Dissertations / Theses on the topic 'Quantum trajectories, Quantum mechanics, statistical mechanics'

The study of the dynamics of open quantum systems sheds light on dissipative processes in quantum mechanics. Any system under continuous measurement is open and the act of measuring induces abrupt changes of the system’s state (collapses). The evolution conditioned to measurement records generates the so-called quantum trajectories. A continuous (unconditioned) evolution of the system is recovered by averaging over a large number of trajectories. Historically this kind of evolution has been the main focus of theoretical investigations. In this dissertation we consider both conditional and unconditional dynamics of quantum optical systems. Unconditioned dynamics is studied through the collision model paradigm. The formalism is described in detail and used for describing generic systems featuring many quantum emitters coupled to a usually one-dimensional field. The negligible-delay regime is widely explored. Collision models are used to unveil the mechanisms underlying the decoherence-free evolution regime typical of these systems, which has received considerable attention in the last years. Then we investigate conditioned dynamics by broadening the study of statistics of quantum trajectories. Specifically, we exploit the information about the emission’s full-counting statistics from large deviations to define a nonclassicality witness. Finally we come back to collision models in order to extend the theory of biased quantum trajectories from Lindblad-like dynamics to sequences of arbitrary dynamical maps, providing at once a transparent physical interpretation.

All fields of physics - be it nuclear, atomic and molecular, solid state, or optical - offer examples of systems which are strongly influenced by the environment of the actual system under investigation. The scope of what is called "the environment" may vary, i.e., how far from the system of interest an interaction between the two does persist. Typically, however, it is much larger than the open system itself. Hence, a fully quantum mechanical treatment of the combined system without approximations and without limitations of the type of system is currently out of reach. With the single assumption of the environment to consist of an internally thermalized set of infinitely many harmonic oscillators, the seminal work of Stockburger and Grabert [Chem. Phys., 268:249-256, 2001] introduced an open system description that captures the environmental influence by means of a stochastic driving of the reduced system. The resulting stochastic Liouville-von Neumann equation describes the full non-Markovian dynamics without explicit memory but instead accounts for it implicitly through the correlations of the complex-valued noise forces. The present thesis provides a first application of the Stockburger-Grabert stochastic Liouville-von Neumann equation to the computation of the dynamics of anharmonic, continuous open systems. In particular, it is demonstrated that trajectory based propagators allow for the construction of a numerically stable propagation scheme. With this approach it becomes possible to achieve the tremendous increase of the noise sample count necessary to stochastically converge the results when investigating such systems with continuous variables. After a test against available analytic results for the dissipative harmonic oscillator, the approach is subsequently applied to the analysis of two different realistic, physical systems. As a first example, the dynamics of a dissipative molecular oscillator is investigated. Long time propagation - until thermalization is reached - is shown to be possible with the presented approach. The properties of the thermalized density are determined and they are ascertained to be independent of the system's initial state. Furthermore, the dependence on the bath's temperature and coupling strength is analyzed and it is demonstrated how a change of the bath parameters can be used to tune the system from the dissociative to the bound regime. A second investigation is conducted for a dissipative tunneling scenario in which a wave packet impinges on a barrier. The dependence of the transmission probability on the initial state's kinetic energy as well as the bath's temperature and coupling strength is computed. For both systems, a comparison with the high-temperature Markovian quantum Brownian limit is performed. The importance of a full non-Markovian treatment is demonstrated as deviations are shown to exist between the two descriptions both in the low temperature cases where they are expected and in some of the high temperature cases where their appearance might not be anticipated as easily.

