Gross-Neveu-XY quantum criticality in moir\'e Dirac materials

Two-dimensional van-der-Waals materials offer a highly tunable platform for engineering electronic band structures and interactions. By employing techniques such as twisting, gating, or applying pressure, these systems enable precise control over the electronic excitation spectrum. In moir\'e bilayer graphene, the tunability facilitates the transition from a symmetric Dirac semimetal phase through a quantum critical point into an interaction-induced long-range ordered phase with a finite band gap. At charge neutrality, the ordered state proposed to emerge from twist-angle tuning is the Kramers intervalley-coherent insulator. In this case, the transition falls into the quantum universality class of the relativistic Gross-Neveu-XY model in 2+1 dimensions. Here, we refine estimates for the critical exponents characterizing this universality class using an expansion around the lower critical space-time dimension of two. We compute the order-parameter anomalous dimension $\eta_\varphi$ and the correlation-length exponent $\nu$ at one-loop order, and the fermion anomalous dimension $\eta_\psi$ at two-loop order. Combining these results with previous findings from the expansion around the upper critical dimension, we obtain improved estimates for the universal exponents in $2+1$ dimensions via Pad\'e interpolation. For $N_\mathrm{f} = 4$ four-component Dirac fermions, relevant to moir\'e bilayer graphene, we estimate $1/\nu=0.916(5)$, $\eta_\varphi=0.926(13)$, and $\eta_\psi=0.0404(13)$. For $N_\mathrm{f} = 2$, potentially relevant to recent tetralayer WSe$_2$ experiments, the Gross-Neveu-XY fixed point may be unstable due to a fixed-point collision at $N_\mathrm{f,c}$, with $N_\mathrm{f,c} = 1 + \sqrt{2} + \mathcal{O}(\epsilon)$ in the expansion around the lower critical dimension.