A Thermodynamic-Limit Pinning Criterion for Two-Dimensional Structural Superlubricity

Abstract

Incommensurability and elastic reconstruction do not by themselves define a structurally superlubric phase. We define fully sliding and pinned zero-temperature phases by $\limsup_{A\to\infty}\tau_{\rm dep}^{\max}(A)=0$ and $\liminf_{A\to\infty}\tau_{\rm dep}^{\min}(A)>0$, respectively; $\Lambda_n=|V_n|G_{n,i}[D_{\rm rel}^{-1}(\mathbf q_n)]_{ij}G_{n,j}$ measures only reconstruction susceptibility. Translational covariance then proves that a clean, smooth, infinite moir\'e continuum can reconstruct without acquiring a bulk sliding barrier. We restore atomic sampling in a two-dimensional discrete model of graphene/hBN and test both a diffusion quantum Monte Carlo first-star potential and a 15-harmonic Leven potential across three rational approximants and five directions. No physical-coupling equilibrium or metastable barrier is resolved. The Leven spectrum raises the largest tested $\Lambda$ from $0.142$ to $0.212$, while artificial scaling through $\Lambda=1$ reaches uncontrolled strain before a size-independent threshold appears. The tested zero-temperature in-plane models are therefore consistent with an elastically relaxed sliding regime; $\Lambda=1$ is a reconstruction scale, not a static phase criterion.