Low temperature acoustic polaron localization
  We calculate the properties of an acoustic polaron in three dimensions in thermal equilibrium at a given low temperature using the path integral Monte Carlo method. The specialized numerical method used is described in full details, thus complementing our previous paper [R. Fantoni, Phys. Rev. B {\bf 86}, 144304 (2012)], and it appears to be the first time it has been used in this context. Our results are in favor of the presence of a phase transition from a localized state to an extended state for the electron as the phonon-electron coupling constant decreases. The phase transition manifests itself with a jump discontinuity in the potential energy as a function of the coupling constant and it affects the properties of the path of the electron in imaginary time: In the weak coupling regime the electron is in an extended state whereas in the strong coupling regime it is found in a self-trapped state. 
Riccardo Fantoni
cond-mat.mtrl-sci physics.comp-ph quant-ph