Abstract: We prove that a simple cubic lattice with nonlocal central-force interactions exhibits emergent macroscopic isotropy in the long-wavelength limit. The key condition is that bond-stretching contributions dominate over bond-bending contributions. The proof is algebraic—the identity Kₗₗ Rᵧ² = Kₗₘ Rₓ Rᵧ holds term-by-term for any central potential—and establishes C₁₂ = C₄₄ as an exact equality. Numerical verification using Gaussian and power-law potentials confirms the result.
卓冰 蒋 (Fri,) studied this question.