Quantum spin models play an important role in theoretical condensed matter physics and quantum information theory. One numerical technique that is frequently used in studies of quantum spin systems is exact diagonalization. In this approach, numerical methods are used to find the lowest eigenvalues and associated eigenvectors of the Hamilton matrix of the quantum system. The computational problem is thus to determine the lowest eigenpairs of an extremely large, sparse matrix. Although many sophisticated iterative techniques for the determination of a small number of lowest eigenpairs can be found in the literature, most exact diagonalization studies of quantum spin systems have employed the Lanczos algorithm. In contrast to this, other methods have been applied very successfully to the similar problem of electronic structure calculations. The well known VASP code for example uses a Block Davidson method as well as the residual-minimization - direct inversion of the iterative subspace algorithm (RMM-DIIS). The Davidson algorithm is closely related to the Lanczos method but usually needs less iterations. The RMM-DIIS method was originally proposed by Pulay and later modified by Wood and Zunger. The RMM-DIIS method is particularly interesting if more than one eigenpair is sought since it does not require orthogonalization of the trial vectors at each step. In this work I study the efficiency of the Lanczos, Block Davidson and RMM-DIIS method when applied to basic quantum spin models like the spin-1/2 Heisenberg chain, ladder and dimerized ladder. I have implemented all three methods and are currently applying the methods to the different models. In our presentation I will compare the three algorithms based on the number of iterations to achieve convergence, the required computational time. An Intel's Many-Integrated Core architecture with Intel Xeon Phi coprocessor 5110P integrates 60 cores with 4 hardware threads per core was used for RMM-DIIS method, the achieved parallel speedups were compared with those obtained on a conventional multi-core system.

The work presented in this thesis detail the measurement of the thermal expansion coefficient of three magnetic spin materials. Thermal expansion coefficient values were measured by capacitive dilatometry in several key low (T < 250 K) temperature regions specific to each material. This thesis is separated into several key parts. The first part establishes the theory behind observing phase transitions through the thermal expansion coefficient. Beginning with the classical definitions of the specific heat, compressibility and thermal expansion coefficient, the three properties are related using a property known as the Grüneisen parameter. To first order, the parameter allows phase transitions to be observed by the thermal expansion coefficient. The second part introduces capacitive dilatometry; a technique used to measure the thermal expansion coefficient. Three capacitive dilatometer devices are presented in this section. The silver compact dilatometer, the fused quartz dilatometer and the copper dilatometer. Each device discusses merits and weaknesses to their designs. Particular focus is made on the fused quartz dilatometer which was built during the duration of this thesis. The third part presents research on three magnetic spin materials; LiHoF4, Tb2Ti2O7 and Ba3NbFe3Si2O14. These materials are studied individually focusing on specific aspects. LiHoF4, a candidate material for the transverse field Ising model, provides insight to quantum phase transitions. Thermal expansion coefficient and magnetostriction along the c-axis for T ≈ 1.3-1.8 K and transverse field Ht ≈ 0-4 T were measured extracting critical points for a Ht-T phase diagram. Existing thermal expansion coefficient measurements had evidence of possible re-entrant behaviour. With a high density of low transverse field critical points it was established that LiHoF4 showed no evidence of re-entrant behaviour. The highly debated material Tb2Ti2O7 has a rich, controversial low temperature behaviour. Originally believed to be a spin liquid, specific heat results propose a scenario involving a sample composition dependent ordered state. Still under considerably attention, thermal expansion coefficient measurements were performed for T < 1 K. The results are interpreted to either fit into the proposed scenario or provide evidence for an alternate scenario. The material Ba3NbFe3Si2O14 exhibits a magnetoelectric multiferroic phase below TN ≈ 27 K; a phase where magnetic and electric order simultaneously exist. The formation of this phase is believed to have a similar structural shift observed in hexagonal perovskite multiferroic materials. The ferroelectric ordering in those materials are brought about through a centrosymmetric to non-centrosymmetric structural shift. The thermal expansion and thermal expansion coefficient coefficient along the a and c axis are measured for T > TN searching for a displacive structural phase transition.

