  Quantum critical phenomena may be qualitatively different when massless Dirac
fermions are present at criticality. Using our recently-discovered
fermion-sign-free Majorana quantum Monte Carlo (MQMC) method introduced by us
in Ref. [1], we investigate the quantum critical phenomena of {\it spinless}
Dirac fermions at their charge-density-wave (CDW) phase transitions on the
honeycomb lattice having $N_s=2L^2$ sites with largest $L=24$. By finite-size
scaling, we accurately obtain critical exponents of this so-called Gross-Neveu
chiral-Ising universality class of {\it two} (two-component) Dirac fermions in
2+1D: $\eta=0.45(2)$, $\nu=0.77(3)$, and $\beta=0.60(3)$, which are
qualitatively different from the mean-field results but are reasonably close to
the ones obtained from renormalization group calculations.
