We study the phase diagram of interacting spinless fermions on the honeycomb bilayer at charge neutrality using large-scale quantum Monte Carlo simulations. In the noninteracting limit, the low-energy spectrum features quadratically dispersing bands that touch at the corners of the hexagonal Brillouin zone. Weak to intermediate interactions induce a splitting of each of the quadratic band touching points into four Dirac points, located along high-symmetry directions of the reciprocal lattice. Strong interactions lead to the formation of a layer-polarized charge density wave, which spontaneously breaks the ${\mathbb{Z}}_2$ layer inversion symmetry and opens an insulating gap in the spectrum. We show that the semimetal-to-insulator quantum phase transition as a function of interaction is continuous and characterized by emergent relativistic symmetry. Our results for the values of the correlation-length exponent $\nu$, the order-parameter anomalous dimension ${\eta }_{\varphi }$, and the fermion anomalous dimension ${\eta }_{\psi }$ agree with those of the theoretically predicted (2+1)-dimension Gross-Neveu-Ising universality class with eight two-component Dirac fermions within less than 5% deviation. We also determine the crossover scale as a function of interaction strength between the nonrelativistic semimetal state at high temperatures, characterized by dynamical critical exponent $z=2$, and the Dirac semimetal state at intermediates temperatures, characterized by $z=1$. Further reducing the temperature below the crossover scale at a fixed value of the interaction strength above the quantum critical point results in a classical ordering transition in the two-dimensional Ising universality class.

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• Charge density waves
• Critical phenomena
• Dirac fermions
• Quantum phase transitions
• Transition temperature
• Dirac semimetal
• Monte Carlo methods