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Orientador: Rogerio Custodio / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Quimica / Made available in DSpace on 2018-08-14T14:18:23Z (GMT). No. of bitstreams: 1 Paschoal_JulianadeLima_M.pdf: 914971 bytes, checksum: 5a3a529f3c0006227e418c9e1629e6a5 (MD5) Previous issue date: 2006 / Resumo: Uma estratégia recente denominada Monte Carlo Quântico (MCQ) permite acessar a função de onda exata de um sistema resolvendo a equação de Schrödinger. Dentre as alternativas de MCQ destacam-se o Monte Carlo Quântico Variacional (MCQV) e o Monte Carlo Quântico de Difusão (MCQD). O MCQV determina o valor médio de qualquer propriedade atômica ou molecular associada a uma função de onda arbitrária empregando o algoritmo de Metropolis. O MCQD, por sua vez, baseia-se na solução da equação de Schrödinger dependente do tempo através de um processo de difusão em equilíbrio com um processo cinético de primeira ordem. Neste trabalho os objetivos são: a) comparar os efeitos de funções de base de Slater com diferentes expoentes nos níveis de teoria do MCQV e MCQD; b) testar funções de onda baseadas no modelo Hartree e Hartree-Fock no MCQV e MCQD e c) avaliar o efeito da localização de orbitais nestes métodos. Esses objetivos foram avaliados em átomos, moléculas diatômicas e alguns hidretos poliatômicos contendo elementos do segundo período da tabela periódica. Inicialmente, usou-se de uma função de onda representada por um determinante de Slater com orbitais obtidos através da combinação linear de funções de Slater através do método Hartree-Fock. Os expoentes do conjunto de base utilizado foram determinados através das Regras de Slater, otimização Hartree-Fock em ambiente molecular e otimizaçãoHartree-Fock em ambiente atômico. O MCQV e o MCQD foram empregados para a obtenção da energia média do sistema. Posteriormente, substituíram-se as funções de Slater por funções STO-6G. Os mesmos expoentes do conjunto de base utilizados nos cálculos com funções de Slater foram empregados para os cálculos STO-6G. Finalmente, utilizou-se o produto de Hartree como função de onda para os cálculos MCQV e MCQD com as funções de base já mencionadas. As principais conclusões desse trabalho são: a) o MCQD, conforme esperado, apresenta menores energias quando comparado ao MCQV; b) Cálculos MCQD usando determinante de Slater, conjunto de base com otimização de expoente para a molécula e átomo e nehum fator de correção forneceu energias comparadas a Gaussianas do tipo double-zeta no método coupled cluster incluindo excitações simples e duplas; c) Funções de base STO-6G devem ser utilizadas com cautela para representar funções STO; d) as energias calculadas através do produto de Hartree apresentam um comportamento que se distancia das funções Hartree-Fock quando orbitais localizados são usados; e) resultados melhores são esperados quando orbitais são auto-consistentes com respeito ao método de Hartree. / Abstract: A recent strategy called Quantum Monte Carlo (QMC) allows to access the exact wave function of a system solving Schrödinger¿s equation. Among the alternatives of QMC, Variational Quantum Monte Carlo (VQMC) and Diffusion Quantum Monte Carlo (DQMC) are distinguished. VQMC determines the average value of any atomic or molecular property associated to an arbitrary wave function using Metropolis algorithm. DQMC, on the other hand, is based on the solution of the time-dependent Schrödinger equation from a diffusion process in equilibrium with a first-order kinetic process. In this work the objectives were: a) to compare the effect of the Slater basis set with exponents adjusted in different environments at the VQMC and DQMC levels of theory; b) to test wave functions based on the Hartree and Hartree-Fock models along with VQMC and DQMC; c) to evaluate the effect of orbital localization within these methods. These objectives are