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Magnetic frustration is a phenomenon arising in spin systems when spin interactions cannot all be satisfied at the same time. A typical example of geometric frustration is a triangle with Ising-spins at its vertices and antiferromagnetic interaction. While we can easily anti-align two neighbouring spins, it is not possible for the third one to simultaneously anti-align with both of them. Another flavour of magnetic frustration is the so called exchange frustration, where different spin components interact in an Ising fashion on different bonds. Moreover, frustrated spin systems give rise to exotic states of matter, such as spin liquids, spin ices and nematic phases. As frustrated systems are rarely analytically solvable, numerical techniques are of the utmost importance in this framework. This dissertation is concerned with a specific class of models, namely one- and quasi-one-dimensional spin systems and studies their properties by making use of the density matrix renormalisation group technique. This method has been shown to be extremely powerful and reliable to study chain and ladder models. We consider examples of both geometric and exchange frustration. For the former, we take into consideration one of the prototypical examples of geometric frustration in one dimension: the J1-J2 model with ferromagnetic nearest-neighbour interaction J1<0 and antiferromagnetic next-nearest-neighbour interaction J2>0. Our results show the existence of a Haldane gap supported by a special AKLT-like valence bond solid state in a specific region of the coupling ratio. Furthermore, we consider the effect of dimerisation of the first-neighbour coupling. This dimerisation affects the critical point and the ground state underlying the spin gap. These models are of interest in the context of cuprate chain materials such as LiVCuO4, LiSbCuO4 and PbCuSO4(OH)2. Concerning exchange frustration, we consider the celebrated Kitaev-Heisenberg model: it is an extension of the exactly solvable Kitaev model with an additional Heisenberg interaction. The Kitaev-Heisenberg model is currently the minimal model for candidate Kitaev materials. The extended model is not analytically solvable and numerics are needed to study the properties of the system. While both the original Kitaev and the Kitaev-Heisenberg models live on a honeycomb lattice, we here perform systematic studies of the Kitaev-Heisenberg chain and of the two-legged ladder. While the chain cannot support a Kitaev spin liquid state, it shows nevertheless a rich phase diagram despite being a one-dimensional system. The long-range ordered states of the honeycomb can be understood in terms of coupled chains within the Kitaev-Heisenberg model. Following this reasoning, we turn our attention to the Kitaev-Heisenberg model on a two-legged ladder. Remarkably, the phase diagram of the ladder is extremely similar to that of the honeycomb model and the differences can be explained in terms of the different dimensionalities. In particular, the ladder exhibits a topologically non-trivial phase with no long-range order, i.e., a spin liquid. Finally, we investigate the low-lying excitations of the Kitaev-Heisenberg model for both the chain and the ladder geometry.

This thesis, comprising three publications, explores the efficacy of novel generalization of the fidelity susceptibility and their numerical application to the study of frustrated quantum phase transitions in two and one dimensions. Specifically, they will be used in exact diagonalization studies of the various limiting cases of the anisotropic next-nearest neighbour triangular lattice Heisenberg model (ANNTLHM). These generalized susceptibilities are related to the order parameter susceptibilities and spin stiffness and are believed to exhibit similar behaviour although with greater sensitivity. This makes them ideal for numerical studies on small systems. Additionally, the utility of the excited-state fidelity and twist boundary conditions will be explored. All studies are done through numerical exact diagonalization. In the limit of interchain couplings going to zero the ANNTLHM reduces to the well studied $J_1-J_2$ chain with a known, difficult to identify, BKT-type transition. In the first publication of this work the generalized fidelity susceptibilities introduced therein are shown to be able to identify this transition as well as characterize the already understood phases it straddles. The second publication of this work then seeks to apply these generalized fidelity susceptibilities, as well as the excited-state fidelity, to the study of the general phase diagram of the ANNTLHM. It is shown that the regular and excited-state fidelities are useful quantities for the mapping of novel phase diagrams and that the generalized fidelity susceptibilities can provide valuable information as to the nature of the phases within the mapped phase regions. The final paper sees the application of twisted boundary conditions to the anisotropic triangular model (next-nearest neighbour interactions are zero). It is demonstrated that these boundary conditions greatly enhance the ability to numerically explore incommensurate physics in small systems. / Thesis / Doctor of Science (PhD)

