As propriedades do estado fundamental do modelo de Heisenberg antiferroinagnético quântico de spin-1/2 na rede quadrada e na rede cúbica espacialmente anisotrópica são investigadas através de um novo método de Monte Carlo, baseado na estimativa do maior autovalor de uma matriz de elementos não negativos. A energia do estado fundamental e a magnetização \"staggered\" destes sistemas são calculadas em redes relativamente grandes com até 24 x 24 sítios para o caso de redes quadradas e 8 x 8 x 8 sítios para o caso de redes cúbicas. O método desenvolvido também pode ser usado como um novo algoritmo para a determinação direta da entropia de sistemas de spins de Ising através de simulações usuais de Monte Carlo. Usando este método, calculamos a entropia do antiferromagneto de Ising na presença de um campo magnético externo nas redes triangular e cúbica de face centrada. / The ground state properties of the antiferromagnetic quantum Heisenberg model with spin-112 defined on a square lattice and on a cubic lattice with spatial anisotropy are investigated through a new Monte Carlo method, based on the estimation of the largest eigenvalue of a matrix with nonnegative elements. The ground state energy and the staggered magnetization of these systems are calculated in relatively large lattices with up to 24 x 24 sites for the square lattices and 8 x 8 x 8 sites for cubic lattices. The method developped can also be used as a new algorithm for the direct determination of the entropy of Ising spin systems through ordinary Monte Car10 simulations. By using this method we calculate the entropy of the Ising antiferromagnetic in the presence of a magnetic field in the triangular and face centered cubic lattices.

Enormous advances in computing power in recent decades have made it possible to perform accurate numerical simulations of a wide range of systems in condensed matter physics. At the forefront of this progress has been density functional theory (DFT), a very popular approach to tackling the complexity of quantum-mechanical systems that very often strikes a good balance between accuracy and tractability in light of the finite computational resources available to researchers. This thesis describes work utilising DFT methods to tackle two distinct problems. Firstly, the theoretical prediction of stable and metastable periodic structures under specified conditions using the ab initio random structure searching (AIRSS) method, which involves a large scale exploration of the Born-Oppenheimer energy surface, and secondly the use of a vibrational self-consistent field (VSCF) approach to investigate the effects of nuclear motion and anharmonicity in crystal systems, which involves a local exploration of the Born-Oppenheimer energy surface. The AIRSS crystal structure prediction method is here applied to a study of defect structures in graphene. It is also applied to a study of the xenon-oxygen binary system under a range of geological pressures (83–200 GPa). Novel xenon oxide structures are predicted and characterised theoretically. This work was carried out in collaboration with an experimental study of the system at the lower end of the pressure range. The VSCF approach to investigating anharmonicity is here applied to the study of tin selenide (SnSe), a material that has recently been shown to demonstrate consider- able promise as a thermoelectric material. In this thesis, the effects of the anharmonic nuclear motion on the vibrational and electronic properties of SnSe are investigated quantitatively.

This thesis considers quantum impurity models that exhibit a quantum phase transition (QPT) between a Fermi liquid strong coupling (SC) phase, and a doubly-degenerate non-Fermi liquid local moment (LM) phase. We focus on what can be said from exact analytic arguments about the LM phase of these models, where the system is characterized by an SU(2) spin degree of freedom in the entire system. Conventional perturbation theory about the non-interacting limit does not hold in the non-Fermi liquid LM phase. We circumvent this problem by reformulating the perturbation theory using a so-called `two self-energy' (TSE) description, where the two self-energies may be expressed as functional derivatives of the Luttinger-Ward functional. One particular paradigmatic model that possesses a QPT between SC and LM phases is the pseudogap Anderson impurity model (PAIM). We use infinite-order perturbation theory in the interaction, U, to self-consistently deduce the exact low-energy forms of both the self-energies and propagators in each of the distinct phases of the model. We analyse the behaviour of the model approaching the QPT from each phase, focusing on the scaling of the zero-field single-particle dynamics using both analytical arguments and detailed numerical renormalization group (NRG) calculations. We also apply two `conserving' approximations to the PAIM. First, second-order self-consistent perturbation theory and second, the fluctuation exchange approximation (FLEX). Within the FLEX approximation we develop a numerical algorithm capable of self-consistently and coherently describing the QPT coming from both distinct phases. Finally, we consider a range of static spin susceptibilities that each probe the underlying QPT in response to coupling to a magnetic field.