General invariance and equilibrium conditions for lattice dynamics in 1D,2D,and 3D materials

The long-wavelength behavior of vibrational modes plays a central role in carrier transport,phonon-assisted optical properties,superconductivity,and thermomechanical and thermoelectric properties of materials.Here,we present general invariance and equilibrium conditions of the lattice potential;these allow to recover the quadratic dispersions of flexural phonons in low-dimensional materials,in agreement with the phenomenological model for long-wavelength bending modes.We also prove that for any low-dimensional material the bending modes can have a purely out-of-plane polarization in the vacuum direction and a quadratic dispersion in the long-wavelength limit.In addition,we propose an effective approach to treat invariance conditions in crystals with non-vanishing Born effective charges where the long-range dipole-dipole interactions induce a contribution to the lattice potential and stress tensor.Our approach is successfully applied to the phonon dispersions of 158 two-dimensional materials,highlighting its critical relevance in the study of phonon-mediated properties of low-dimensional materials.