  The phonon Casimir effect describes the phonon-mediated interaction between
defects in condensed-matter systems. Using the path-integral formalism, we
derive a general method for calculating the Helmholtz free energy due to
vibrational modes in systems of arbitrary dimensionality and composition. Our
results make it possible to extract the defect interaction energy at any
temperature for various defect configurations. We demonstrate our approach in
action by performing numerical calculations for mono- and diatomic chains, as
well as a diatomic molecule, at zero and finite temperatures and validate our
results using exact diagonalization.
