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The dynamical properties of nuclei, carried by the concept of phononquasiparticles (QP), are central to the field of condensed matter. While theharmonic approximation can reproduce a number of properties observed in realcrystals, the inclusion of anharmonicity in lattice dynamics is essential toaccurately predict properties such as heat transport or thermal expansion. Forhighly anharmonic systems, non perturbative approaches are needed, which resultin renormalized theories of lattice dynamics. In this article, we apply theMori-Zwanzig projector formalism to derive an exact generalized Langevinequation describing the quantum dynamics of nuclei in a crystal. By projectingthis equation on quasiparticles in reciprocal space, and with results fromlinear response theory, we obtain a formulation of vibrational spectra thatfully accounts for the anharmonicity. Using a mode-coupling approach, weconstruct a systematic perturbative expansion in which each new order is builtto minimize the following ones. With a truncation to the lowest order, we showhow to obtain a set of self-consistent equations that can describe thelineshapes of quasiparticles. The only inputs needed for the resulting set ofequations are the static Kubo correlation functions, which can be computedusing (fully quantum) path-integral molecular dynamics or approximated with(classical or ab initio) molecular dynamics. We illustrate the theory with anapplication on fcc 4He, an archetypal quantum crystal with very stronganharmonicity.

Mode-coupling theory of lattice dynamics for classical and quantum crystals