Recently, the possibility of investigating single molecule, or single spin observables, as well as the necessity of a better understanding of the mechanisms underlying quantum dynamics in order to obtain nanoscale devices and nanostructered materials suitable for quantum computing tasks, have revived the interest in foundational aspects of quantum statistical mechanics. This thesis aims to give a contribution to this field by re-considering the statistical characterization of a quantum system at the light of some paradigmatic changes in our understanding of quantum theory which have taken place in the last two decades. In particular the impressive development of quantum information theory has changed the perceptions of quantum entanglement: for a long time it has been considered a somewhat paradoxical property of the matter at the atomic scale, but now it is regarded as an essential and ubiquitous phenomenon whose consequences are affecting the very macroscopic world that we experience. Still the decoherence program has brought out the importance of considering a quantum system together with its environment in order to clarify some key aspects of quantum dynamics. Thus, we start from the idea that quantum correlations are ubiquitous and somewhat uncontrollable in systems with many degrees of freedom which are typically considered in statistical mechanics. As a consequence we assume the standpoint that quantum statistical mechanics has not to be based on the underlying idea of a collection of many, independent quantum systems but rather it has to emerge at the level of a global wavefunction (pure state) which describes the system as well as its environment as a whole. In order to investigate the consequences of these assumptions we study the equilibrium distribution of an isolated quantum system. This is defined, in analogy with the ergodic foundations of classical statistical mechanics, on the basis of the time evolution of the quantum state. Then, we study the emergence of thermodynamic properties in a quantum system by studying the probability distribution of some function of interests, as the entropy and the equilibrium state of a subsystem, on Ensembles of Pure States. Such a probability distribution is derived from the geometry of the Hilbert space, and the theoretical tools suitable for its characterization are developed. On the one hand we perform a numerical sampling of the ensemble distributions by employing Monte Carlo techniques, on the other hand simpler analytical approximation of the geometrical distributions are derived by means of a maximum entropy principle. Model systems composed of an ensemble of spins are chosen to illustrate the salient features which emerge from the developed theoretical framework: the main point is that the Ensemble Distributions of “thermodynamic observables” (entropy or equilibrium state of a subsystem) are sharply peaked around a typical value. From the analysis it emerges that each of the overwhelming majority of the wavefunctions which has appreciable weight in the considered ensemble, is characterized by the same value of the “macroscopic” functions. This is a striking evidence of the “typicality” of these properties. In the essence, our impossibility to know the state of the system in detail does not matter, just for the remarkable fact that almost all quantum states behave essentially in the same way. By virtue of this typicality the study of the behaviour of the typical values of the thermodynamic function become meaningful. Notably, under certain conditions, one recovers the results of standard statistical mechanics, that is, the equilibrium average of the state of a subsystem can be cast in the Boltzmann canonical form at the temperature given by the usual thermodynamical relation . In the second part of the thesis we consider the dynamical aspects of the equilibrium state of a subsystem interacting with its environment. The fluctuations around the equilibrium average critically depends on the entanglement between the system and the environment and on the form of the interaction Hamiltonian. The connection between the dynamics of the fluctuations of an observable at the equilibrium and the relaxation toward the equilibrium from a “non typical” initial value is also investigated with the aid of simple model systems. The study presented in this thesis was partly motivated by a critical analysis of the statistical methods available for the theoretical modelling of magnetic resonance experiments. One of these, the Stochastic Liouville Equation, has been employed in a work completed during the first year of my Ph.D. program in order to interpret some feature of a two dimensional electron spin resonance experiment, [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].
Nuove tecnologie hanno reso possibile lo studio spettroscopico di proprietà di singola molecola e di singolo spin, inoltre, gli avanzamenti nel campo delle nanotecnologie, mettono costantemente alla prova la nostra comprensione dei meccanismi che governano la dinamica a livello quantistico. Questi recenti sviluppi stanno rinnovando l’interesse intorno a questioni fondamentali non pienamente comprese e risolte; una di queste questioni riguarda i fondamenti della meccanica statistica quantistica. Lo scopo della presente tesi è quello di dare un contributo in questo affascinante campo, alla luce degli importanti cambiamenti avvenuti negli ultimi vent’ anni nella nostra comprensione della meccanica quantistica. In particolare gli studi condotti nell’ambito della teoria dell’informazione hanno profondamente modificato la nostra percezione dell’ entanglement quantistico. Questo è stato per lungo tempo considerato una proprietà quasi paradossale della materia su scala atomica mentre oggi è ritenuto un fenomeno essenziale e onnipresente importante per comprendere l’emergere del mondo macroscopico così come lo conosciamo. Inoltre, la formulazione e lo sviluppo del cosiddetto “decoherence program” ha introdotto un nuovo paradigma nella descrizione dell’evoluzione temporale dei sistemi quantistici riconoscendo il ruolo fondamentale dell’interazione con l’ambiente nel determinare aspetti essenziali della dinamica. Assumendo una prospettiva in linea con questi progressi, in questa tesi si parte dall’idea che la correlazione quantistica, l’entanglement, non possa essere ignorata nel derivare una descrizione statistica coerente dei sistemi complessi tradizionalmente considerati in meccanica statistica. La logica conseguenza di questo punto di vista è che la meccanica statistica quantistica non possa essere basata sull’idea dell’esistenza di insiemi di sistemi quantistici fra loro indipendenti, ma al contrario debba emergere dalla descrizione in termini di una singola funzione d’onda (stato puro) che descrive il sistema nella sua globalità, i.e. il sottosistema di interesse insieme con il suo ambiente (“environment”). Allo scopo di costruire tale descrizione, in questa tesi si considera in primo luogo la distribuzione di probabilità che descrive lo stato di equilibrio di un sistema quantistico isolato. Essa è definita, in analogia con la teoria ergodica classica, sulla base dell’evoluzione temporale del sistema. Per studiare l’emergere delle proprietà termodinamiche si introducono poi distribuzioni di probabilità su insiemi di stati puri (“Ensemble Distributions”). Tali distribuzioni sono derivate sulla base della geometria dello spazio di Hilbert che descrive il sistema nella sua interezza. Inoltre si sono sviluppati gli strumenti teorici che permettono la caratterizzazione di tali distribuzioni di probabilità: essi consistono da un lato nell’implementazione di metodi numerici di tipo Monte Carlo che permettono il campionamento statistico diretto delle distribuzioni, d’altro canto sono state sviluppate approssimazioni analitiche delle distribuzioni sulla base del principio di massima entropia. I risultati fondamentali che emergono dal quadro teorico sviluppato sono illustrati mediante lo studio della statistica in sistemi di spin: il messaggio fondamentale è che le funzioni termodinamiche, come l’entropia del sistema globale e lo stato di equilibrio di un sottosistema, sono caratterizzate da distribuzioni sull’ ensemble che risultano molto concentrate intorno ad un valore tipico. Dall’analisi condotta si deduce quindi che ognuno dei singoli stati puri considerati nell’insieme è caratterizzato dallo stesso valore delle funzioni termodinamiche studiate. Questa è una chiara evidenza della proprietà di tipicalità, (“typicality”), di queste funzioni. L’essenza di questo risultato è che la nostra incapacità di conoscere i dettagli dello stato quantistico del sistema non è così importante dal momento che la grande maggioranza dei possibili stati che appartengono all’insieme considerato sono caratterizzati dallo stesso valore delle proprietà termodinamiche alle quali siamo interessati. In virtù di tale proprietà risulta sensato studiare gli andamenti dei valori tipici delle proprietà termodinamiche. Sotto certe condizioni si ritrovano i risultati della meccanica statistica standard: in particolare lo stato di equilibrio di un sottosistema risulta essere in media lo stato canonico di Boltzmann alla temperatura definita dall’usuale relazione termodinamica . Nella seconda parte della tesi, invece, si illustra la dinamica associata allo stato di equilibrio di un sistema in interazione con il suo ambiente. Le caratteristiche delle fluttuazioni intorno ai valori medi di equilibrio dipendono sia dall’entanglement tra il sistema e l’ambiente che dal tipo di interazione considerato. Per finire si considera la connessione fra la dinamica delle fluttuazioni all’equilibrio e i processi di rilassamento da uno stato iniziale di non equilibrio. Il lavoro presentato in questa tesi è stato in parte motivato da un analisi critica dei metodi stocastici utilizzati nella modellizzazione teorica delle spettroscopie magnetiche. Durante il primo anno di dottorato tali metodologie sono state impiegate per l’interpretazione di alcune osservabili in esperimenti di risonanza magnetica elettronica bidimensionale. [Fresch B., Frezzato D., Moro G. J., Kothe G., Freed J. H.; J. Phys. Chem. B., 110, 24238, (2006)].

