Dirac quantum criticality in twisted double bilayer transition metal dichalcogenides

We investigate the phase diagram of moir\'e double bilayer transition metal dichalcogenides with ABBA stacking as a function of twist angle and applied pressure. At hole filling $\nu = 2$ per moir\'e unit cell, the noninteracting system hosts a Dirac semimetal with graphene-like low-energy bands in the moir\'e Brillouin zone. At small twist angles, the Fermi velocity is reduced and interactions dominate the low-temperature behavior. A strong-coupling analysis identifies insulating ferromagnetic and antiferromagnetic ground-state candidates, characterized by spin-density modulations set by the moir\'e scale. Using a realistic continuum model with long-range Coulomb interactions, we perform self-consistent Hartree-Fock calculations to study the competition between these states. Varying the twist angle or pressure drives a transition from a Dirac semimetal to an antiferromagnetic insulator, which breaks SU(2) spin rotation and two-fold lattice rotation symmetries. Within a renormalization group analysis of the most general symmetry-allowed low-energy field theory, we show that this semimetal-to-insulator transition is continuous and belongs to the (2+1)D relativistic Gross-Neveu-Heisenberg universality class with $N = 2$ four-component Dirac fermions. Finite heterostrain, relevant in realistic samples, induces a crossover from Gross-Neveu-Heisenberg universality at intermediate temperatures to conventional (2+1)D Heisenberg criticality at the lowest temperatures. Further decreasing the twist angle can cause a level crossing from the antiferromagnetic insulator into a ferromagnetic insulator with spin-split bands. Our results provide a comprehensive theoretical framework that complements and elucidates recent experiments in twisted double bilayer WSe$_2$.