This dissertation is devoted to theoretical studies of strongly interacting one-dimensional and quasi one-dimensional electron systems. The properties of one-dimensional electron systems can be studied within the bosonization technique, which presents fermions as collective bosonic density excitations. The power of this approach is the ability to treat electron-electron interaction exactly in the low-energy limit. The approach predicts the failure of Fermi liquid and an absence of long-range order in one-dimensions. The low-energy description of one-dimensional interacting systems is called the Tomonaga-Luttinger liquid theory. For example, the edges of quantum Hall systems are one-dimensional and described by a chiral Tomonaga-Luttinger liquid. Another example is a quantum spin Hall system with helical edge states, which are also described by a Tomonaga-Luttinger liquid. In our first work, a study of magnetized edge states of quantum spin-Hall system is presented. A magnetic field dependent signature of such edges is obtained, which can be verified in a Coulomb drag experiment. The second part of the dissertation is devoted to quasi-one dimensional antiferromagnetic lattices. A spatially anisotropic lattice antiferromagnet can be viewed as an array of one dimensional spin chains coupled in a way to match the lattice symmetry. This allows to use the non-Abelian bosonization technique to describe the low-energy physics of spin chains and study the inter-chain interactions perturbatively. The work presented in the dissertation studies the effect of Dzyaloshinskii-Moriya interaction on the magnetic phase diagram of the spatially anisotropic kagome antiferromagnet. We predict a Dzyaloshinskii-Moriya interaction driven phase transition from Neel to Neel+dimer state. In the third work, a novel model of the fractional quantum Hall effect is given. Wave functions of two-dimensional electrons in strong and quantizing magnetic field are essentially one-dimensional. That invites one to use the one-dimensional phenomenological bosonization to describe the density fluctuations of the two-dimensional interacting electrons in magnetic field. Remarkably, the constructed trial bosonized fermion operator describing the electron states with a fixed Landau gauge momentum is effectively two-dimensional. / text

Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.

We study the conjectured equivalence between the O(3) Gross-Neveu model and the supersymmetric sine-Gordon model under a naive application of the bosonization rules. We start with a review of the equivalence between sine-Gordon model and the massive Thirring model. We study the models by perturbation theory and then determine the equivalence. We find that the dependence of the identifications on the couplings can change according to the definition of the vector current. With the operator identifications of the special case corresponding to a free fermionic theory, known as the bosonization rules, we describe the equivalence between the massless Thirring model and the model of a compactified free boson field. For the massless Thirring model, or equivalently the O(2) Gross-Neveu model, we study the conservation laws for the vector current and the axial current by employing a generalized point-splitting method which allows a one-parameter family of definitions of the vector current. With this parameter, we can make contact with different approaches that can be found in the literature; these approaches differ mainly because of the specific definition of the current that was used. We also find the Sugawara form of the stress-energy tensor and its commutation relations. Further, we rewrite the identifications between sine-Gordon and Thirring models in our generalized framework. For the O(3) Gross-Neveu model, we extend our point-splitting method to determine the exact expression for the supercurrent. Using this current, we compute the superalgebra which determines three quantum components of the stress-energy tensor. With an Ansatz for the undetermined component, we find the trace anomaly and the first beta-function coefficient. The central charge which can be computed without using our point-splitting method is independent of the coupling constant, in fact, it is always zero. For the supersymmetric sine-Gordon model, we review its supersymmetry in the context of models derived from a scalar multiplet in two dimensions. We then obtain the central charge and discover an extra term that was missing in the original derivation. We also analyze how normal ordering modifies the central charge. Finally, we discuss the conjectured equivalence of the O(3) Gross-Neveu model and the supersymmetric sine-Gordon model under the naive application of the bosonization rules. Comparing our results of the central charges and the supercurrents for these models, we find that they disagree; consequently the models should be generically inequivalent. We also conclude that the naive application of the bosonization rules at the Lagrangian level does not always lead to an equivalent theory.

Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.

Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

This Ph.D. thesis in mathematical physics concerns systems of interacting fermions with strong correlations. For these systems the physical properties can only be described in terms of the collective behavior of the fermions. Moreover, they are often characterized by a close competition between fermion localization versus delocalization, which can result in complex and exotic physical phenomena. Strongly correlated fermion systems are usually modelled by many-body Hamiltonians for which the kinetic- and interaction energy have the same order of magnitude. This makes them challenging to study as the application of conventional computational methods, like mean field- or perturbation theory, often gives unreliable results. Of particular interest are Hubbard-type models, which provide minimal descriptions of strongly correlated fermions. The research of this thesis focuses on such models defined on two-dimensional square lattices. One motivation for this is the so-called high-Tc problem of the cuprate superconductors. A main hypothesis is that there exists an underlying Fermi surface with nearly flat parts, i.e. regions where the surface is straight. It is shown that a particular continuum limit of the lattice system leads to an effective model amenable to computations. This limit is partial in that it only involves fermion degrees of freedom near the flat parts. The result is an effective quantum field theory that is analyzed using constructive bosonization methods. Various exactly solvable models of interacting fermions in two spatial dimensions are also derived and studied. / QC 20111207