evaluated in atoms, diatomic molecules and some polyatomic hydrates containing elements from the second period of the Periodic Table. Initially, a conventional wave function represented by a single Slater determinant is used with orbitals from the linear combination of Slater¿s functions from the Hartree- Fock method. The basis set exponents are determined from the Slater rules, Hartree-Fock atom optimized and Hartree-Fock molecule optimized. VQMC and DQMC yielded the average energy of each system. Later, Slater¿s functions are changed to the STO-6G basis functions. The same basis set exponents are applied for the STO-6G calculations. Finally, the Hartree product is used as a wave function for the VQMC and DQMC calculations with the same basis functions already mentioned. The main conclusions fro this work are: a) DQMC, as expected, presents lower energies when compared to VQMC; b) DQMC calculations using single Slater determinant and basis set with molecule and atom optimized exponents and no correlation factor provided energies compared to a Gaussian double zeta basis set at the coupled cluster including singles and doubles excitations level of theory; c) STO-6G must be used with caution in order to represent STO functions; d) the energies calculated with the Hartree product presented a behavior not far from the Hartree-Fock wave functions when localized orbitals were used; e) better results are expected if orbitals are self-consistent with respect to the Hartree method. / Mestrado / Físico-Química / Mestre em Química

Phase transitions and their associated critical phenomena are of fundamental importance and play a crucial role in the development of statistical physics for both classical and quantum systems. Phase transitions embody diverse aspects of physics and also have numerous applications outside physics, e.g., in chemistry, biology, and combinatorial optimization problems in computer science. Many problems can be reduced to a system consisting of a large number of interacting agents, which under some circumstances (e.g., changes of external parameters) exhibit collective behavior; this type of scenario also underlies phase transitions. The theoretical understanding of equilibrium phase transitions was put on a solid footing with the establishment of the renormalization group. In contrast, non-equilibrium phase transition are relatively less understood and currently a very active research topic. One important milestone here is the Kibble-Zurek (KZ) mechanism, which provides a useful framework for describing a system with a transition point approached through a non-equilibrium quench process. I developed two efficient Monte Carlo techniques for studying phase transitions, one is for classical phase transition and the other is for quantum phase transitions, both are under the framework of KZ scaling. For classical phase transition, I develop a non-equilibrium quench (NEQ) simulation that can completely avoid the critical slowing down problem. For quantum phase transitions, I develop a new algorithm, named quasi-adiabatic quantum Monte Carlo (QAQMC) algorithm for studying quantum quenches. I demonstrate the utility of QAQMC quantum Ising model and obtain high-precision results at the transition point, in particular showing generalized dynamic scaling in the quantum system. To further extend the methods, I study more complex systems such as spin-glasses and random graphs. The techniques allow us to investigate the problems efficiently. From the classical perspective, using the NEQ approach I verify the universality class of the 3D Ising spin-glasses. I also investigate the random 3-regular graphs in terms of both classical and quantum phase transitions. I demonstrate that under this simulation scheme, one can extract information associated with the classical and quantum spin-glass transitions without any knowledge prior to the simulation.

