<p>In this thesis I present three publications about the use of<br />valence bonds to gain information about quantum spin systems.<br />Valence bonds are an essential ingredient of low energy states present<br />in many compounds.<br /><br />The first part of this thesis is dedicated to<br />two studies of the antiferromagnetic J₁-J₂ chain with<br />S=1/2. We show how automated variational calculations based on<br />valence bond states can be performed close to the Majumdar-Ghosh point<br />(MG-point). At this point, the groundstate is a product state of<br />dimers (valence bonds between nearest neighbours). In the dimerized<br />region surrounding the MG-point, we find such variational computations<br />to be reliable.<br /><br />The first publication is about<br />the entanglement properties of an impurity attached to the chain. We show<br />how to use the variational method to calculate the negativity, an<br />entanglement measure between the impurity and a distant part of the<br />chain. We find that increasing the impurity coupling and a<br />minute explicit dimerization, suppress the long-ranged entanglement<br />present in the system for small impurity coupling at the MG-point. <br /><br />The second publication is about a<br />transition from commensurate to incommensurate behaviour and how its<br />characteristics depend on the parity of the length of the chain. The<br />variational technique is used in a parameter regime inaccessible to<br />DMRG. We find that in odd chains, unlike in even chains, a very<br />intricate and interesting pattern of level crossings can be observed. <br /><br />The publication of the second part is about novel worm algorithms for<br />a popular quantum Monte Carlo method called valence bond quantum Monte<br />Carlo (VBQMC). The algorithms are based on the notion of a worm<br />moving through a decision tree. VBQMC is entirely formulated in<br />terms of valence bonds. In this thesis, I explain how the approach<br />of VBQMC can be translated to the S_z-basis. The algorithms explained<br />in the publication can be applied to this S_z-method.</p> / Doctor of Philosophy (PhD)

Condensed matter systems undergoing phase transitions rarely allow exact solutions. The presence of disorder renders the situation  even worse but collective Monte Carlo methods and parallel algorithms allow numerical descriptions. This thesis considers classical phase transitions in disordered spin systems in general and in effective models of superfluids with disorder and novel interactions in particular. Quantum phase transitions are considered via a quantum to classical mapping. Central questions are if the presence of defects changes universal properties and what qualitative implications follow for experiments. Common to the cases considered is that the disorder maps out correlated structures. All results are obtained using large-scale Monte Carlo simulations of effective models capturing the relevant degrees of freedom at the transition. Considering a model system for superflow aided by a defect network, we find that the onset properties are significantly altered compared to the $\lambda$-transition in $^{4}$He. This has qualitative implications on expected experimental signatures in a defect supersolid scenario. For the Bose glass to superfluid quantum phase transition in 2D we determine the quantum correlation time by an anisotropic finite size scaling approach. Without a priori assumptions on critical parameters, we find the critical exponent $z=1.8 \pm 0.05$ contradicting the long standing result $z=d$. Using a 3D effective model for multi-band type-1.5 superconductors we find that these systems possibly feature a strong first order vortex-driven phase transition. Despite its short-range nature details of the interaction are shown to play an important role. Phase transitions in disordered spin models exposed to correlated defect structures obtained via rapid quenches of critical loop and spin models are investigated. On long length scales the correlations are shown to decay algebraically. The decay exponents are expressed through known critical exponents of the disorder generating models. For cases where the disorder correlations imply the existence of a new long-range-disorder fixed point we determine the critical exponents of the disordered systems via finite size scaling methods of Monte Carlo data and find good agreement with theoretical expectations. / <p>QC 20150306</p>