The asymptotic state of a quantum system, which is in contact with a heat bath, is strongly disturbed by a time-periodic driving in comparison to a time-independent system. In this thesis an extensive picture of the asymptotic state of time-periodic quantum systems is drawn by relating it to the structure of the corresponding classical phase space. To this end the occupation probabilities of the Floquet states are analyzed with respect to their semiclassical property of being either regular or chaotic. The regular Floquet states are occupied with exponential weights e^{-betaeff Ereg} similar to the canonical weights e^{-beta E} of time-independent systems. The regular energies Ereg are defined by the quantization of the time-periodic system, whose classical properties also determine the effective temperature 1/betaeff. In contrast, the chaotic Floquet states acquire almost equal probabilities, irrespective of their time-averaged energy. Beyond these semiclassical properties the existence of avoided crossings in the spectrum is an intrinsic quantum property of time-periodic systems. Avoided crossings can strongly influence the entire occupation distribution. As an impressive application a novel switching mechanism is proposed in a periodically driven double well potential coupled to a heat bath. By a weak variation of the driving amplitude its asymptotic state is switched from the ground state in one well to a state with higher average energy in the other well
Der asymptotische Zustand eines Quantensystems, das in Kontakt mit einem Wärmebad steht, wird durch einen zeitlich periodischen Antrieb gegenüber einem zeitunabhängigen System nachhaltig verändert. In dieser Arbeit wird ein umfassendes Bild über den asymptotischen Zustand zeitlich periodischer Quantensysteme entworfen, indem es diesen zur Struktur des zugehörigen klassischen Phasenraums in Beziehung setzt. Dazu werden die Besetzungswahrscheinlichkeiten der Floquet-Zustände hinsichtlich ihrer semiklassischen Eigenschaft analysiert, nach welcher sie entweder regulär oder chaotisch sind. Die regulären Floquet-Zustände sind mit exponentiellen Gewichten e^{-betaeff Ereg} ähnlich der kanonischen Verteilung e^{-beta E} zeitunabhängiger Systeme besetzt. Dabei sind die reguläre Energien Ereg durch die Quantisierung des Systems vorgegeben, dessen klassische Eigenschaften auch die effektive Temperatur 1/betaeff bestimmen. Die chaotischen Zustände dagegen haben fast einheitliche Besetzungswahrscheinlichkeiten, welche unabhängig von ihrer mittleren Energie sind. Über diese semiklassischen Eigenschaften hinaus ist das Auftreten von vermiedenen Kreuzungen im Spektrum eine intrinsisch quantenmechanische Eigenschaft zeitlich periodischer Systeme. Diese können die gesamte Besetzungsverteilung nachhaltig beeinflussen und finden eine eindrucksvolle Anwendung in Form eines neuartigen Schaltmechanismus in einem harmonisch modulierten Doppelmuldenpotential in Kontakt mit einem Wärmebad. Der asymptotische Zustand kann unter geringer Variation der Antriebsamplitude vom Grundzustand der einen Mulde in einen Zustand höherer mittlerer Energie in der anderen Mulde geschaltet werden

The pseudospectral method is a family of numerical methods for the solution of differential equations based on the expansion of basis functions defined on a set of grid points. In this thesis, the relationship between the distribution of grid points and the accuracy and convergence of the solution is emphasized. The polynomial and sinc pseudospectral methods are extensively studied along with many applications to quantum and statistical mechanics involving the Fokker-Planck and Schroedinger equations. The grid points used in the polynomial methods coincide with the points of quadrature, which are defined by a set of polynomials orthogonal with respect to a weight function. The choice of the weight function plays an important role in the convergence of the solution. It is observed that rapid convergence is usually achieved when the weight function is chosen to be the square of the ground-state eigenfunction of the problem. The sinc method usually provides a slow convergence as the grid points are uniformly distributed regardless of the behaviour of the solution. For both polynomial and sinc methods, the convergence rate can be improved by redistributing the grid points to more appropriate positions through a transformation of coordinates. The transformation method discussed in this thesis preserves the orthogonality of the basis functions and provides simple expressions for the construction of discretized matrix operators. The convergence rate can be improved by several times in the evaluation of loosely bound eigenstates with an exponential or hyperbolic sine transformation. The transformation can be defined explicitly or implicitly. An explicit transformation is based on a predefined mapping function, while an implicit transformation is constructed by an appropriate set of grid points determined by the behaviour of the solution. The methodologies of these transformations are discussed with some applications to 1D and 2D problems. The implicit transformation is also used as a moving mesh method for the time-dependent Smoluchowski equation when a function with localized behaviour is used as the initial condition.

Thermal (canonical) condensed-phase systems are of considerable interest in computational science, and include for example reactions in solution. Time-independent properties of these systems include free energies and thermally averaged geometries - time-dependent properties include correlation functions and thermal reaction rates. Accounting for quantum effects in such simulations remains a considerable challenge, especially for large systems, due to the quantum nature and high dimensionality of the phase space. Additionally time-dependent properties require treatment of quantum dynamics. Most current methods rely on semi-classical trajectories, path integrals or imaginary-time propagation of wave packets. Trajectory based approaches use continuous phase-space trajectories, similar to classical molecular dynamics, but lack a direct link to a wave packet and so the time-dependent schrodinger equation. Imaginary time propagation methods retain the wave packet, however the imaginary-time trajectory cannot be used as an approximation for real-time dynamics. We present a new approach that combines aspects of both. Using a generalisation of the coherent-state basis allows for mapping of the quantum canonical statistical average onto a phase-space average of the centre and width of thawed Gaussian wave packets. An approximate phase-space density that is exact in the low-temperature harmonic limit, and is a direct function of the phase space is proposed, defining the Gaussian statistical average. A novel Nose-Hoover looped chain thermostat is developed to generate the Gaussian statistical average via the ergodic principle, in conjunction with variational thawed Gaussian wave-packet dynamics. Numerical tests are performed on simple model systems, including quartic bond stretching modes and a double well potential. The Gaussian statistical average is found to be accurate to around 10% for geometric properties at room temperature, but gives energies two to three times too large. An approach to correct the Gaussian statistical average and ensure classical statistics is retrieved at high temperature is then derived, called the switched statistical average. This involves transitioning the potential surface upon which the Gaussian wave packet propagates, and the system property being averaged. Switching functions designed to perform these tasks are derived and tested on model systems. Bond lengths and their uncertainties calculated using the switched statistical average were found to be accurate to within 1% relative to exact results, and similarly for energies. The switched statistical average, calculated with Nose- Hoover looped chain thermostatted Gaussian dynamics, forms a new platform for evaluating statistical properties of quantum condensed-phase systems using an explicit real-time wave packet, whilst retaining appealing features of trajectory based approaches.

