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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.

By numerical methods, two models of anisotropic mixed quantum spin chains, consisting of spins of two different sizes, Sa = 1/2 and Sb = 1 as well as Sb = 3/2, are studied with respect to their critical properties at quantum phase transitions in a selected region of parameter space. The quantum spin chains are made up of basecells of four spins, according to the structure Sa − Sa − Sb − Sb. They are described by the XXZ Hamiltonian, that extends the quantum Heisenberg model by a variable anisotropic exchange interaction. As additional control parameter, an alternating exchange constant between nearest-neighbour spins is introduced. Insight gained by complementary application of exact diagonalization and quantum Monte Carlo simulations, as well as appropriate methods of analysis, is embedded in the broad existing knowledge on homogeneous quantum spin chains. In anisotropic homogeneous quantum spin chains, there exist phase boundaries with continuously varying critical exponents, the Gaussian critical lines, along which, in addition to standard scaling relations, further extended scaling relations hold. Reweighting methods, also applied to improved quantum Monte Carlo estimators, and finite-size scaling analysis of simulation data deliver a wealth of numerical results confirming the existence of a Gaussian critical line also in the mixed spin models considered. Extrapolation of exact data offers, apart from confirmation of simulation data, furthermore, insight into the conformal operator content of the model with Sb = 1. / Mittels numerischer Methoden werden zwei Modelle anisotroper gemischter Quantenspinketten, bestehend aus Spins zweier unterschiedlicher Größen, Sa = 1/2 und Sb = 1 sowie Sb = 3/2, hinsichtlich ihrer kritischen Eigenschaften an Quanten-Phasenübergängen in einem ausgewählten Parameterbereich untersucht. Die Quantenspinketten sind aus Basiszellen zu vier Spins, gemäß der Struktur Sa − Sa − Sb − Sb, aufgebaut. Sie werden durch den XXZ Hamiltonoperator beschrieben, der das isotrope Quanten-Heisenberg Modell um eine variable anistrope Austauschwechselwirkung erweitert. Als zusätzlicher Kontrollparameter wird eine alterniernde Kopplungskonstante zwischen unmittelbar benachbarten Spins eingeführt. Die durch komplementäre Anwendung exakter Diagonalisierung und Quanten-Monte-Carlo Simulationen, sowie entsprechender Analyseverfahren, gewonnenen Erkenntnisse werden in das umfangreiche existierende Wissen über homogene Quantenspinketten eingebettet. Im Speziellen treten in anisotropen homogenen Quantenspinketten Phasengrenzen mit kontinuierlich variierenden kritischen Exponenten auf, die Gaußschen kritischen Linien, auf denen neben den herkömmlichen auch erweiterte Skalenrelationen Gültigkeit besitzen. Umgewichtungsmethoden, speziell auch angewandt auf verbesserte Quanten-Monte-Carlo Schätzer, und Endlichkeitsskalenanalyse von Simulationsdaten liefern eine Fülle von numerischen Ergebnissen, die das Auftreten der Gaußschen kritischen Linie auch in den untersuchten gemischten Quantenspinketten bestätigen. Die Extrapolation exakter Daten bietet, neben der Bestätigung der Simulationsdaten, darüber hinaus Einblick in einen Teil des konformen Operatorinhalts des Modells mit Sb = 1.

By numerical methods, two models of anisotropic mixed quantum spin chains, consisting of spins of two different sizes, Sa = 1/2 and Sb = 1 as well as Sb = 3/2, are studied with respect to their critical properties at quantum phase transitions in a selected region of parameter space. The quantum spin chains are made up of basecells of four spins, according to the structure Sa − Sa − Sb − Sb. They are described by the XXZ Hamiltonian, that extends the quantum Heisenberg model by a variable anisotropic exchange interaction. As additional control parameter, an alternating exchange constant between nearest-neighbour spins is introduced. Insight gained by complementary application of exact diagonalization and quantum Monte Carlo simulations, as well as appropriate methods of analysis, is embedded in the broad existing knowledge on homogeneous quantum spin chains. In anisotropic homogeneous quantum spin chains, there exist phase boundaries with continuously varying critical exponents, the Gaussian critical lines, along which, in addition to standard scaling relations, further extended scaling relations hold. Reweighting methods, also applied to improved quantum Monte Carlo estimators, and finite-size scaling analysis