The Monte Carlo Method and Quantum Path Integrals

Abstract

Cubic structured perovskites is a family of solids displaying a wide range of interesting physical properties, including several types of structural phase transitions. Frequently, these transitions give rise to competing crystal strucutres separated by small energy barriers in which case it is not obvious that zero-point fluctuations and other nuclear quantum effects can be neglected. One such perovskite is barium zirconate where the possible presence of antiferrodistortive (AFD) phases related to tilting of the oxygen octahedra have been a matter of some debate. The interest in this material is motived by its application as a proton conducting electrolyte in solid oxide fuel cells. To take quantum effects into account when calculating properties a path integral formulation may be used. This approach leads to a multi-dimensional integral which can be calculated using Metropolis Monte Carlo, resulting in the path integral Monte Carlo method (PIMC). In this study PIMC simulations are used to calculate various properties of barium zirconate, most notably the momentum distribution which is sensitive to nuclear quantum effects. To describe the inter-atomic interactions a rigid ion model is adopted with a simple pair potential consisting of a short-range Buckingham potential and the long-range Coulomb potential. Tuning of the parameters in the Buckingham potential allows for the introduction of AFD instabilities to the system. In this way two different model systems are established, one stable cubic system and one with AFD phases. Comparing the momentum distributions for these model system excludes the possibility that an oxygen atom can simultaneously occupy two different sites in the unstable system. The thesis also serves as an introduction to PIMC simulations in general. Different algorithms for sampling new paths and calculating the momentum distribution are investigated and compared. All algorithms in the project have been implemented from scratch and the source code is made available in an appendix.