The project concerns the study of the interplay among quantum mechanics, statistical mechanics and thermodynamics, in isolated quantum systems. The goal of this research is to improve our understanding of the concept of thermal equilibrium in quantum systems. First, I investigated the role played by observables and measurements in the emergence of thermal behaviour. This led to a new notion of thermal equilibrium which is specific for a given observable, rather than for the whole state of the system. The equilibrium picture that emerges is a generalization of statistical mechanics in which we are not interested in the state of the system but only in the outcome of the measurement process. I investigated how this picture relates to one of the most promising approaches for the emergence of thermal behaviour in quantum systems: the Eigenstate Thermalization Hypothesis. Then, I applied the results to study the equilibrium properties of peculiar quantum systems, which are known to escape thermalization: the many-body localised systems. Despite the localization phenomenon, which prevents thermalization of subsystems, I was able to show that we can still use the predictions of statistical mechanics to describe the equilibrium of some observables. Moreover, the intuition developed in the process led me to propose an experimentally accessible way to unravel the interacting nature of many-body localised systems. Then, I exploited the "Concentration of Measure" and the related "Typicality Arguments" to study the macroscopic properties of the basis states in a tentative theory of quantum gravity: Loop Quantum Gravity. These techniques were previously used to explain why the thermal behaviour in quantum systems is such an ubiquitous phenomenon at the macroscopic scale. I focused on the local properties, their thermodynamic behaviour and interplay with the semiclassical limit. The ultimate goal of this line of research is to give a quantum description of a black hole which is consistent with the expected semiclassical behaviour. This was motivated by the necessity to understand, from a quantum gravity perspective, how and why an horizon exhibits thermal properties.

Molecular Dynamics (MD) simulations based on classical statistical mechanics always allow the atom to have thermal heat capacity. Quantum mechanics (QM) differs in that the heat capacity of atoms in submicron nanostructures vanishes. Nevertheless, MD simulations of heat transfer in discrete nanostructures are routlinely performed and abound in the literature. Not only are discrete MD sumultions invalid by QM, but give unphysical results, e.g., thermal conducitvity in nanofluids is found to exceed standard mixing rules while in solid metal films depends on thickness. QM negates the heat capacity of atoms in discrete nanostructures, thereby precluding the usual conservation of absorbed electromagnetic (EM) energy by an increase in temperature. Instead, conservation proceeds by QED inducing the absorbed EM energy to create non-thermal EM radiation inside the nanostructure that by the photoelectric effect chargea the nanostructure, or is emitted to the surroundings. QED stands for quantum electrodynamics. Unphysical results occur because QED induced radiation is not included in the nanoscale heat balance, but if included physical results for discrete nanostructures are found. Examples of unphysical MD simulatons are presented. When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/35192

Thesis (Ph. D.)--University of Oregon, 2002.
Typescript. Includes vita and abstract. Includes bibliographical references (leaves 223-225). Also available for download via the World Wide Web; free to University of Oregon users.

Es conocido que a escalas nanométricas se debe tratar con en el problema de muchas partículas a la hora de estudiar dispositivos electrónicos. Es estos escenarios, la ecuación de Schrödinger dependiente del tiempo para muchas partículas solo se puede resolver para unos pocos grados de libertad. En este sentido, diferentes formalismos han sido desarrollados en la literatura (tales como time-dependent Density Functional Theory, Green's functions técnicas o Quantum Monte Carlo técnicas) para tratar sistemas cuánticos de muchos electrones. Estas aproximaciones modelizan de forma razonable el transporte electrónico en sistemas de muchas partículas. Una propuesta alternativa ha sido desarrollada por el Dr. Oriols para descomponer la ecuación de Schrödinger de N-partículas en un sistema de N-ecuaciones de Schrödinger para una sola partícula usando trayectorias (cuánticas) de Bohm. Basado en esta propuesta se presenta un 3D, general, versátil y dependiente del tiempo simulador de transporte de dispositivos electrónicos llamado BITLLES (Bohmian Interacting Transport for non-equiLibrium eLEctronic Structures). Las novedades que aporta el simulador BITLLES se basan en dos puntos. El primero, éste representa un modelo de transporte cuántico de electrones para muchas partículas en el cual se tiene en cuenta de forma explicita las correlaciones de Coulomb y de intercambio entre electrones usando trayectorias de Bohm. En segundo lugar, el simulador proporciona una completa información de los momentos de la corriente (i.e., DC, AC, fluctuaciones o incluso momentos mayores). A continuación resumimos las contribuciones que esta tesis aporta al desarrollo del simulador BITLLES. De esta forma, introducimos de forma explicita la interacción de intercambio entre electrones. En este contexto, mostramos como la interacción de intercambio es la responsable final para determinar la corriente total a través del sistema. Además presentamos una nueva aproximación para estudiar sistemas de muchas partículas donde los espines de los electrones tienen diferente orientación. Hasta donde llega nuestro conocimiento, es la primera vez que la interacción de intercambio es introducida de forma práctica en un simulador de transporte de electrones. Además presentamos la computación de la corriente total dependiente del tiempo en un contexto de alta frecuencia donde se tienen que tener en cuenta las variaciones del campo eléctrico dependientes del tiempo (i.e., la corriente de desplazamiento) para asegurar la conservación de la corriente. También discutimos el cálculo de la corriente total (conducción más desplazamiento) usando los teoremas de Ramo-Shockley-Pellegrini. Diferentes capacidades del simulador BITLLES como AC y fluctuaciones de la corriente se presentan para el diodo túnel resonante. También hemos usado el simulador BITLLES para testear un nuevo tipo de dispositivo nanoeléctronico diseñado para procesar señales dentro del espectro de los THz. Hemos llamado a este dispositivo Driven Tunneling Device. Se trata de un dispositivo de tres terminales donde la conductancia entre el drain y el source se controla por el terminal del gate el cual oscila a frecuencias de THz. También presentamos ejemplos prácticos de la funcionalidad de este dispositivo como un rectificador y un multiplicador de frecuencia. Finalmente, hemos desarrollado una aproximación numérica para resolver la ecuación de Schrödinger usando el modelo de tight-binding con el propósito de mejorar la descripción de la estructura de bandas del simulador BITLLES.
It is known that at nanoscale regime we must deal with the many-particle problem in order to study electronic devices. In this scenario, the time-dependent many-particle Schrödinger equation is only directly solvable for very few degrees of freedom. However, there are many electrons (degrees of freedom) in any electron device. In this sense, many-particle quantum electron formalisms (such as time-dependent Density Functional Theory, Green's functions techniques or Quantum Monte Carlo techniques) have been developed in the literature to provide reasonable approximations to model many-particle electron transport. An alternative proposal has been developed by Dr. Oriols to decompose the N-particle Schrödinger equation into a N-single particle Schrödinger equation using Bohmian trajectories. Based on this proposal a general, versatile and time-dependent 3D electron transport simulator for nanoelectronic devices, named BITLLES (Bohmian Interacting Transport for non-equiLibrium eLEctronic Structures) is presented. The novelty of the BITLLES simulator is based on two points. First, it presents a many-particle quantum electron transport model taking into account explicitly the Coulomb and exchange correlations among electrons using Bohmian trajectories. Second, it provides full information of the all current distribution moments (i.e. DC, AC, fluctuations and even higher moments). We summarize the important contributions of this thesis to the development of BITLLES simulator. Thus, we introduce explicitly the exchange correlations among electrons. In this context, we show how exchange interaction is the final responsible for determining the total current across the system. We also present a new approximation to study many-particle systems with spin of different orientations. Some practical examples are studied taking into account the exchange interaction. To the best of our knowledge, it is the first time that the exchange interaction is introduced explicitly (imposing the exchange symmetry properties directly into the many-particle wavefunction) in practical electron transport simulators. We present the computation of the time-dependent total current in the high-frequency regime where one has to compute time-dependent variations of the electric field (i.e. the displacement current) to assure current conservation. We discuss the computation of the total (conduction plus displacement) current using Bohmian trajectories and the Ramo-Shockley-Pellegrini theorems. Different capabilities of BITLLES simulator such as AC and current fluctuations are presented for Resonant Tunneling Devices. We have used the BITLLES simulator to test a new type of nanoelectronic device designed to process signals at THz regime named Driven Tunneling Device. It is a three terminal device where the drain-source conductance is controlled by a gate terminal that can oscillate at THz frequencies. We also present practical examples on the functionality of this device such as rectifier and frequency multiplier. Finally, we have developed a numerical approximation to solve the Schrödinger equation using tight-binding model to improve the band structure description of the BITLLES simulator.

