Dispersion of Flexural Modes in Soft 2D Lattices

We have constructed a consistent theory of flexural phonon mode spectra of simple 2D crystal lattice. Analytic expressions have been obtained for the dispersion relations of 2D lattices with different configurations. It is shown that the propagation of flexural modes with a quadratic dispersion relation becomes possible during the interaction of each atom with not only nearest neighbors, but also with more distant atoms. It turns out that the signs of the effective force constants must be different to provide mechanical stability of the system. Moreover, there exists a relation between force constants of the lattice, which depends on its geometrical configuration and for which the account for the influence of more distant coordination spheres reproduces the square of the dispersion relation of the first sphere, preserving its angular isotropy in a wide range of wavevectors.