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Este trabalho visa a um estudo de bosonização de sistemas fermiônicos. As relações entre a bosonização e matéria condensada, notoriamente o caso do Grafeno, são apresentados e depois discutimos as possíveis implicações do mecanismo de i-particles na existência de correlatores de 4 correntes. Cálculos em D = 4 também são realizados para que se possa observar o diferente comportamento da ação bosonizada em diferentes dimensões. / This work aims to study the bosonization of fermionic systems. Relations between bosonization and condensed matter, notoriously the case of graphene, are presented and then discussed. The possible implications of the mechanism of it i-particles in 4 current correlators. Calculations in D = 4 are also performed in order to observe the different behavior of bosonized action in different dimensions.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Este trabalho visa a um estudo de bosonização de sistemas fermiônicos. As relações entre a bosonização e matéria condensada, notoriamente o caso do Grafeno, são apresentados e depois discutimos as possíveis implicações do mecanismo de i-particles na existência de correlatores de 4 correntes. Cálculos em D = 4 também são realizados para que se possa observar o diferente comportamento da ação bosonizada em diferentes dimensões. / This work aims to study the bosonization of fermionic systems. Relations between bosonization and condensed matter, notoriously the case of graphene, are presented and then discussed. The possible implications of the mechanism of it i-particles in 4 current correlators. Calculations in D = 4 are also performed in order to observe the different behavior of bosonized action in different dimensions.

We study three different kinds of mesoscopic systems – in the intermediate region between macroscopic and microscopic scales consisting of many interacting constituents: We consider particle entanglement in one-dimensional chains of interacting fermions. By employing a field theoretical bosonization calculation, we obtain the one-particle entanglement entropy in the ground state and its time evolution after an interaction quantum quench which causes relaxation towards non-equilibrium steady states. By pushing the boundaries of the numerical exact diagonalization and density matrix renormalization group computations, we are able to accurately scale to the thermodynamic limit where we make contact to the analytic field theory model. This allows to fix an interaction cutoff required in the continuum bosonization calculation to account for the short range interaction of the lattice model, such that the bosonization result provides accurate predictions for the one-body reduced density matrix in the Luttinger liquid phase. Establishing a better understanding of how to control entanglement in mesoscopic systems is also crucial for building qubits for a quantum computer. We further study a popular scalable qubit architecture that is based on Majorana zero modes in topological superconductors. The two major challenges with realizing Majorana qubits currently lie in trivial pseudo-Majorana states that mimic signatures of the topological bound states and in strong disorder in the proposed topological hybrid systems that destroys the topological phase. We study coherent transport through interferometers with a Majorana wire embedded into one arm. By combining analytical and numerical considerations, we explain the occurrence of an amplitude maximum as a function of the Zeeman field at the onset of the topological phase – a signature unique to MZMs – which has recently been measured experimentally [Whiticar et al., Nature Communications, 11(1):3212, 2020]. By placing an array of gates in proximity to the nanowire, we made a fruitful connection to the field of Machine Learning by using the CMA-ES algorithm to tune the gate voltages in order to maximize the amplitude of coherent transmission. We find that the algorithm is capable of learning disorder profiles and even to restore Majorana modes that were fully destroyed by strong disorder by optimizing a feasible number of gates. Deep neural networks are another popular machine learning approach which not only has many direct applications to physical systems but which also behaves similarly to physical mesoscopic systems. In order to comprehend the effects of the complex dynamics from the training, we employ Random Matrix Theory (RMT) as a zero-information hypothesis: before training, the weights are randomly initialized and therefore are perfectly described by RMT. After training, we attribute deviations from these predictions to learned information in the weight matrices. Conducting a careful numerical analysis, we verify that the spectra of weight matrices consists of a random bulk and a few important large singular values and corresponding vectors that carry almost all learned information. By further adding label noise to the training data, we find that more singular values in intermediate parts of the spectrum contribute by fitting the randomly labeled images. Based on these observations, we propose a noise filtering algorithm that both removes the singular values storing the noise and reverts the level repulsion of the large singular values due to the random bulk.

O presente trabalho consiste na expansão1/N de uma versão estendida do modelo de Nambu-Jona-Lasinio SU(3) no contexto de integrais funcionais. A equação de gap propagadores mesônicos, decaimentos e espalhamentos, aparecem naturalmente como ordens diferentes na expansão. A característica nova nesta abordagem é a inclusão de correções de ordem superior à ordem dominante, que nunca foram consideradas anteriormente. O método também permite a construção de uma densidade de Lagrangeana quiral mésons interagentes baseada no modelo Nambu-JonaLasinio SU(3) aqui obtida pela primeira vez. / The present work consists in 1/N expansion extended version of the SU(3) Nambu-Jona-Lasinio model in the context of the Functional Integral. The gap equations, meson propagators, triangle diagram, etc, appear quite naturally as different orders in the expansion. The new features of this approach in the inclusion of high order corrections in the I/N leading orders, which have never been included in the previous one. The method also allows for the construction of a chical Lagrangean of interacting mesons based on the SU(3) NJL model, here obtained for the first time.