Desenvolvemos no presente trabalho um tratamento perturbativo para a matriz densidade reduzida de forma similar à regra áurea de Fermi para espalhamento. Aplicamos a teoria a vários exemplos e em particular reproduzimos os resultados experimentais obtidos no laboratório Kastler Brossel e obtivemos uma relação entre os tempos característicos de dissipação e decoerência. Por outro lado, desenvolvemos um modelo simples para duas moléculas interagindo com um reservatório. Mostramos resultados surpreendentes quando temos mais que dois subsistemas interagindo: no caso particular em que as moléculas estão num estado inicial coerente, bombardeadas por fontes de mesma intensidade, o estado assintótico apresenta urna concentração de energia no modo de menor frequência. Este resultado dá suporte a um modelo fenomenológico de sistemas biológicos onde a condensação de Bose- Einstein é produzida e o estado final também exibe uma concentração de energia no modo de frequência mais baixa.
In the present work we developed a perturbative treatment of reduced density matrices which is similar in spirit to Fermi\'s Golden Rule for scattering. We applied the theory to several e- xamples and in particular reproduced the results obtained in the laboratory Kastler Brossel experiment quantifying the relation between decoherence and dissipation characteristic times. On the other hand we developed a simple model for two molecules interacting through a re- servoir. We show that rather surprising results may arise when we have more than two subsys- tems in interaction: in the particular case where both molecules are initially in coherent states, if they are pumped with the same strength, the asymptotic state shows a concentraction of energy on the mode with smallest frequency. This result gives support to a phenomenological model for biological systems where a Bose- Einstein condensation is predicted and the final state also exhibits a concentration of energy in the lowest frequency mode.

This dissertation addresses the delicate problem of aging in complex systems characterized by non-Poissonian statistics. With reference to a generic two-states system interacting with a bath it is shown that to properly describe the evolution of such a system within the formalism of the continuous time random walk (CTRW), it has to be taken into account that, if the system is prepared at time t=0 and the observation of the system starts at a later time ta>0, the distribution of the first sojourn times in each of the two states depends on ta, the age of the system. It is shown that this aging property in the fractional derivative formalism forces to introduce a fractional index depending on time. It is shown also that, when a stationary condition exists, the Onsager regression principle is fulfilled only if the system is aged and consequently if an infinitely aged distribution for the first sojourn times is adopted in the CTRW formalism used to describe the system itself. This dissertation, as final result, shows how to extend to the non-Poisson case the Kubo Anderson (KA) lineshape theory, so as to turn it into a theoretical tool adequate to describe the time evolution of the absorption and emission spectra of CdSe quantum dots. The fluorescence emission of these single nanocrystals exhibits interesting intermittent behavior, namely, a sequence of "light on" and "light off" states, departing from Poisson statistics. Taking aging into account an exact analytical treatment is derived to calculate the spectrum. In the regime fitting experimental data this final result implies that the spectrum of the "blinking" quantum dots must age forever.

Monodromy fields on I3 are a family of lattice fields in two dimensions which are a natural generalisation of the two dimensional Ising field occurring in the C*-algebra approach to Statistical Mechanics. A criterion for the critical limit one point correlation of the monodromy field tra(M) at a 6 l3, Um(#.(M)). is deduced for matrices M € GL(p, C) having non-negative eigenvalues. Using this criterion a non-identity 2x2 matrix is found with a finite critical limit one point correlation. The general set of p x p matrices with finite critical limit one point correlations is also considered and a conjecture for the critical limit n point correlations postulated. The boson-fermion correspondence for the representation of the CAR algebra over L3(Sl, C) defined by the (t,B) KMS state with chemical potential p is considered and the non-bijectivity shown. Using an alternative formulation the correlations are recalculated leading to a determinant identity reminiscent of Saego’s Theorem.

