We present in this thesis a series of studies on the physical properties of some one dimensional systems. In particular we study the low energy properties of various spin chains and a junction of Luttinger wires. For spin chains we specifically look at the role of perturbations like frustrating interactions and dimerisation in a nearest neighbour chain and the formation of magnetisation plateaus in two kinds of models; one purely theoretical and the other motivated by experiments. In our second subject of interest we study using a renormalisation group analysis the effect of spin dependent scattering at a junction of Luttinger wires. We look at the physical effects caused by the interplay of electronic interactions in the wires and the scattering processes at the junction. The thesis begins with an introductory chapter which gives a brief glimpse of the ideas and techniques used in the specific problems that we have worked on. Our work on these problems is then described in detail in chapters 25. We now present a brief summary of each of those chapters. In the second chapter we look at the ground state phase diagram of the mixed-spin sawtooth chain, i.e a system where the spins along the baseline are allowed to be different from the spins on the vertices. The spins S1 along the baseline interact with a coupling strength J1(> 0). The coupling of the spins on the vertex (S2) to the baseline spins has a strength J2. We study the phase diagram as a function of J2/J1 [1]. The model exhibits a rich variety of phases which we study using spinwave theory, exact diagonalisation and a semi-numerical perturbation theory leading to an effective Hamiltonian. The spinwave theory predicts a transition from a spiral state to a ferrimagnetic state at J2S2/2J1S1 = 1 as J2/J1 is increased. The spectrum has two branches one of which is gapless and dispersionless (at the linear order) in the spiral phase. This arises because of the infinite degeneracy of classical ground states in that phase. Numerically, we study the system using exact diagonalisation of up to 12 unit cells and S1 = 1 and S2 =1/2. We look at the variation of ground state energy, gap to the lowest excitations, and the relevant spin correlation functions in the model. This unearths a richer phase diagram than the spinwave calculation. Apart from revealing a possibility of the presence of more than one kind of spiral phases, numerical results tell us about a very interesting phase for small J2. The spin correlation function (for the spin1/2s) in this region have a property that the nextnearest-neighbour correlations are much larger than the nearest neighbour correlations. We call this phase the NNNAFM (nextnearest neighbour antiferromagnet) phase and provide an understanding of this phase by deriving an effective Hamiltonian between the spin1/2s. We also show the existence of macroscopic magnetisation jumps in the model when one looks at the system close to saturation fields. The third chapter is concerned with the formation of magnetisation plateaus in two different spin models. We show how in one model the plateaus arise because of the competition between two coupling constants, and in the other because of purely geometrical effects. In the first problem we propose [2] a class of spin Hamiltonians which include as special cases several known systems. The class of models is defined on a bipartite lattice in arbitrary dimensions and for any spin. The simplest manifestation of such models in one dimension corresponds to a ladder system with diagonal couplings (which are of the same strength as the leg couplings). The physical properties of the model are determined by the combined effects of the competition between the ”rung” coupling (J’ )and the ”leg/diagonal” coupling (J ) and the magnetic field. We show that our model can be solved exactly in a substantial region of the parameter space (J’ > 2J ) and we demonstrate the existence of magnetisation plateaus in the solvable regime. Also, by making reasonable assumptions about the spectrum in the region where we cannot solve the model exactly, we prove the existence of first order phase transitions on a plateau where the sublattice magnetisations change abruptly. We numerically investigate the ladder system mentioned above (for spin1) to confirm all our analytical predictions and present a phase diagram in the J’/J - B plane, quite a few of whose features we expect to be generically valid for all higher spins. In the second problem concerning plateaus (also discussed in chapter 3) we study the properties of a compound synthesised experimentally [3]. The essential feature of the structure of this compound which gives rise to its physical properties is the presence of two kinds of spin1/2 objects alternating with each other on a helix. One kind has an axis of anisotropy at an inclination to the helical axis (which essentially makes it an Ising spin) whereas the other is an isotropic spin1/2 object. These two spin1/2 objects interact with each other but not with their own kind. Experimentally, it was observed that in a magnetic field this material exhibits magnetisation plateaus one of which is at 1/3rd of the saturation magnetisation value. These plateaus appear when the field is along the direction of the helical axis but disappear when the field is perpendicular to that axis. The model being used for the material prior to our work could not explain the existence of these plateaus. In our work we propose a simple modification in the model Hamiltonian which is able to qualitatively explain the presence of the plateaus. We show that the existence of the plateaus can be explained using a periodic variation of the angles of inclination of the easy axes of the anisotropic spins. The experimental temperature and the fields are much lower than the magnetic coupling strength. Because of this quite a lot of the properties of the system can be studied analytically using transfer matrix methods for an effective theory involving only the anisotropic spins. Apart from the plateaus we study using this modified model other physical quantities like the specific heat, susceptibility and the entropy. We demonstrate the existence of finite entropy per spin at low temperatures for some values of the magnetic field. In chapter 4 we investigate the longstanding problem of locating the gapless points of a dimerised spin chain as the strength of dimerisation is varied. It is known that generalising Haldane’s field theoretic analysis to dimerised spin chains correctly predicts the number of the gapless points but not the exact locations (which have determined numerically for a few low values of spins). We investigate the problem of locating those points using a dimerised spin chain Hamiltonian with a ”twisted” boundary condition [4]. For a periodic chain, this ”twist” consists simply of a local rotation about the zaxis which renders the xx and yy terms on one bond negative. Such a boundary condition has been used earlier for numerical work whereby one can find the gapless points by studying the crossing points of ground states of finite chains (with the above twist) in different parity sectors (parity sectors are defined by the reflection symmetry about the twisted bond). We study the twisted Hamiltonian using two analytical methods. The modified boundary condition reduces the degeneracy of classical ground states of the chain and we get only two N´eel states as classical ground states. We use this property to identify the gapless points as points where the tunneling amplitude between these two ground states goes to zero. While one of our calculations just reproduces the results of previous field theoretic treatments, our second analytical treatment gives a direct expression for the gapless points as roots of a polynomial equation in the dimerisation parameter. This approach is found to be more accurate. We compare the two methods with the numerical method mentioned above and present results for various spin values. In the final chapter we present a study of the physics of a junction of Luttinger wires (quantum wires) with both scalar and spin scattering at the junction ([5],[6]). Earlier studies have investigated special cases of this system. The systems studied were two wire junctions with either a fully transmitting scattering matrix or one corresponding to disconnected wires. We extend the study to a junction of N wires with an arbitrary scattering matrix and a spin impurity at the junction. We study the RG flows of the Kondo coupling of the impurity spin to the electrons treating the electronic interactions and the Kondo coupling perturbatively. We analyse the various fixed points for the specific case of three wires. We find a general tendency to flow towards strong coupling when all the matrix elements of the Kondo coupling are positive at small length scales. We analyse one of the strong coupling fixed points, namely that of the maximally transmitting scattering matrix, using a 1/J perturbation theory and we find at large length scales a fixed point of disconnected wires with a vanishing Kondo coupling. In this way we obtain a picture of the RG at both short and long length scales. Also, we analyse all the fixed points using lattice models to gain an understanding of the RG flows in terms of specific couplings on the lattice. Finally, we use to bosonisation to study one particular case of scattering (the disconnected wires) in the presence of strong interactions and find that sufficiently strong interactions can stabilise a multichannel fixed point which is unstable in the weak interaction limit.