Usando uma transformação de dualidade generalizada, considerações de simetria e supondo que as superfície críticas sejam contínuas, obtivemos o dia grama de fase para sistemas de spins Z (N) bidimensionais e sistemas com invariança de calibre Z (N) a quatro dimensões. Caracterizamos as diversas fases dos sistemas de spins pelo valor esperado das potências dos operadores de ordem e desordem. No sistema com invariança de calibre, por outro lado, estas fases caracterizadas pelo comportamento do valor esperado das potências das alças de Wilson e de \'t Hooft. Obtivemos para ambos os sistemas fases moles em que no caso de spins 2D (calibre 4D) todas as potências dos parâmetros de ordem e desordem ( todas as potências das alças de Wilson e \'t Hooft) são nulas (exibem decaimento com o perímetro da alça). Enquanto no sistema com invariança de calibre todas as combinações de decaimento (área ou perímetro) das alças de Wilson e \'t Hooft são permitidas, as relações de comutação no sistema de spins proíbe a existência de fases em que tanto o parâmetro de ordem como o de desordem são não nulos (exceto quando estes operadores comutam). Apresentamos por completeza as relações de dualidade para sistemas de calibre Z (N) com campos de Higgs a três dimensões. / Using a generalized duality transformation, symetry considerations and assuming that criticality is continuous in the system?s parameters, we obtain the phase diagram for two-dimensional Z (N) spins system?s and four-dimensional gauge Z (N) system\'s. For spins system we characterize the various phases by the expectation value of powers of the order and disorder operators. For gauge systems, on the other hand, the characterization is via decay law of powers of Wilson and \'t Hooft loops. We obtain soft phases for both systems, with the folowing, behaviour: for spins system all powers of order and disorder parameters vanish, whereas for gauge systems all powers of Wilson and \'t Hooft loops decay like the perimeter. Whereas all combinations of area and perimeter decay are allowed for Wilson\'s and \'t Hooft\'s loops, the Z (N) commutation relations for spin systems forbid the simultaneous non-vanishing of order and disorder parameters (except when these operators commute). For completeness we include the duality relations for three-dimensional gauge plus Higgs Z(N) systems.

Usando uma transformação de dualidade generalizada, considerações de simetria e supondo que as superfície críticas sejam contínuas, obtivemos o dia grama de fase para sistemas de spins Z (N) bidimensionais e sistemas com invariança de calibre Z (N) a quatro dimensões. Caracterizamos as diversas fases dos sistemas de spins pelo valor esperado das potências dos operadores de ordem e desordem. No sistema com invariança de calibre, por outro lado, estas fases caracterizadas pelo comportamento do valor esperado das potências das alças de Wilson e de \'t Hooft. Obtivemos para ambos os sistemas fases moles em que no caso de spins 2D (calibre 4D) todas as potências dos parâmetros de ordem e desordem ( todas as potências das alças de Wilson e \'t Hooft) são nulas (exibem decaimento com o perímetro da alça). Enquanto no sistema com invariança de calibre todas as combinações de decaimento (área ou perímetro) das alças de Wilson e \'t Hooft são permitidas, as relações de comutação no sistema de spins proíbe a existência de fases em que tanto o parâmetro de ordem como o de desordem são não nulos (exceto quando estes operadores comutam). Apresentamos por completeza as relações de dualidade para sistemas de calibre Z (N) com campos de Higgs a três dimensões. / Using a generalized duality transformation, symetry considerations and assuming that criticality is continuous in the system?s parameters, we obtain the phase diagram for two-dimensional Z (N) spins system?s and four-dimensional gauge Z (N) system\'s. For spins system we characterize the various phases by the expectation value of powers of the order and disorder operators. For gauge systems, on the other hand, the characterization is via decay law of powers of Wilson and \'t Hooft loops. We obtain soft phases for both systems, with the folowing, behaviour: for spins system all powers of order and disorder parameters vanish, whereas for gauge systems all powers of Wilson and \'t Hooft loops decay like the perimeter. Whereas all combinations of area and perimeter decay are allowed for Wilson\'s and \'t Hooft\'s loops, the Z (N) commutation relations for spin systems forbid the simultaneous non-vanishing of order and disorder parameters (except when these operators commute). For completeness we include the duality relations for three-dimensional gauge plus Higgs Z(N) systems.