We have used ab initio computational simulation techniques to investigate both intra- and intermolecular interactions in a novel family of ionic organophosphonate molecular crystals. We have examined the influence of various numerical approximations on the computed geometry and binding energies of a selection of well-characterised hydrogen bonded systems. It was found that numerical basis sets provided the efficiency required to study the large hydrogen bonded dimer anions present in the organophosphonate system, while also producing accurate geometries and binding energies. We then calculated the relaxed structures and binding energies of phenylphosphonic acid dimer in the two arrangements in which it is present in the bulk crystal. The computed geometries were in excellent agreement with the experimental structures and the binding energies were consistent with those found for other ionic hydrogen bonded systems. Electron density maps were used to gain insight into the nature of the hydrogen bonding interaction between phenylphosphonic acid dimers. We also examined the effect of aromatic ring substituents on the geometry and energetics of the hydrogen bonding interaction. The nitro-substituted dimer was predicted to have a stronger binding energy than its unsubstituted parent while the methyl-substituted dimer was predicted to have a similar binding energy to its unsubstituted parent. An analysis of crystal field effects showed that the structure of the phenylphosphonic acid dimers in the organophosphonates is a complex product of competing intra- and intermolecular forces and crystal field effects. Cooperative effects in the organophosphonate system were also investigated and it was found that the interactions were mostly one-body (local) in nature. We have examined the intramolecular charge-transfer interaction between copper-halogen cations in the organophosphonate materials. The origin of geometric differences between the Cu(I) starting material and Cu(II) product cations was attributed to the electronic configuration of the Cu ion, not crystal field effects. To gain further insight into the difference in electronic structure between the starting material and product, we attempted to simulate the step-by-step dissociation of the [CuI]+ system. Although this investigation was not successful, we were able to expose some of the pitfalls of simulating dissociating odd-electron systems. We also analysed and compared the charge-transfer interaction in the chloro-, bromo- and iodo-forms of the organophosphonate family. The charge-transfer interaction was predicted to increase on going from the chloro- to the iodo-form, consistent with solid-state UV-visible data. Finally, we used the highly accurate Quantum Monte Carlo (QMC) method to investigate the hydrogen bonding interaction in water dimer and to calculate the dissociation energy. The accuracy of the experimental estimate for the dissociation energy has recently been questioned and an alternative value has been put forward. Our results lend support to the validity of the alternative value and are also in excellent agreement with those from other high-level calculations. Our results also indicate that QMC techniques are a promising alternative to traditional wavefunction techniques in situations where both high accuracy and efficiency are important.

Over the past 20 years a great deal of progress has been made towards understanding the physics of the high-temperature (high-Tc) cuprate superconductors. Much of the low- energy physics of these materials appears to be captured by two-dimensional Hubbard or t-J models which have provided significant insight into a number of properties such as the pseudogap, antiferromagnetism and superconductivity itself. However, intrinsically planar models are unable to account for the large variations in Tc observed across materials nor do they capture the electron-phonon (el-ph) interaction, the importance of which a number of experimental probes now indicate. This thesis examines the el-ph interaction in cuprates using a combination of analytical and numerical techniques. Starting from the microscopic mechanism for coupling to in-plane and c-axis polarized oxygen phonons, the theory of el-ph coupling is presented. The el-ph self-energy is derived in the context of Migdal-Eliashberg theory and then applied to understanding the detailed temperature and doping dependence of the renormalizations observed by Angle-resolved photoemission spectroscopy. The qualitative signatures of el- boson coupling in the density of states of a d-wave superconductor are also examined on general grounds and a model calculation is presented for el-ph coupling signatures in the density of states. Following this, the theory is extended to include the effects of screening and the consequences of this theory are explored. Due to the quasi-2D nature of the cuprates, screening is found to anomalously enhance the el-ph contribution to d-wave pairing. This result is then considered in light of the material and doping dependence of Tc and a framework for understanding the materials variations in Tc is presented. From these studies, a detailed picture of the role of the el-ph interaction in the doped cuprates emerges where the interaction, working in conjunction with a dominant pairing interaction, provides much of the materials variations in Tc observed across the cuprate families. Turning towards numerical techniques, small cluster calculations are presented which examine the effects of a local oxygen dopant in an otherwise ideal Bi2Sr2CaCu2O8+δ crystal. Here, it is demonstrated that the dopant locally enhances electronic properties such as the antiferromagnetic exchange energy J via local el-ph coupling to planar local oxygen vibrations. Finally, in an effort to extend the scope of this work to the underdoped region of the phase diagram, an examination of the properties of the single-band Hubbard and Hubbard-Holstein model is carried out using Determinant Quantum Monte Carlo. Here focus is placed on the spectral properties of the model as well as the competition between the the antiferromagnetic and charge-density-wave orders. As with the small cluster calculations, a strong interplay between the magnetic and lattice properties is observed.

