs
exactly in the physical 2+1 dimensions. We compute the critical exponents and
predict sizable corrections to scaling for $N_f =2$.
