  The effect of electron-phonon coupling (EPC) on Dirac fermions has recently
been explored numerically on a honeycomb lattice, leading to precise
quantitative values for the finite temperature and quantum critical points. In
this paper, we use the unbiased determinant Quantum Monte Carlo (DQMC) method
to study the Holstein model on a half-filled staggered-flux square lattice, and
compare with the honeycomb lattice geometry, presenting results for a range of
phonon frequencies $0.1 \leqslant \omega \leqslant 2.0$. We find that the
interactions give rise to charge-density wave (CDW) order, but only above a
finite coupling strength $\lambda_{\rm crit}$. The transition temperature is
evaluated and presented in a $T_c$-$\lambda$ phase diagram. An accompanying
mean-field theory (MFT) calculation also predicts the existence of quantum
phase transition (QPT), but at a substantially smaller coupling strength.