Developments are made on the quantum Monte Carlo methods towards increasing the precision and the stability of the non fixed-node projector calculations of fermions. In the first part of the developments, the wavefunction correction scheme, which was developed to increase the precision of the diusion Monte Carlo (DMC) method, is applied to non fixed-node DMC to increase the precision of such fermion calculations which do not have nodal error. The benchmark calculations indicate a significant decrease of statistical error due to the usage of the correction scheme in such non fixed-node calculations. The second part of the developments is about the modifications of the wavefunction correction scheme for having a stable non fixed-node DMC algorithm for fermions. The minus signed walkers of the non fixed-node calculations are avoided by these modifications in the developed stable algorithm. However, the accuracy of the method decreases, especially for larger systems, as a result of the discussed modifications to overcome the sign instability.

Over the past 20 years a great deal of progress has been made towards understanding the physics of the high-temperature (high-Tc) cuprate superconductors. Much of the low- energy physics of these materials appears to be captured by two-dimensional Hubbard or t-J models which have provided significant insight into a number of properties such as the pseudogap, antiferromagnetism and superconductivity itself. However, intrinsically planar models are unable to account for the large variations in Tc observed across materials nor do they capture the electron-phonon (el-ph) interaction, the importance of which a number of experimental probes now indicate. This thesis examines the el-ph interaction in cuprates using a combination of analytical and numerical techniques. Starting from the microscopic mechanism for coupling to in-plane and c-axis polarized oxygen phonons, the theory of el-ph coupling is presented. The el-ph self-energy is derived in the context of Migdal-Eliashberg theory and then applied to understanding the detailed temperature and doping dependence of the renormalizations observed by Angle-resolved photoemission spectroscopy. The qualitative signatures of el- boson coupling in the density of states of a d-wave superconductor are also examined on general grounds and a model calculation is presented for el-ph coupling signatures in the density of states. Following this, the theory is extended to include the effects of screening and the consequences of this theory are explored. Due to the quasi-2D nature of the cuprates, screening is found to anomalously enhance the el-ph contribution to d-wave pairing. This result is then considered in light of the material and doping dependence of Tc and a framework for understanding the materials variations in Tc is presented. From these studies, a detailed picture of the role of the el-ph interaction in the doped cuprates emerges where the interaction, working in conjunction with a dominant pairing interaction, provides much of the materials variations in Tc observed across the cuprate families. Turning towards numerical techniques, small cluster calculations are presented which examine the effects of a local oxygen dopant in an otherwise ideal Bi2Sr2CaCu2O8+δ crystal. Here, it is demonstrated that the dopant locally enhances electronic properties such as the antiferromagnetic exchange energy J via local el-ph coupling to planar local oxygen vibrations. Finally, in an effort to extend the scope of this work to the underdoped region of the phase diagram, an examination of the properties of the single-band Hubbard and Hubbard-Holstein model is carried out using Determinant Quantum Monte Carlo. Here focus is placed on the spectral properties of the model as well as the competition between the the antiferromagnetic and charge-density-wave orders. As with the small cluster calculations, a strong interplay between the magnetic and lattice properties is observed.

At the interface of quantum information and condensed matter physics, the study of entanglement in quantum many-body systems requires a new toolset which combines concepts from each. This thesis introduces a set of computational methods to study phases and phase transitions in lattice models of quantum systems, using the Renyi entropies as a means of quantifying entanglement. The scaling of entanglement entropy can give valuable insight into the phase of a condensed matter system. It can be used to detect exotic types of phases, to pinpoint transitions between phases, and can give us universal information about a system. The first approach in this thesis is a technique to measure entanglement in finite size lattice systems using zero-temperature quantum Monte Carlo simulations. The algorithm is developed, implemented, and used to explore anomalous entanglement scaling terms in the spin-1/2 Heisenberg antiferromagnet. In the second part of this thesis, a new and complementary numerical technique is introduced to study entanglement not just in finite size systems, but as we approach the thermodynamic limit. This “numerical linked-cluster expansion” is used to study two different systems at their quantum critical points — continuous phase transitions occurring at zero temperature, at which these systems exhibit universal properties. Remarkably, these universal properties can be reflected in the scaling of entanglement. Entanglement offers a new perspective on condensed matter systems, one which takes us closer to genuinely understanding what goes on in these materials at the quantum mechanical level. This thesis demonstrates the first steps in developing an extensive list of computational tools that can be used to study entanglement over a wide range of interacting quantum many-body systems. With the ever increasing computational power available, it may be only a matter of time before these tools are used to create a comprehensive framework for the characterization of condensed matter phases and phase transitions.

