This thesis focuses on the study of the thermodynamics of fermionic systems on lattices, from the point of view of the C*-algebraic formulation of quantum mechanics, and on the application of the so-called \"catastrophe theory\" to the analysis of the phase diagram of an explicit fermionic model. The first chapter is devoted to the introduction of some of the basic and most important properties of C*-algebras, and also to the study of its representations and its set of states. The second chapter is devoted to the development of some important results of catastrophe theory. The results derived in the chapter allows one to analyze the behavior of the minima of members of a family of functions around a degenerate critical point, and they will be used to study the behavior of the thermodynamic pressure for a given fermionic lattice model. The third chapter presents some of the formalism developed in [4], that concerns the existence of the thermodynamics and equilibrium states of lattice fermi systems subject to a suitable set of long-range interactions (i.e., interactions containing mean-field terms). Finally, in the last chapter, using all the formalisms and results already studied in the previous chapters, the thermodynamics of an explicit BCS-like model is analyzed. With the help of catastrophe theory, it is shown that such model exhibits a coexistence of magnetic and superconducting phases for a suitable choice of parameters, and through a perturbative analysis it is also shown that the coexistence still holds when a small kinetic term is added into the model.
This thesis focuses on the study of the thermodynamics of fermionic systems on lattices, from the point of view of the C*-algebraic formulation of quantum mechanics, and on the application of the so-called "catastrophe theory" to the analysis of the phase diagram of an explicit fermionic model. The first chapter is devoted to the introduction of some of the basic and most important properties of C*-algebras, and also to the study of its representations and its set of states. The second chapter is devoted to the development of some important results of catastrophe theory. The results derived in the chapter allows one to analyze the behavior of the minima of members of a family of functions around a degenerate critical point, and they will be used to study the behavior of the thermodynamic pressure for a given fermionic lattice model. The third chapter presents some of the formalism developed in [4], that concerns the existence of the thermodynamics and equilibrium states of lattice fermi systems subject to a suitable set of long-range interactions (i.e., interactions containing mean-field terms). Finally, in the last chapter, using all the formalisms and results already studied in the previous chapters, the thermodynamics of an explicit BCS-like model is analyzed. With the help of catastrophe theory, it is shown that such model exhibits a coexistence of magnetic and superconducting phases for a suitable choice of parameters, and through a perturbative analysis it is also shown that the coexistence still holds when a small kinetic term is added into the model.
This thesis focuses on the study of the thermodynamics of fermionic systems on lattices, from the point of view of the C*-algebraic formulation of quantum mechanics, and on the application of the so-called "catastrophe theory" to the analysis of the phase diagram of an explicit fermionic model.The first chapter is devoted to the introduction of some of the basic and most important properties of C*-algebras, and also to the study of its representations and its set of states.The second chapter is devoted to the development of some important results of catastrophe theory.The results derived in the chapter allows one to analyze the behavior of the minima of members of a family of functions around a degenerate critical point, and they will be used to study the behavior of the thermodynamic pressure for a given fermionic lattice model.The third chapter presents some of the formalism developed in [4], that concerns the existence of the thermodynamics and equilibrium states of lattice fermi systems subject to a suitable set of long-range interactions (i.e., interactions containing mean-field terms).Finally, in the last chapter, using all the formalisms and results already studied in the previous chapters, the thermodynamics of an explicit BCS-like model is analyzed.With the help of catastrophe theory, it is shown that such model exhibits a coexistence of magnetic and superconducting phases for a suitable choice of parameters, and through a perturbative analysis it is also shown that the coexistence still holds when a small kinetic term is added into the model.
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.