of simulation data deliver a wealth of numerical results confirming the existence of a Gaussian critical line also in the mixed spin models considered. Extrapolation of exact data offers, apart from confirmation of simulation data, furthermore, insight into the conformal operator content of the model with Sb = 1. / Mittels numerischer Methoden werden zwei Modelle anisotroper gemischter Quantenspinketten, bestehend aus Spins zweier unterschiedlicher Größen, Sa = 1/2 und Sb = 1 sowie Sb = 3/2, hinsichtlich ihrer kritischen Eigenschaften an Quanten-Phasenübergängen in einem ausgewählten Parameterbereich untersucht. Die Quantenspinketten sind aus Basiszellen zu vier Spins, gemäß der Struktur Sa − Sa − Sb − Sb, aufgebaut. Sie werden durch den XXZ Hamiltonoperator beschrieben, der das isotrope Quanten-Heisenberg Modell um eine variable anistrope Austauschwechselwirkung erweitert. Als zusätzlicher Kontrollparameter wird eine alterniernde Kopplungskonstante zwischen unmittelbar benachbarten Spins eingeführt. Die durch komplementäre Anwendung exakter Diagonalisierung und Quanten-Monte-Carlo Simulationen, sowie entsprechender Analyseverfahren, gewonnenen Erkenntnisse werden in das umfangreiche existierende Wissen über homogene Quantenspinketten eingebettet. Im Speziellen treten in anisotropen homogenen Quantenspinketten Phasengrenzen mit kontinuierlich variierenden kritischen Exponenten auf, die Gaußschen kritischen Linien, auf denen neben den herkömmlichen auch erweiterte Skalenrelationen Gültigkeit besitzen. Umgewichtungsmethoden, speziell auch angewandt auf verbesserte Quanten-Monte-Carlo Schätzer, und Endlichkeitsskalenanalyse von Simulationsdaten liefern eine Fülle von numerischen Ergebnissen, die das Auftreten der Gaußschen kritischen Linie auch in den untersuchten gemischten Quantenspinketten bestätigen. Die Extrapolation exakter Daten bietet, neben der Bestätigung der Simulationsdaten, darüber hinaus Einblick in einen Teil des konformen Operatorinhalts des Modells mit Sb = 1.

Ce mémoire étudie le modèle quantique d’Ising-Kawasaki en une dimension. Cette chaîne quantique de spin-1/2 décrit la dynamique de Kawasaki hors d’équilibre d’une chaîne d’Ising classique couplée à un bain thermique. L’Hamiltonien est obtenu pour le cas général désor- donné avec des couplages d’Ising et champs magnétiques non-uniformes. Quand les champs magnétiques sont nuls, la chaîne de spin quantique est stochastique, et dépend des couplages d’Ising normalisés par la température du bain de chaleur. Dans le cas de couplages uniformes, nous donnons les états fondamentaux exacts de la chaîne de spin, ainsi que ses excitations à 1-magnon. Les solutions pour les spectres à deux magnons sont dérivées via une variante de l’Ansatz de Bethe. Dans le régime antiferromagnétique, les états de branche à deux magnons présentent un comportement complexe, notamment en ce qui concerne l’hybridation avec le continuum. L’analyse faite dans ce mémoire, combinée aux études précédentes, suggère que le système manifeste des dynamiques multiples à basse énergie, comme le montre la présence de plusieurs exposants critiques dynamiques. La distribution de l’espacement de l’ensemble des niveaux d’énergie est évaluée en fonction du couplage d’Ising. On conclut que le sys- tème est non-intégrable pour des paramètres génériques, ou de manière équivalente, que la dynamique classique hors équilibre correspondante est ergodique. / We study a quantum spin-1/2 chain that is dual to the non-equilibrium Kawasaki dynamics of a classical Ising chain coupled to a thermal bath. The Hamiltonian is obtained for the general disordered case with non-uniform Ising couplings. The quantum spin chain is stoquastic, and depends on the Ising couplings normalized by the bath’s temperature. Proceeding with uniform couplings, we give the exact groundstates of the gapless spin chain, as well as its single-magnon excitations. Solutions for the two-magnon spectra are derived via a Bethe Ansatz scheme. In the antiferromagnetic regime, the two-magnon branch states show intricate behavior, especially regarding hybridization with the continuum. Our analysis, when combined with previous studies, suggests that the system hosts multiple dynamics at low energy as seen via the presence of multiple dynamical critical exponents. Finally, we analyze the full energy level spacing distribution as a function of the Ising coupling. We conclude that the system is non-integrable for generic parameters, or equivalently, that the corresponding non-equilibrium classical dynamics are ergodic.

