Cutoff nonlinearities in the low-temperature vibrations of glasses and crystals

We present a computer simulation study of glassy and crystalline states using the standard Lennard-Jones interaction potential that is truncated at a finite cutoff distance, as is typical of many computer simulations. We demonstrate that the discontinuity at the cutoff distance in the first derivative of the potential (corresponding to the interparticle force) leads to the appearance of cutoff nonlinearities. These cutoff nonlinearities persist into the very-low-temperature regime thereby affecting low-temperature thermal vibrations, which leads to a breakdown of the harmonic approximation for many eigenmodes, particularly for low-frequency vibrational modes. Furthermore, while expansion nonlinearities which are due to higher order terms in the Taylor expansion of the interaction potential are usually ignored at low temperatures and show up as the temperature increases, cutoff nonlinearities can become most significant at the lowest temperatures. Anharmonic effects readily show up in the elastic moduli which not only depend on the eigenfrequencies, but are crucially sensitive to the eigenvectors of the normal modes. In contrast, those observables that rely mainly on static structural information or just the eigenfrequencies, such as the vibrational density of states, total potential energy, and specific heat, show negligible dependence on the presence of the cutoff. Similar aspects of nonlinear behavior have recently been reported in model granular materials, where the constituent particles interact through finite-range, purely repulsive potentials. These nonlinearities have been ascribed to the nature of the sudden cutoff at contact in the force law. As a consequence, we demonstrate that cutoff nonlinearities emerge as a general feature of ordered and disordered solid state systems interacting through truncated potentials.