We investigate the effects of thermal fluctuations in graphene bilayers by means of a nonperturbative renormalization group (NPRG) approach, following the pioneering work of Mauri et al. [Phys. Rev. B 102, 165421 (2020)] based on a self-consistent screening approximation (SCSA). We consider a model of two continuum polymerized membranes, separated by a distance $\ell$, in their flat phase, coupled by interlayer shear, compression/dilatation, and elastic terms. Within a controlled truncation of the effective average action, we retain only the contributions that generate a pronounced crossover of the effective bending rigidity along the renormalization group flow between two regimes: at high running scale $k$, the rigidity is dominated by the in-plane elastic properties, with ${\kappa }_{\mathrm{eff}}\sim {\ell }^2(\lambda +2\mu )/2$, whereas at low $k$ it is controlled by the bending rigidity of two independent monolayers, ${\kappa }_{\mathrm{eff}}\sim 2\kappa$. This crossover is reminiscent of that observed by Mauri et al. as a function of the wavevector scale $q$, but here it is obtained within a renormalization group framework. This has several advantages. First, although approximations are performed, the NPRG approach allows one, in principle, to take into account all nonlinearities present in the elastic theory, in contrast to the SCSA treatment, which requires, already at the formal level, significant simplifications. Second, it demonstrates that the bilayer problem can be treated as a straightforward extension of the monolayer case, with flow equations that keep the same structure and differ only by bilayer-specific adjustments. Third, unlike the SCSA, the NPRG framework admits a controlled, systematically improvable, hierarchy of approximations.

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• Critical phenomena
• Membrane structures
• Functional renormalization group
• Renormalization group