The de Broglie-Bohm causal interpretation of quantum mechanics is discussed, and applied to the hydrogen atom in several contexts. Prominent critiques of the causal program are noted and responses are given; it is argued that the de Broglie-Bohm theory is of notable interest to physics. Using the causal theory, electron trajectories are found for the conventional Schrödinger, Pauli and Dirac hydrogen eigenstates. In the Schrödinger case, an additional term is used to account for the spin; this term was not present in the original formulation of the theory but is necessary for the theory to be embedded in a relativistic formulation. In the Schrödinger, Pauli and Dirac cases, the eigenstate trajectories are shown to be circular, with electron motion revolving around the z-axis. Electron trajectories are also found for the 1s-2p0 transition problem under the Schrödinger equation; it is shown that the transition can be characterized by a comparison of the trajectory to the relevant eigenstate trajectories. The structures of the computed trajectories are relevant to the question of the possible evolution of a quantum distribution towards the standard quantum distribution (quantum equilibrium); this process is known as quantum relaxation. The transition problem is generalized to include all possible transitions in hydrogen stimulated by semi-classical radiation, and all of the trajectories found are examined in light of their implications for the evolution of the distribution to the standard distribution. Several promising avenues for future research are discussed.

Les comportements critiques des systèmes de mécanique statistique en 2 dimensions ou de mécanique quantique en 1+1 dimensions, ainsi que certains aspects des systèmes sans interactions en 2+1 dimensions, sont efficacement décrits par les méthodes de la théorie des champs conforme et de l'intégrabilité, dont le développement a été spectaculaire au cours des 40 dernières années. Plusieurs problèmes résistent cependant toujours à une compréhension exacte, parmi lesquels celui de la transition entre plateaux dans l'Effet Hall Quantique Entier. La raison principale en est que de tels problèmes sont généralement associés à des théories non unitaires, ou théories conformes logarithmiques, dont la classification se révèle être d'une grande difficulté mathématique. Se tournant vers la recherche de modèles discrets (chaînes de spins, modèles sur réseau), dans l'espoir en particulier d'en trouver des représentations en termes de modèles exactement solubles (intégrables), on se heurte à la deuxième difficulté représentée par le fait que les théories associées sont la plupart du temps non compactes, ou en d'autres termes qu'elles donnent lieu à un continuum d'exposants critiques. En effet, le lien entre modèles discrets et théories des champs non compactes est à ce jour loin d'être compris, en particulier il a longtemps été cru que de telles théories ne pouvaient pas émerger comme limites continues de modèles discrets construits à partir d'un ensemble compact de degrés de libertés, par ailleurs les seuls qui donnent a accès à une construction systématique de solutions exactes.Dans cette thèse, on montre que le monde des modèles discrets compacts ayant une limite continue non compacte est en fait beaucoup plus grand que ce que les quelques exemples connus jusqu'ici auraient pu laisser suspecter. Plus précisément, on y présente une solution exacte par ansatz de Bethe d'une famille infinie de modèles(les modèles $a_n^{(2)}$, ainsi que quelques résultats sur les modèles $b_n^{(1)}$, où il est observé que tous ces modèles sont décrits dans un certain régime par des théories conformes non compactes. Parmi ces modèles, certains jouent un rôle important dans la description de phénomènes physiques, parmi lesquels la description de polymères en deux dimensions avec des interactions attractives et des modèles de boucles impliqués dans l'étude de modèles de Potts couplés ou dans une tentative de description de la transition entre plateaux dans l'Effet Hall par un modèle géométrique compact.On montre que l'existence insoupçonnéede limite continues non compacts pour de tels modèles peut avoir d'importantes conséquences pratiques, par exemple dans l'estimation numérique d'exposants critiques ou dans le résultats de simulations de Monte Carlo. Nos résultats sont appliqués à une meilleure compréhension de la transition theta décrivant l'effondrement des polymères en deux dimensions, et des perspectives pour une potentielle compréhension de la transition entre plateaux en termes de modèles sur réseaux sont présentées
The critical points of statistical mechanical systems in 2 dimensions or quantum mechanical systems in 1+1 dimensions (this also includes non interacting systems in 2+1 dimensions) are effciently tackled by the exact methods of conformal fieldtheory (CFT) and integrability, which have witnessed a spectacular progress during the past 40 years. Several problems have however escaped an exact understanding so far, among which the plateau transition in the Integer Quantum Hall Effect,the main reason for this being that such problems are usually associated with non unitary, logarithmic conformal field theories, the tentative classification of which leading to formidable mathematical dificulties. Turning to a lattice approach, andin particular to the quest for integrable, exactly sovable representatives of these problems, one hits the second dificulty that the associated CFTs are usually of the non compact type, or in other terms that they involve a continuum of criticalexponents. The connection between non compact field theories and lattice models or spin chains is indeed not very clear, and in particular it has long been believed that the former could not arise as the continuum limit of discrete models built out of acompact set of degrees of freedom, which are the only ones allowing for a systematic construction of exact solutions.In this thesis, we show that the world of compact lattice models/spin chains with a non compact continuum limit is much bigger than what could be expected from the few particular examples known up to this date. More precisely we propose an exact Bethe ansatz solution of an infinite family of models (the so-called $a_n^{(2)}$ models, as well as some results on the $b_n^{(1)}$ models), and show that all of these models allow for a regime described by a non compact CFT. Such models include cases ofgreat physical relevance, among which a model for two-dimensional polymers with attractive interactions and loop models involved in the description of coupled Potts models or in a tentative description of the quantum Hall plateau transition by somecompact geometrical truncation. We show that the existence of an unsuspected non compact continuum limit for such models can have dramatic practical effects, for instance on the output of numerical determination of the critical exponents or ofMonte-Carlo simulations. We put our results to use for a better understanding of the controversial theta transition describing the collapse of polymers in two dimensions, and draw perspectives on a possible understanding of the quantum Hall plateautransition by the lattice approach