We present a comprehensive study of a low-dimensional spin-half quantum antiferromagnet, the J-Q model, in the presence of an external (Zeeman) magnetic field using numerical methods, chiefly stochastic series expansion quantum Monte Carlo with directed loop updates and quantum replica exchange. The J-Q model is a many-body Hamiltonian acting on a lattice of localized spin-half degrees of freedom; it augments the Heisenberg exchange with a four-spin interaction of strength Q. This model has been extensively studied at zero field, where the Q term drives a quantum phase transition from a Néel-like state to a valence-bond solid (a nonmagnetic state consisting of a long-range-ordered arrangement of local singlet bonds between sites). This transition is believed to be an example of deconfined quantum criticality, where the excitations are spinons—exotic spin-half bosons. We study the J-Q model in the presence of a magnetic field in both one and two dimensions. In one dimension, there is metamagnetism above a critical coupling ratio (Q/J)min. Metamagnetism is a first-order quantum phase transition characterized by discontinuities in the magnetization as a function of field (magnetization jumps). We derive an exact expression for (Q/J)min = 2/9, and show that the metamagnetism is caused by the onset of attractive interactions between magnons (flipped spins on a polarized background). We predict that the same mechanisms will produce metamagnetism in the unfrustrated antiferromagnetic J1-J2 model with anisotropy. Below (Q/J)min, the saturation transition is continuous and we show that it is governed by the expected zero-scale-factor universality. In two dimensions, we also find metamagnetism above a critical coupling ratio (Q/J)min=0.417, caused by the same mechanism as in the one-dimensional case. In two dimensions we also show evidence of an anomalous temperature dependence of specific heat arising from field-induced Bose-Einstein condensation of spinons at the deconfined quantum critical point. / 2019-11-06T00:00:00Z

The properties of the macroscopic world around us, and of which we are a part, are largely determined by the low energy, collective behaviour of many interacting particles, including the nuclei and, especially, the electrons present. Although the fundamental laws governing the behaviour of these many-body systems are believed to be known in principle, the practical solution of the equations of quantum mechanics remains a challenging area of research. This thesis is concerned with the application of quantum Monte Carlo methods to two model systems: the spin-polarised homogeneous electron gas, and a hole-doped electron gas. Electronic structure theory is briefly reviewed before discussing in more detail the quantum Monte Carlo methods used in this thesis. A study of the three-dimensional spin-polarised homogeneous electron gas (HEG) is then reported, where the relatively new technique of twist averaging is investigated in detail and accurate energies and pair correlation functions are obtained over densities $r_s = 0.5 – 20$ a.u. and the full range of spin-polarisation, allowing comparison with the Perdew-Zunger interpolation scheme used in local spin density approximation exchange-correlation functionals. Following this, an impurity is added to the electron gas in the form of a positively charged hole, and the interaction is studied. Relaxation energies, pair correlation functions and momentum densities are reported. Trion formation is observed over a range of carrier densities and electron-hole mass ratios in agreement with experiment. Isolated trions are also studied, where the diffusion Monte Carlo method is exact. Methodological innovations developed while carrying out this work are discussed, including a variance reduction technique for twist-averaged calculations and a new trial wave function for impurity-in-HEG calculations.