  We present a vibrational dynamical mean-field theory (VDMFT) of the dynamics
of atoms in solids with anharmonic interactions. Like other flavors of DMFT,
VDMFT maps the dynamics of a periodic anharmonic lattice of atoms onto those of
a self-consistently defined impurity problem with local anharmonicity and
coupling to a bath of harmonic oscillators. VDMFT is exact in the harmonic and
molecular limits, nonperturbative, systematically improvable through its
cluster extensions, usable with classical or quantum impurity solvers
(depending on the importance of nuclear quantum effects), and can be combined
with existing low-level diagrammatic theories of anharmonicity. When tested on
models of anharmonic optical and acoustic phonons, we find that classical VDMFT
gives good agreement with classical molecular dynamics, including the
temperature dependence of phonon frequencies and lifetimes. Using a quantum
impurity solver, signatures of nuclear quantum effects are observed at low
temperatures. We test the description of nonlocal anharmonicity via cellular
VDMFT and the combination with self-consistent phonon (SCPH) theory, yielding
the powerful SCPH+VDMFT approach.