This work presents macroscopic and microscopic experiments of the disordered hexagonal double perovskite BaTi1/2Mn1/2O3, in order to characterize its electronic, magnetic and structural properties to support the possibility that this system hosts a spin liquid phase. Such assumption is based on the absence of a transition to a magnetically ordered phase in the magnetic and thermodynamic measurements, which points to a strong magnetic frustration in this material. In addition, it is observed the formation of a correlated spin state. To characterize this correlation, we resorted to Muon Spin Resonance (µSR) experiments to measure the low temperature spin dynamics. The zero field µSR relaxation regime displays dynamic magnetism down to T = 0.019 K and longitudinal field experiments support as well that dynamic magnetism persists at low temperatures, a behavior expected for a spin liquid system. The magnetic behavior of BaTi1/2Mn1/2O3 consists in the high temperature physics being dominated by the presence of magnetic trimers, magnetic dimers, and orphan spins. At lower temperatures, the effective magnetic degrees of freedom, composed by orphan spins and magnetic trimers, are correlated but no phase transition is detected down to T = 0.1 K, despite the effective exchange couplings between magnetic trimers and orphan spins being -8.5 K, resulting in a magnetic frustration parameter of at least 85. The possibility that disorder is responsible for the spin liquid ground state is discussed, however, other scenarios are not totally discarded. For example, the possibility that the measured state is not the true ground state, which could lie at even lowers temperatures or the possible formation of a spin glass state. This work raises questions that are not easy to answer. Ultimately, the growth of a single crystal is necessary to continue the characterization of BaTi1/2Mn1/2O3. Besides, theoretical and experimental developments in this field of research are needed to find a more direct and conclusive way to characterize the magnetic phases in this complex material. / Neste trabalho apresento dados experimentais macroscópicos e microscópicos da peroviskita hexagonal dupla BaTi1/2Mn1/2O3, a fim de caracterizar sua estrutura eletrônica, magnética e cristalina para embasar a possibilidade deste sistema apresentar uma fase de líquido de spin. Esta hipótese está baseada na ausência de transição para uma fase magneticamente ordenada nas medidas magnéticas e termodinâmicas, que apontam para uma forte frustração magnética neste material. Além disso, é observada a formação de um estado de spins correlacionados. Para caracterizar esta correlação, recorremos para experimentos de ressonância de múons (µSR) para medir a dinâmica de spins em baixas temperaturas. Dados de µSR para campo magnético nulo mostram em seu regime de relaxamento um magnetismo dinâmico para temperaturas tão baixas quanto T = 0.019 K, adicionalmente, experimentos com campo magnético longitudinal aplicado apontam também que o magnetismo dinâmico persiste em baixas temperaturas, um comportamento esperado para um sistema de líquido de spin. O comportamento magnético do BaTi1/2Mn1/2O3 consiste na física de altas temperaturas sendo dominada pela presença de trimers magnéticos, dimers magnéticos e spins órfãos. Para temperaturas mais baixas, os graus de liberdade magnéticos são efetivamente compostos por spins órfãos e trimers magnéticos, que estão correlacionados mas nenhuma transição de fase é detectada para temperaturas tão baixas quanto T = 0.1 K, mesmo que a constante de interação efetiva entre os spins órfãos e os trimers magnéticos seja -8.5 K, resultando num fator de frustração magnética de ao menos 85. A possibilidade da desordem ser responsável pelo estado fundamental de líquido de spin é discutida, no entanto, outros cenários não estão totalmente descartados, por exemplo, a possibilidade de que o estado medido não seja o verdadeiro estado fundamental, e que este estaria em temperaturas ainda mais baixas ou a possível formação de um estado de vidro de spin. Este trabalho levanta questões que não são fáceis de responder. Por fim, o crescimento de uma amostra monocristalina é necessário para continuar a caracterização do BaTi1/2Mn1/2O3. Ademais, desenvolvimentos de cunho teórico e experimental neste campo de pesquisa são necessários para encontrar um método mais direto e conclusivo para caracterizar a fase magnética neste material complexo.