Complex networks are an important class of models used to describe the behaviour of a very broad category of systems which appear in different fields of science ranging from physics, biology and statistics to computer science and other disciplines. This set of models includes spin systems on a graph, neural networks, decision networks, spreading disease, financial trade, social networks and all systems which can be represented as interacting agents on some sort of graph architecture. In this thesis, by using the theoretical framework of statistical mechanics, the equilibrium and the dynamical behaviour of such systems is studied. For the equilibrium case, after presenting the region graph free energy approximation, the Survey Propagation method, previously used to investi- gate the low temperature phase of complex systems on tree-like topologies, is extended to the case of loopy graph architectures. For time-dependent behaviour, both discrete-time and continuous-time dynamics are considered. It is shown how to extend the cavity method ap- proach from a tool used to study equilibrium properties of complex systems to the discrete-time dynamical scenario. A closure scheme of the dynamic message-passing equation based on a Markovian approximations is presented. This allows to estimate non-equilibrium marginals of spin models on a graph with reversible dynamics. As an alternative to this approach, an extension of region graph variational free energy approximations to the non-equilibrium case is also presented. Non-equilibrium functionals that, when minimized with constraints, lead to approximate equations for out-of-equilibrium marginals of general spin models are introduced and discussed. For the continuous-time dynamics a novel approach that extends the cav- ity method also to this case is discussed. The main result of this part is a Cavity Master Equation which, together with an approximate version of the Master Equation, constitutes a closure scheme to estimate non-equilibrium marginals of continuous-time spin models. The investigation of dynamics of spin systems is concluded by applying a quasi-equilibrium approach to a sim- ple case. A way to test self-consistently the assumptions of the method as well as its limits is discussed. In the final part of the thesis, analogies and differences between the graph- ical model approaches discussed in the manuscript and causal analysis in statistics are presented. / <p>QC 20160904</p>

Les opérateurs de Berezin--Toeplitz permettent de quantifier des fonctions, ou des symboles, sur des variétés kähleriennes compactes, et sont définies à partir du noyau de Bergman (ou de Szeg\H{o}). Nous étudions le spectre des opérateurs de Toeplitz dans un régime asymptotique qui correspond à une limite semiclassique. Cette étude est motivée par le comportement magnétique atypique observé dans certains cristaux à basse température. Nous étudions la concentration des fonctions propres des opérateurs de Toeplitz, dans des cas où les effets sous-principaux (du même ordre que le paramètre semiclassique) permet de différencier entre plusieurs configurations classiques, un effet connu en physique sous le nom de sélection quantique Nous exhibons un critère général pour la sélection quantique et nous donnons des développements asymptotiques précis de fonctions propres dans le cas Morse et Morse--Bott, ainsi que dans un cas dégénéré. Nous développons également un nouveau cadre pour le traitement du noyau de Bergman et des opérateurs de Toeplitz en régularité analytique. Nous démontrons que le noyau de Bergman admet un développement asymptotique, avec erreur exponentiellement petite, sur des variétés analytiques réelles. Nous obtenons aussi une précision exponentiellement fine dans les compositions et le spectre d'opérateurs à symbole analytique, et la décroissance exponentielle des fonctions propres. / Berezin-Toeplitz operators allow to quantize functions, or symbols, on compact Kähler manifolds, and are defined using the Bergman (or Szeg\H{o}) kernel. We study the spectrum of Toeplitz operators in an asymptotic regime which corresponds to a semiclassical limit. This study is motivated by the atypic magnetic behaviour observed in certain crystals at low temperature. We study the concentration of eigenfunctions of Toeplitz operators in cases where subprincipal effects (of same order as the semiclassical parameter) discriminate between different classical configurations, an effect known in physics as quantum selection . We show a general criterion for quantum selection and we give detailed eigenfunction expansions in the Morse and Morse-Bott case, as well as in a degenerate case. We also develop a new framework in order to treat Bergman kernels and Toeplitz operators with real-analytic regularity. We prove that the Bergman kernel admits an expansion with exponentially small error on real-analytic manifolds. We also obtain exponential accuracy in compositions and spectra of operators with analytic symbols, as well as exponential decay of eigenfunctions.