Thesis presented December 13, 2022

Abstract:
Historically, the motion of crystal lattice was treated from the perspective of classical mechanics. However, the nuclear quantum effects (NQE), namely zero-point energy and tunneling through the potential barrier, can alter drastically the behaviour of a crystal. One can study these effects by using path-integral Monte-Carlo techniques. This approach, however, poses certain difficulties. Computing the imaginary time correlation functions for the operators of interest, one is then required to perform analytical continuation to real times. In addition, the transformation to the frequency space is often desired, in particular for the comparison with experiment. This inversion, however, is ill-defined and requires a careful treatment. In the present work we address these questions and indicate the possible way around for the inversion problem which is suitable for various calculation schemes. We demonstrate the utility of this approach on several simple oscillator-like models.
Using this machinery, we turn our attention to the analysis of the NQE in a crystal. They are of particular importance for the different transport properties: unlike metals, the transport in semiconductors and insulators is governed by the vibrations of the crystal lattice, which are described by the normal modes. In the real crystal the modes interact and scatter with each other which shows in their lifetime and, ultimately, leads to the finite heat conductance. We demonstrate that the presence of NQE alters the temperature behaviour of each mode along with it's decay rate. We also study the corresponding changes in the heat conductance in comparison to the classical predictions.

Keywords:
Theory, transport, path integral