This work presents macroscopic and microscopic experiments of the disordered hexagonal double perovskite BaTi1/2Mn1/2O3, in order to characterize its electronic, magnetic and structural properties to support the possibility that this system hosts a spin liquid phase. Such assumption is based on the absence of a transition to a magnetically ordered phase in the magnetic and thermodynamic measurements, which points to a strong magnetic frustration in this material. In addition, it is observed the formation of a correlated spin state. To characterize this correlation, we resorted to Muon Spin Resonance (µSR) experiments to measure the low temperature spin dynamics. The zero field µSR relaxation regime displays dynamic magnetism down to T = 0.019 K and longitudinal field experiments support as well that dynamic magnetism persists at low temperatures, a behavior expected for a spin liquid system. The magnetic behavior of BaTi1/2Mn1/2O3 consists in the high temperature physics being dominated by the presence of magnetic trimers, magnetic dimers, and orphan spins. At lower temperatures, the effective magnetic degrees of freedom, composed by orphan spins and magnetic trimers, are correlated but no phase transition is detected down to T = 0.1 K, despite the effective exchange couplings between magnetic trimers and orphan spins being -8.5 K, resulting in a magnetic frustration parameter of at least 85. The possibility that disorder is responsible for the spin liquid ground state is discussed, however, other scenarios are not totally discarded. For example, the possibility that the measured state is not the true ground state, which could lie at even lowers temperatures or the possible formation of a spin glass state. This work raises questions that are not easy to answer. Ultimately, the growth of a single crystal is necessary to continue the characterization of BaTi1/2Mn1/2O3. Besides, theoretical and experimental developments in this field of research are needed to find a more direct and conclusive way to characterize the magnetic phases in this complex material. / Neste trabalho apresento dados experimentais macroscópicos e microscópicos da peroviskita hexagonal dupla BaTi1/2Mn1/2O3, a fim de caracterizar sua estrutura eletrônica, magnética e cristalina para embasar a possibilidade deste sistema apresentar uma fase de líquido de spin. Esta hipótese está baseada na ausência de transição para uma fase magneticamente ordenada nas medidas magnéticas e termodinâmicas, que apontam para uma forte frustração magnética neste material. Além disso, é observada a formação de um estado de spins correlacionados. Para caracterizar esta correlação, recorremos para experimentos de ressonância de múons (µSR) para medir a dinâmica de spins em baixas temperaturas. Dados de µSR para campo magnético nulo mostram em seu regime de relaxamento um magnetismo dinâmico para temperaturas tão baixas quanto T = 0.019 K, adicionalmente, experimentos com campo magnético longitudinal aplicado apontam também que o magnetismo dinâmico persiste em baixas temperaturas, um comportamento esperado para um sistema de líquido de spin. O comportamento magnético do BaTi1/2Mn1/2O3 consiste na física de altas temperaturas sendo dominada pela presença de trimers magnéticos, dimers magnéticos e spins órfãos. Para temperaturas mais baixas, os graus de liberdade magnéticos são efetivamente compostos por spins órfãos e trimers magnéticos, que estão correlacionados mas nenhuma transição de fase é detectada para temperaturas tão baixas quanto T = 0.1 K, mesmo que a constante de interação efetiva entre os spins órfãos e os trimers magnéticos seja -8.5 K, resultando num fator de frustração magnética de ao menos 85. A possibilidade da desordem ser responsável pelo estado fundamental de líquido de spin é discutida, no entanto, outros cenários não estão totalmente descartados, por exemplo, a possibilidade de que o estado medido não seja o verdadeiro estado fundamental, e que este estaria em temperaturas ainda mais baixas ou a possível formação de um estado de vidro de spin. Este trabalho levanta questões que não são fáceis de responder. Por fim, o crescimento de uma amostra monocristalina é necessário para continuar a caracterização do BaTi1/2Mn1/2O3. Ademais, desenvolvimentos de cunho teórico e experimental neste campo de pesquisa são necessários para encontrar um método mais direto e conclusivo para caracterizar a fase magnética neste material complexo.

