From Incommensurate Bilayer Heterostructures to Allen–Cahn: An Exact Thermodynamic Limit

We give a complete and rigorous derivation of the mechanical energy for twisted 2D bilayer heterostructures without any approximation beyond the existence of an empirical many-body site energy. Our results apply to both the continuous and discontinuous continuum limit. Approximating the intralayer Cauchy–Born energy by linear elasticity theory and assuming an interlayer coupling via pair potentials, our model reduces to a modified Allen–Cahn functional. We rigorously control the error, and, in the case of sufficiently smooth lattice displacements, provide a rate of convergence for twist angles satisfying a Diophantine condition.