Quantum Anharmonic Phonons in the Fermi-Pasta-Ulam Chain

Abstract

By dealing with the classical equation of motion of a Fermi-Pasta-Ulam chain as if the atomic displacement coordinates were quantum operators and requiring that some special linear combinations of the displacements and momenta pertaining to two nearest neighbor atoms obey the Bloch theorem, an effective hermitian one-body potential is worked out at each wave-vector throughout the Brillouin zone. The associated Schrödinger equation is solved to yield the exact full spectrum of quantum anharmonic phonons as the set of bound eigenstates. The anharmonic dispersion curve differs barely from the harmonic one close to the Brillouin zone center even at strong anharmonicity, the deviation being the largest in the middle of the Brillouin zone.