This thesis examines various reductive case studies in thermal physics. In particular, I argue that according to my account of reduction-as-construction, there are two suc- cessful examples of reduction. Thermodynamics reduces to statistical mechanics, and statistical mechanics reduces to the underlying microdynamics - be they quantum or classical. The reduction of a given theory alters that theory's scope, that is: its domain of applicability. The scope of thermodynamics will be central to this thesis - and I will argue for a narrower scope than some authors. This thesis consists of four Chapters, together with an introduction and a conclusion. In Chapter 1, I discuss how different levels of description relate to one another. I argue that a higher-level of description is reduced to the lower level, if the higher-level quantities and their behaviour can be constructed or captured by the lower-level theory. I claim that 'functionalism' can be helpful in securing reductions. In this Chapter I also argue that the aim of reduction is to vindicate, not eliminate, the higher-level theory. In Chapter 2, I tackle the reduction of thermodynamics to statistical mechanics. I articulate the functional, or nomological, role of various thermodynamic quantities that are implicitly defined by the zeroth, first and second laws of thermodynamics: temperature, energy and entropy respectively. I then argue that there are quantities in statistical mechanics that realise these roles: though finding them sometimes requires us to focus on quantum, rather than classical, statistical mechanics. In Chapter 3, I consider the reductive relationship between statistical mechanics and the underlying microdynamics. I demonstrate how the irreversible equations of statistical mechanics can be constructed from the underlying microdynamics using what I label the 'Zwanzig-Zeh-Wallace' framework. Yet this framework uses a procedure called 'coarse-graining' which has been heavily criticised in the literature; so in this Chapter I offer a justification of coarse-graining. One upshot is that the time-asymmetry in statistical mechanics is weakly emergent. In Chapter 4, I consider a question about the domain of applicability of thermal physics. Namely: does it apply to self-gravitating systems, such as elliptical galaxies? Much controversy surrounds this question: some argue yes, others argue no. I deflate the dispute by claiming that thermodynamics does not apply, but statistical mechanics does. Thus, my delineation of thermodynamics and statistical mechanics earlier in this thesis not only makes headway with the question of reduction, but also sheds light on this dispute. I argue that this situation - statistical mechanics, but without thermodynamics - can be understood in terms of a central notion in thermal physics: the thermodynamic limit. But as I also discuss: justifying this idealisation has been philosophically controversial.

Cette thèse traite la dynamique hors équilibre des systèmes quantiques ouverts couplés à un réservoir externe. Un modèle spécifique exactement soluble, le modèle sphérique, sert comme exemple paradigmatique. Ce modèle se résout exactement en toute dimension spatiale et pour des interactions très générales. Malgré sa simplicité technique, ce modèle est intéressant car ni son comportement critique d’équilibre ni celui hors équilibre est du genre champ moyen. La présentation débute avec une revue sur la mécanique statistique des transitions de phases classique et quantique, et sur les propriétés du modèle sphérique. Sa dynamique quantique ne se décrit point à l’aide d’une équation de Langevin phénoménologique. Une description plus complète à l’aide de la théorie de l’équation de Lindblad est nécessaire. Les équations de Lindblad décrivent la relaxation d’un système quantique vers son état d’équilibre. En tant que premier exemple, le diagramme de phases dynamique d’un seul spin sphérique quantique est étudié. Réinterprétant cette solution en tant qu’une approximation champ moyen d’un problème de N corps, le diagramme de phases quantique est établi et un effet « congeler en réchauffant » quantique est démontré. Ensuite, le formalisme de Lindblad est généralisé au modèle sphérique quantique de N particules: primo, la forme précise de l’équation de Lindblad est obtenue des conditions que (i) l’état quantique d’équilibre exacte est une solution stationnaire de l’équation de Lindblad et (ii) dans le limite classique, l’équation Langevin de mouvement est retrouvée. Secundo, le modèle sphérique permet la réduction exacte du problème de N particules à une seule équation intégro-différentielle pour le paramètre sphérique. Tertio, en résolvant pour le comportement asymptotique des temps longs de cette équation, nous démontrons que dans la limite semi-classique, la dynamique quantique effective redevient équivalente à une dynamique classique, à une renormalisation quantique de la température T près. Quarto, pour une trempe quantique profonde dans la phase ordonnée, nous démontrons que la dynamique quantique dépend d’une manière non triviale de la dimension spatiale. L’émergence du comportement d’échelle dynamique et des corrections logarithmiques est discutée en détail. Les outils mathématiques de cette analyse sont des nouveaux résultats sur le comportement asymptotique de certaines fonctions hypergéométriques confluentes en deux variables
This study deals with the dynamic properties of open quantum systems far from equilibrium in d dimensions. The focus is on a special, exactly solvable model, the spherical model (SM), which is technically simple. The analysis is of interest, since the critical behaviour in and far from equilibrium not of mean-field type. We begin with a résumé of the statistical mechanics of phase transitions and treat especially the quantum version of the SM. The quantum dynamics (QD) of the model cannot be described by phenomenological Langevin equation and must be formulated with Lindblad equations.First we examine the dynamic phase diagram of a single spherical quantum spin and interpret the solution as a mean-field approximation of the N-body problem. Hereby, we find a quantum mechanical ‘freezing by heating’ effect. After that, we extend the formalism to the N-body problem, determining first the form of the Lindblad equation from consistency conditions. The SM then allows the reduction to a single integro-differential equation whose asymptotic solution shows, that the effective QD in the semi-classical limit is fully classical. For a deep quench in the ordered phase, we show that the QD strongly and non-trivially depends on d and derive the dynamic scaling behaviour and its corrections. The mathematical tools for this analysis are new results on the asymptotic behaviour of certain confluent hypergeometric functions in two variables

In this thesis we study three examples of interacting many-body systems undergoing a non equilibrium time evolution. Firstly we consider the time evolution in an integrable system: the sine-Gordon field theory in the repulsive regime. We will focus on the one point function of the semi-local vertex operator e^iβφ(x)/2 on a specific class of initial states. By analytical means we show that the expectation value considered decays exponentially to zero at late times and we determine the decay time. The method employed is based on a form-factor expansion and uses the "Representative Eigenstate Approach" of Ref. [73] (a.k.a. "Quench Action"). In a second example we study the time evolution in models close to "special" integrable points characterised by hidden symmetries generating infinitely many local conservation laws that do not commute with one another, in addition to the infinite commuting family implied by integrability. We observe that both in the case where the perturbation breaks the integrability and when it breaks only the additional symmetries maintaining integrability, the local observables show a crossover behaviour from an initial to a final quasi stationary plateau. We investigate a weak coupling limit, identify a time window in which the effects of the perturbations become significant and solve the time evolution through a mean-field mapping. As an explicit example we study the XYZ spin-1/2 chain with additional perturbations that break integrability. Finally, we study the effects of integrability breaking perturbations on the non-equilibrium evolution of more general many-particle quantum systems, where the unperturbed integrable model is generic. We focus on a class of spinless fermion models with weak interactions. We employ equation of motion techniques that can be viewed as generalisations of quantum Boltzmann equations. We benchmark our method against time dependent density matrix renormalisation group computations and find it to be very accurate as long as interactions are weak. For small integrability breaking, we observe robust prethermalisation plateaux for local observables on all accessible time scales. Increasing the strength of the integrability breaking term induces a "drift" away from the prethermalisation plateaux towards thermal behaviour. We identify a time scale characterising this crossover.