Mon travail de thèse est consacré à l’étude de quelques aspects de la physique quantique des systèmes quantiques à N corps en interaction. Il est orienté vers l’étude des chaînes de spins quantiques. Je me suis intéressé à plusieurs questions relatives aux chaînes de spins quantiques, du point de vue numérique et analytique à la fois. J'aborde en particulier les questions relatives à la structure des fonctions d'onde, la forme de la densité d'états et les propriétés spectrales des Hamiltoniens de chaînes de spins. Dans un premier temps, je présenterais très rapidement les techniques numériques de base pour le calcul des vecteurs et valeurs propres des Hamiltonien de chaînes de spins. Les densités d’états des modèles quantiques constituent des quantités importantes et très simples qui permettent de caractériser les propriétés spectrales des systèmes avec un grand nombre de degrés de liberté. Alors que dans la limite thermodynamique, les densités d'états de la plupart des modèles intégrables sont bien décrites par une loi gaussienne, dans certaines limites de couplage de la chaîne de spins au champ magnétique et pour un nombre de spins N fini sur la chaîne, on observe l’apparition de pics dans la densité d’états. Je montrerais que la connaissance des deux premiers moments du Hamiltonien dans le sous-espace dégénéré associé à chaque pics donne une bonne approximation de la densité d’états. Dans un deuxième temps je m'intéresserais aux propriétés spectrales des Hamiltoniens de chaînes de spins quantiques. L’un des principal résultats sur la statistique spectrale des systèmes quantiques concerne le comportement universel des fluctuations des mesures telles que l’espacement entre valeurs propres consécutives. Ces fluctuations sont bien décrites par la théorie des matrices aléatoires mais la comparaison avec les prédictions de cette théorie nécessite généralement une opération sur le spectre du Hamiltonien appelée unfolding. Dans les problèmes quantiques de N corps, la taille de l’espace de Hilbert croît généralement exponentiellement avec le nombre de particules, entraînant un manque de données pour pouvoir faire une statistique. Ces limitations ont amené l’introduction d’une nouvelle mesure se passant de la procédure d’unfolding basée sur le rapport d’espacements successifs plutôt que les espacements. En suivant l’idée du “surmise” de Wigner pour le calcul de la distribution de l’espacement, je montre comment calculer une approximation de la distribution du rapport d’espacements dans les trois ensembles gaussiens invariants en faisant le calcul pour des matrices 3x3. Les résultats obtenus pour les différents ensembles de matrices aléatoires se sont révélés être en excellent accord avec les résultats numériques. Enfin je m’intéresserais à la structure des fonctions d’ondes fondamentales des modèles de chaînes de spins quantiques. Les fonctions d’onde constituent, avec le spectre en énergie, les objets fondamentaux des systèmes quantiques : leur structure est assez compliquée et n’est pas très bien comprise pour la plupart des systèmes à N corps. En raison de la croissance exponentielle de la taille de l’espace de Hilbert avec le nombre de particules, l’étude des vecteurs propres est une tâche très difficile, non seulement du point de vue analytique mais aussi du point de vue numérique. Je démontrerais en particulier que l’état fondamental de tous les modèles que nous avons étudiés est multifractal avec en général une dimension fractale non triviale. / My thesis is devoted to the study of some aspects of many body quantum interacting systems. In particular we focus on quantum spin chains. I have studied several aspects of quantum spin chains, from both numerical and analytical perspectives. I addressed especially questions related to the structure of eigenfunctions, the level densities and the spectral properties of spin chain Hamiltonians. In this thesis, I first present the basic numerical techniques used for the computation of eigenvalues and eigenvectors of spin chain Hamiltonians. Level densities of quantum models are important and simple quantities that allow to characterize spectral properties of systems with large number of degrees of freedom. It is well known that the level densities of most integrable models tend to the Gaussian in the thermodynamic limit. However, it appears that in certain limits of coupling of the spin chain to the magnetic field and for finite number of spins on the chain, one observes peaks in the level density. I will show that the knowledge of the first two moments of the Hamiltonian in the degenerate subspace associated with each peak give a good approximation to the level density. Next, I study the statistical properties of the eigenvalues of spin chain Hamiltonians. One of the main achievements in the study of the spectral statistics of quantum complex systems concerns the universal behaviour of the fluctuation of measure such as the distribution of spacing between two consecutive eigenvalues. These fluctuations are very well described by the theory of random matrices but the comparison with the theoretical prediction generally requires a transformation of the spectrum of the Hamiltonian called the unfolding procedure. For many-body quantum systems, the size of the Hilbert space generally grows exponentially with the number of particles leading to a lack of data to make a proper statistical study. These constraints have led to the introduction of a new measure free of the unfolding procedure and based on the ratio of consecutive level spacings rather than the spacings themselves. This measure is independant of the local level density. By following the Wigner surmise for the computation of the level spacing distribution, I obtained approximation for the distribution of the ratio of consecutive level spacings by analyzing random 3x3 matrices for the three canonical ensembles. The prediction are compared with numerical results showing excellent agreement. Finally, I investigate eigenfunction statistics of some canonical spin-chain Hamiltonians. Eigenfunctions together with the energy spectrum are the fundamental objects of quantum systems: their structure is quite complicated and not well understood. Due to the exponential growth of the size of the Hilbert space, the study of eigenfunctions is a very difficult task from both analytical and numerical points of view. I demonstrate that the groundstate eigenfunctions of all canonical models of spin chain are multifractal, by computing numerically the Rényi entropy and extrapolating it to obtain the multifractal dimensions.