Systematic approach to the statics and dynamics of phonon systems with quartic anharmonicity

A microscopic theory of the statics and dynamics of one-component phonon systems with quartic anharmonicity is presented, based on our recently developed generalization of the Blume-Hubbard formalism. Introducing as a set of representative microscopic variables the local displacements $Q_i$, their squares $Q_i^2$, and momenta $P_i$, exact equations for the corresponding relaxation functions are derived involving static susceptibilities and time-dependent memory functions. Both the static susceptibilities and the memory functions are expressed in terms of cumulants of $Q_i$, $P_i$, and $Q_i^2$, thus allowing systematic approximations to be made by terminating the cumulant expansion at a given order. It is shown that the lowest approximation involving first-order cumulants leads to the conventional undamped soft modes. The next approximation including second-order cumulants results in complicated nonlinear integro-differential equations for the static and dynamic quantities which have been solved numerically for the exactly solvable case of uncoupled classical oscillators and for the linear lattice, respectively. The static and dynamical results are discussed and found to be in substantial agreement with the exact solutions (when available) and with molecular dynamics calculations. Finally, the applicability of our method to phonon systems with $n$ components, to local motion of isolated defects in crystals, to molecular crystals, and to superionic conductors is emphasized.