We use determinant quantum Monte Carlo (DQMC) simulations to study the role of electron-electron interactions on three-dimensional (3D) Dirac fermions based on the $pi$-flux model on a cubic lattice. We show that the Hubbard interaction drives the 3D Dirac semimetal to an antiferromagnetic (AF) insulator only above a finite critical interaction strength and the long-ranged AF order persists up to a finite temperature. We evaluate the critical interaction strength and temperatures using finite-size scaling of the spin structure factor. The critical behaviors are consistent with the (3+1)d Gross-Neveu universality class for the quantum critical point and 3D Heisenberg universality class for the thermal phase transitions. We further investigate correlation effects in birefringent Dirac fermion system. It is found that the critical interaction strength $U_c$ is decreased by reducing the velocity of the Dirac cone, quantifying the effect of velocity on the critical interaction strength in 3D Dirac fermion systems. Our findings unambiguously uncover correlation effects in 3D Dirac fermions, and may be observed using ultracold atoms in an optical lattice.
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