Study of nuclear quantum effects in insulating solids by means of path-integral Monte-Carlo

Historically, the motion of a 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 which can be analyzed via various thermodynamic properties.
These properties can be computed exactly by using path-integral Monte-Carlo (PIMC) techniques. However, employing this approach in order to determine dynamical quantities, one encounters difficulties that are intrinsic to the method. Within the formalism of Green and Kubo, which is a common tool for analysis of transport properties, linear response functions can in principle be calculated by analytical continuation of imaginary time correlation function obtained with PIMC. In practice, it corresponds to an inverse Laplace transform, which becomes ill-defined for numerical data suffering from finite precision. In the thesis 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. A description, which properly accounts for the quantum phenomena, is of particular practical importance for the transport properties: unlike metals, the transport in semiconductors and insulators is governed by the vibrations of it's crystal lattice, which are described by the normal modes. In the real crystal the modes interact and scatter with each other resulting in a finite lifetime which, ultimately, determines 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.