In this thesis, several cases of non-equilibrium phenomena and quantum phase transitions in strongly correlated electron systems are analyzed. The unconventional critical behavior near magnetic quantum phase transitions in various heavy-fermion metals has triggered proposals on the breakdown of the Kondo effect at the critical point. In part I, we investigate, within one specific scenario, the fate of such a zero-temperature transition upon coupling of the electronic to lattice degrees of freedom. We study a Kondo-Heisenberg model with volume-dependent Kondo coupling � this model displays both Kondo volume collapse and Kondo-breakdown transitions. Within a large-N treatment, we find that the Kondo breakdown transition remains of second order except for very soft lattices. Finally, we relate our findings to current heavy-fermion experiments. Using non-equilibrium Green�s functions, we derive transport equations for the degrees of freedom participating in the quantum critical region of the Kondo breakdown transition. We discuss conditions under which the transport of electrical charge is described by the independent motion of conduction electrons and auxiliary bosons. Under these conditions, we derive a semiclassical transport equation for the bosons and quantitatively discuss the electrical conductivity of the whole system. Motivated by pressure experiments on FeAs-122 superconductors, in part II we propose a scenario based on local-moment physics to explain salient features at the magnetic phase boundary of CaFe2As2. In this scenario, the low-pressure magnetic phase derives from Fe moments, which become screened in the paramagnetic high-pressure phase. The quantum phase transition can be described as an orbital-selective Mott transition, which is rendered first order by coupling to the lattice. These ideas are illustrated by a suitable mean-field analysis of an Anderson lattice model. An analytical description of non-equilibrium phenomena in interacting quantum systems is rarely possible. In part III we present one example where such a description can be achieved, namely the ferromagnetic Kondo model. In equilibrium, this model is tractable via perturbative renormalization-group techniques. We employ a recently developed extension of the flow-equation method to calculate the non-equilibrium decay of the local magnetization at zero temperature. The flow equations admit analytical solutions which become exact at short and long times, in the latter case revealing that the system always retains a memory of its initial state. Finally, in part IV we analyze the Nernst effect resulting from normal state quasiparticles in the cuprates in presence of various types of translational symmetry breaking. In the electron-doped cuprates, the Nernst signal resulting from a reconstruction of the Fermi surface due to spin density wave order is discussed. An order parameter consistent with the reconstruction of the Fermi surface detected in electron-doped materials is shown to sharply enhance the Nernst signal close to optimal doping. Within a semiclassical treatment, the obtained magnitude and position of the enhanced Nernst signal agrees with Nernst measurements in electron-doped cuprates. In the hole-doped cuprates, we discuss relations between the normal-state Nernst effect and stripe order. We find that Fermi pockets caused by translational symmetry breaking lead to a strongly enhanced Nernst signal with a sign depending on the modulation period of the ordered state and other details of the Fermi surface. This implies differences between antiferromagnetic and charge-only stripes. We compare our findings with recent data from La1.6−xNd0.4SrxCuO4 and YBa2Cu3Oy.
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.
The present thesis is focused on the study of the models of Ising and Heisenberg (classical) for ferromagnetism, the theory of phase transitions and the Monte Carlo method as an efficient computational scheme for dealing with these kinds of strongly interacting systems. With the aid of this method, computational simulations were performed for the above mentioned models in 2 and 3 dimensions (2D and 3D) in order to compare these results with the predictions of some theoretical models. The method followed to model these systems is indicated as well as the process for constructing phase diagrams for the main thermodynamic quantities of interest (mean magnetization, mean energy, specific heat, magnetic susceptibility) and their behavior in a vicinity of the critical point is analyzed. Furthermore, a quantitative analysis is carried out to calculate the corresponding critical exponents (_; _; and _) and to prove the phenomenon of universality they fulfill for different types of crystalline structure (3D case) and for the 2 models previously cited (Ising and Heisenberg). The programs written down in C++ that were constructed with the aim of performing the respective simulations are also included. \nKeywords: models of Ising and Heisenberg, ferromagnetism, phase transitions, Monte Carlo method, critical point, critical exponents, universality.
The study of quantum phase transitions provides a new route to ¯nd and understand unconventional phases in Condensed Matter Physics. The past decades have seen a sub- stantial rejuvenation of interest in this area of study, driven by experiments on the cuprate superconductors, heavy fermions' materials, organic conductors and related compounds. Nevertheless, the transitions involving superconducting phases are not as well explored theoretically as transitions involving magnetic phases. Thus, the study of superconduct- ing systems with treatments beyond mean ¯eld theories seems to be necessary to verify the behavior of °uctuations in the normal phase near the transition to the superconductor state. We propose a generalization of the equation of motion method for equilibrium Green's functions to handle systems with time-dependent perturbations. This generalization is based on the properties of Green's functions and in the adiabatic application of the disturbance. We study a two band s-wave superconductor with hybridization, based on an attractive Hubbard model. We consider the system under the action of a fictitious field, time and wave vector dependent, which couples to the superconducting order parameter. We calculate the zero order propagators in the field, using the conventional equation of motion method, and the first order ones, using the generalized equation of motion. The response to the field is given by a generalized susceptibility, which in the one-band case reduces to the one obtained by Thouless. We associate the emergence of superconductivity with the divergence of the static part of the generalized susceptibility. The advantage of this method is its simplicity, because we only deal with single-particle propagators. We analyze this superconducting susceptibility, which allows us to find a critical hybridization above which there is no superconducting phase, determine the critical and the crossover lines and determine the dynamical critical exponent z for this system. Therefore we can predict the behavior of some quantities measured experimentally at low temperatures close to the superconducting quantum critical point.