Nesta dissertação estudamos uma probabilidade obtida a partir de conceitos da Mecânica Estatística Quântica do ponto de vista da Teoria Ergódica. A probabilidade é obtida a partir de um estado KMS sobre um lattice unidimensional de spins quânticos. Mostramos que esta probabilidade é mixing para o shift. Além disso, mostramos que vale um princípio dos grandes desvios para uma certa classe de funções e exploramos algumas propriedades do Jacobiano. Iremos considerar o estado KMS associado a um certo Hamiltoniano específico agindo sobre o lattice de spins quânticos. Nas seções iniciais vamos apresentar alguns conceitos e prerequisitos básicos (como operadores densidade, produto tensorial, C*-algebras e estados KMS) para o entendimento do resultado principal / In this dissertation we study a probability derived from Quantum Statistical Mechanics through the viewpoint of Ergodic Theory. The probability is obtained from a KMS state acting on a one dimensional lattice of quantum spins. We show that this probability is mixing for the shift map. Moreover, we show that a large deviation principle is true for a certain class of functions and we explore some properties of the Jacobian. We will consider the KMS state associated to a certain specific Hamiltonian acting on the quantum spin lattice. In the initial sections we will present some concepts and prerequisites (such as density operators, tensor product, C*-algebras and KMS states) for the understanding of our main results.

