Renormalization group approach to graphene bilayers

TL;DR

This work extends the nonperturbative renormalization-group (NPRG) treatment of polymerized membranes to graphene bilayers, formulating a rotationally invariant action in terms of the mean and relative fields to preserve full elastic nonlinearities. The authors derive bilayer-specific NPRG flow equations that reproduce a pronounced crossover in the effective bending rigidity κ_eff(k) from a short-distance elastic-bilayer contribution κ_el ∝ ℓ^2(λ+2μ)/2 to a long-distance regime κ_eff ∼ 2κ, confirming a mechanism for enhanced bilayer stiffness at intermediate scales. Key results include the identification of a mechanical crossover scale k_c, graphene parameter initialization showing a strong elastic contribution in the UV, and a consistent RG trajectory toward the flat-phase fixed point with η ≈ 0.85; the approach also yields a large, temperature-dependent Ginzburg scale and characteristic crossovers that align with experimental and simulation insights on bilayer graphene. The NPRG framework offers a systematically improvable, symmetry-preserving alternative to SCSA, enabling extensions such as wavevector-dependent couplings and asymmetric bilayers to more accurately capture realistic systems. This provides a coherent, scale-resolved picture of bilayer elasticity across regimes relevant to experiments and simulations of few-layer graphene.

Abstract

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 $κ_{\mathrm{eff}}\sim \ell^{2}(λ+2μ)/2$, whereas at low $k$ it is controlled by the bending rigidity of two independent monolayers, $κ_{\mathrm{eff}}\sim 2κ$. 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.

Figures (10)

• Figure 1: Bilayer model: two coupled membranes, separated by a distance $\ell$. A point on membrane 1 (respectively, membrane 2) is parameterized by the position vector ${\bf R}_1({\bf x})$ (respectively, ${\bf R}_2({\bf x})$).
• Figure 2: A typical shape of the cut-off function $R_k({\bf q})$.
• Figure 3: Behaviour of $\bar{c}_k$ -- dashed-dot curve -- as a function of $t=-\ln ka$ for $T=500$ K. One has displayed the typical RG-time $t_c$ (dashed-dot vertical line) associated with the bending rigidity crossover and the RG-times $t_{1c}$ and $t_{2c}$ (full vertical lines).
• Figure 4: Behaviour of $\eta_k$ -- dashed curve -- as a function of $t=-\ln ka$ for $T=500$ K; One gives also the typical Ginzburg RG-time scale $t_G$ (dashed vertical line) and the bare Ginzburg RG-time scales $t_{G1}$ and $t_{G2}$ (full vertical lines).
• Figure 5: The flow of the anomalous dimension at $T=10\text{K}$. Dashed-dot curve : $\eta_k^B$ of a bilayer; dashed curve : $\eta_k^m$ of a monolayer; dotted vertical lines : the Ginzburg RG-time $t_{G}$; dashed-dot vertical line : the mechanical crossover RG-time $t_c$.