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2016-06-29T13:52:59Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) tese_final.pdf: 5635071 bytes, checksum: b300efb627e9ece412ad5936ab67e8e2 (MD5) / Made available in DSpace on 2016-06-29T13:52:59Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) tese_final.pdf: 5635071 bytes, checksum: b300efb627e9ece412ad5936ab67e8e2 (MD5) Previous issue date: 2013-08-26 / CNPq / No presente trabalho duas classes de problemas são abordadas. Primeiramente, são apresentados estudos computacionais sobre o modelo O(n) de spins na rede quadrada, e em seguida apresentamos novas soluções exatas para a dinâmica de bolhas na célula de Hele-Shaw. O estudo do modelo O(n) é feito utilizando sua representação em laços (cadeias fechadas), a qual é obtida a partir de uma expansão para altas temperaturas. Nesse representação, a função de partição do modelo possui uma expansão diagramática em que cada termo depende do número e comprimento total de laços e do número de (auto)interseções entre esses laços. Propriedades críticas do modelo de laços O(n) são obtidas através de conceitos oriundos da teoria de percolação. Para executar as simulações Monte Carlo, usamos o eficiente algoritmo WORM, o qual realiza atualizações locais através do movimento da extremidade de uma cadeia aberta denominada de verme e não sofre com o problema de "critical slowing down". Para implementar esse algoritmo de forma eficiente para o modelo O(n) na rede quadrada, fazemos uso de um nova estrutura de dados conhecida como listas satélites. Apresentamos estimativas para o ponto crítico do modelo para vários valores de n no intervalo de 0 < n ≤ 2. Usamos as estatísticas de laços e vermes para extrair, respectivamente, os expoentes críticos térmicos e magnéticos do modelo. No estudo de dinâmica de interfaces, apresentamos uma solução exata bastante geral para um arranjo periódico de bolhas movendo-se com velocidade constante ao longo de uma célula de Hele-Shaw. Usando a periodicidade da solução, o domínio relevante do problema pode ser reduzido a uma célula unitária que contém uma única bolha. Nenhuma imposição de simetria sobre forma da bolha é feita, de modo que a solução é capaz de produzir bolhas completamente assimétricas. Nossa solução é obtida por métodos de transformações conformes entre domínios duplamente conexos, onde utilizamos a transformação de Schwarz-Christoffel generalizada para essa classe de domínios. / In this thesis two classes of problems are discussed. First, we present computational studies of the O(n) spin model on the square lattice and determine its critical properties, whereas in the second part of the thesis we present new exact solutions for bubble dynamics in a Hele-Shaw cell. The O(n) model is investigated by using its loop representation which is obtained from a high-temperature expansion of the original model. In this representation, the partition function admits an diagrammatic expansion in which each term depends on the number and total length of loops (closed graphs) as well as on the number of intersections between these loops. Critical properties of the O(n) model are obtained by employing concepts from percolation theory. To perform Monte Carlo simulations of the model, we use the WORM algorithm, which is an efficient algorithm that performs local updates through the motion of one of the ends (called head) of an open chain (called worm) and hence does not suffer from “critical slowing down”. To implement this algorithm efficiently for the O(n) model on the square lattice, we make use of a new data structure known as a satellite list. We present estimates for the critical point of the model for various values of n in the range 0 < n ≤ 2. We use the statistics about the loops and the worm to extract the thermal and magnetic critical exponents of the model, respectively. In our study about interface dynamics, we present a rather general exact solution for a periodic array of bubbles moving with constant velocity in a Hele-Shaw cell. Using the periodicity of the solution, the relevant domain of the problem can be reduced to a unit cell containing a single bubble. No symmetry requirement is imposed on the bubble shape, so that the solution is capable of generating completely asymmetrical bubbles. Our solution is obtained by using conformal mappings between doubly-connected domains and employing the generalized Schwarz-Christoffel formula for this class of domains.

The quantum phase transition between groundstates of a system with correlated disorder near absolute zero is studied. The computations are based on Monte Carlo methods and the worm algorithm which is an effective method to simulate basic models like the Ising and XY model by making use of global Monte Carlo moves given by modified random walks. Random quenched disorder modeled as a correlated distribution of two values of the coupling constant gives rise to an additional phase transition with a not before seen intermediate phase. / Kvantfasövergången mellan grundtillstånd av ett system med korrelerad oordning nära nolltemperaturen studeras. Beräkningarna är baserade på Monte Carlo metoder och worm algoritmen som är en effektiv metod för att simulera grundläggande modeller som Ising och XY modellen genom att använda sig av globala Monte Carlo steg som ges av modifierade slumpmässiga vandringar. Slumpmässig infrusen oordning modellerad som en korrelerad fördelning av två värden på kopplingsstyrkan ger upphov till en ny mellanliggande fas.