This thesis, divided into two parts, is concerned with the analysis of spectral and dynamical characteristics of certain quantum spin systems in the presence of either I) quenched disorder, or II) dynamical noise. In the first part, the quantum random energy model (QREM), a mean-field spin glass model with a many-body localisation transition, is studied. In Chapter 2, we attempt a diagrammatic perturbative analysis of the QREM from the ergodic side, proceeding by analogy to the single-particle theory of weak localisation. Whilst we are able to describe diffusion, the analogy breaks down and a description of the onset of localisation in terms of quantum corrections quickly becomes intractable. Some progress is possible by deriving a quantum kinetic equation, namely the relaxation of the one-spin reduced density matrix is determined, but this affords little insight and extension to two-spin quantities is difficult. We change our approach in Chapter 3, studying instead a stroboscopic version of the model using the formalism of quantum graphs. Here, an analytic evaluation of the form factor in the diagonal approximation is possible, which we find to be consistent with the universal random matrix theory (RMT) result in the ergodic regime. In Chapter 4, we replace the QREM's transverse field with a random kinetic term and present a diagrammatic calculation of the average density of states, exact in the large-N limit, and interpret the result in terms of the addition of freely independent random variables. In the second part, we turn our attention to noisy quantum spins. Chapter 5 is concerned with noninteracting spins coupled to a common stochastic field; correlations arising from the common noise relax only due to the spins' differing precession frequencies. Our key result is a mapping of the equation of motion of n-spin correlators onto the (integrable) non-Hermitian Richardson-Gaudin model, enabling exact calculation of the relaxation rate of correlations. The second problem, addressed in Chapter 6, is that of the dynamics of operator moments in a noisy Heisenberg model; qualitatively different behaviour is found depending on whether or not the noise conserves a component of spin. In the case of nonconserving noise, we report that the evolution of the second moment maps onto the Fredrickson-Andersen model - a kinetically constrained model originally introduced to describe the glass transition. This facilitates a rigorous study of operator spreading in a continuous-time model, providing a complementary viewpoint to recent investigations of random unitary circuits.

No description available.

In this thesis we present results from quantum Monte Carlo for the fully-frustrated honeycomb lattice. The XXZ model is of interest in the classical limit, as there is a mapping between the classical fully-frustrated honeycomb Ising model groundstates and the classical hard-core dimer model groundstate. The aim of this work is to explore the effect of quantum fluctuations on the fully-frustrated honeycomb model to see what sort of interesting physics arises. One might expect unusual physics due to the quantum hard-core dimer model, where interesting physics are known to exist. This is because there is a duality mapping between the classical dimer model and the classical fully-frustrated honeycomb Ising model. Indeed, by studying the fully-frustrated honeycomb XXZ model we find that in some cases the system orders into crystal-like structures, a case of order-by-disorder. The most interesting case, when the frustrating bonds are chosen randomly, reveals to us a novel state without any discernible order while at the same time avoiding the freezing one would expect of a glass. This state is a featureless system lacking low temperature magnetic susceptibility---a candidate ``quantum spin liquid''. Future work that might more easily measure quantum spin liquid criteria is suggested.

No description available.

In this thesis we present results from quantum Monte Carlo for the fully-frustrated honeycomb lattice. The XXZ model is of interest in the classical limit, as there is a mapping between the classical fully-frustrated honeycomb Ising model groundstates and the classical hard-core dimer model groundstate. The aim of this work is to explore the effect of quantum fluctuations on the fully-frustrated honeycomb model to see what sort of interesting physics arises. One might expect unusual physics due to the quantum hard-core dimer model, where interesting physics are known to exist. This is because there is a duality mapping between the classical dimer model and the classical fully-frustrated honeycomb Ising model. Indeed, by studying the fully-frustrated honeycomb XXZ model we find that in some cases the system orders into crystal-like structures, a case of order-by-disorder. The most interesting case, when the frustrating bonds are chosen randomly, reveals to us a novel state without any discernible order while at the same time avoiding the freezing one would expect of a glass. This state is a featureless system lacking low temperature magnetic susceptibility---a candidate ``quantum spin liquid''. Future work that might more easily measure quantum spin liquid criteria is suggested.

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.