1ere partie : le travail justifie un code numerique de resolution de l'equation de liouville classique dans l'espace des phases (x,v), qui consiste a "transporter" cette equation dans l'espace des configurations q (ou p) de la mecanique quantique (m. Q. ). Le but recherche est de ralentir le developpement des microstructures de la fonction de distribution du fluide. Le schema propose consiste a construire une fonction d'onde correspondant a la condition initiale (c. I. ) macroscopique classique (c'est le probleme de la synthese), a faire evoluer cette fonction d'onde selon l'equation de schroedinger. Nous proposons un code de resolution numerique qui consiste a discretiser en temps le potentiel par une suite de distributions de dirac qui reduise la dynamique a une suite de propagations libres (fpm) et approximations soudaines (s. A. ), que nous savons resoudre. En fin de calcul, le passage de la fonction d'onde a la distribution classique se fait en prenant la limite classique de la transformee de wigner. L'etude de la limite classique nous a conduit a considerer une version lissee de la representation de wigner par un noyau doublement gaussien minimal qui assure la positivite de la distribution, et qui peut etre interprete comme lie a "l'instrument de mesure" qui doit etre symetrique. La synthese de la fonction d'onde correspondant a la c. I. (qui necessite une condition de faisabilite), est faite relativement a cette representation conjointe et dans sa limite classique. La dynamique de la distribution lissee nous a permis de mettre en evidence deux classes de c. I. : celles qui proviennent d'un etat initial pur (conditions ric) auquel cas, tous les diagrammes de dynamique commutent et les autres (melanges statistiques ou c. I. Macroscopiques) dites gic, qui n'ont par la propriete de commutation et qui necessitent une preparation differente de la phase d'une fonction g pour assurer l'obtention de la limite classique de chacune des dynamiques fpm ou sa. 2eme partie : nous utilisons une analogie formelle entre la mecanique quantique (m. Q. ) et le traitement du signal (t. S. ) pour construire une representation temps-frequence des signaux certains, dite de wigner ville lissee qui donne correctement la frequence instantanee des signaux unimodulaires h. F. (ce qui est l'analogue t. S. De la limite classique de la m. Q. ) et qui reduit de facon optimale les parasites que donnerait la transformee de wigner ville originelle dus a sa nature sesquilineaire. Sous reserve de condition de faisabilite, nous realisons deux types de synthese de signaux ayant une representation donnee

The aim of this article is to give a pedagogical introduction to the exact equilibrium and nonequilibrium properties of free fermionic quantum spin chains. In a first part we present in full details the canonical diagonalisation procedure and review quickly the equilibrium dynamical properties. The phase diagram is analysed and possible phase transitions are discussed. The two next chapters are concerned with the effect of aperiodicity and quenched disorder on the critical properties of the quantum chain. The remaining part is devoted to the nonequilibrium dynamical behaviour of such quantum chains relaxing from a nonequilibrium pure initial state. In particular, a special attention is made on the relaxation of transverse magnetization. Two-time linear response functions and correlation functions are also considered, giving insights on the nature of the final nonequilibrium stationnary state. The possibility of aging is also discussed.

O estudo de matrizes aleatórias na física tradicionalmente ocorre no contexto dos modelos de Wigner e na estatística por modelos de Wishart, que se conectam através do threefold way de Dyson para matrizes aleatórias reais, complexas e de quaternios indexadas respectivamente pelo índice B = 1; 2; 4 de Dyson. Estudos recentes mostraram o caminho para que estes modelos fossem generalizados para valores reais de B, permitindo o estudo de ensembles com índice arbitrário. Neste trabalho, estudamos as propriedades estatísticas destes sistemas e exploramos a física subjacente nos modelos de Wigner e Wishart e investigamos, através de cálculos numéricos, os efeitos de localização nos modelos de geral. Também introduzimos quebras na simetria desta nova forma e estudamos numericamente os resultados da estatística dos sistemas perturbados.
The study of random matrices in physics has traditionally occurred in the context of Wigner models and in statistics by Wishart models, which are connected through Dyson\'s threefold way for real, complex and quaternion random matrices index by the Dyson _ = 1; 2; 4 index, respectively. Recent studies have shown the way by which these models are generalized for real values of _, allowing for the study the ensembles with arbitrary index. In this work, we study the statistical properties of these systems and explore the underlying physics in Wigner\'s and Wishart\'s models through and investigate through numerical calculations the e_ects of localization in general _ models. We also introduce symmetry breaks in this new form and study numerically the results of the statistics of the disturbed systems.

Nós trabalhamos em três problemas relacionados com as teorias de gauge a temperatura finita. O primeiro discute a invariância de gauge da massa física do elétron num espaço de dimensão arbitrária a temperatura zero. Obtivemos a massa física a partir do polo do propagador fermiônico e demonstramos que a maneira usual de definir este propagador funciona para gauges covariantes mas não para gauges não covariantes. Em seguida propusemos um novo propagador e verificamos de duas formas diferentes que a massa física obtida a partir deste funciona para um gauge definido com parâmetros de controle tais que ele possa ser generalizado para as duas classes estudadas. O segundo problema é sobre a interação de n fótons num espaço de (1+1) dimensões no limite de altas temperaturas. Usando o formalismo de tempo imaginário e o modelo de Schwinger, mostramos que todos os termos das amplitudes causais retardadas com um ou mais loops têm contribuição nula. Interpretamos fisicamente este resultado e fizemos um paralelo de como ele se relaciona com a invariância CPT da teoria. A última parte é relacionada à gravitação quântica em (3+1) dimensões. Discutimos a possibilidade de obtermos as funções de n grávitons 1PI nos limites estático e de comprimento de onda longo em função de polinômios que podem ser escritos e relacionados de uma maneira simples. Para tanto, usamos as identidades de Ward e a invariância de Weyl de forma a relacionar as funções de n e (n+1) grávitons. Em seguida, utilizamos o formalismo da equação de transporte de Boltzmann para compreender melhor os resultados.
We have worked in three problems related to finite temperature gauge theory. The first one discusses the gauge invariance of the electron physical mass in an arbitrary dimension space at zero temperature. We have obtained the physical mass from the pole of the fermion propagator and we have demonstrated that the usual form to define this propagator works well for covariant gauges, but not for non covariant gauges. Then, we have proposed anew fermion propagator and we verified in two different ways that the physical mass obtained from this new one works for a gauge defined with control parameters so that it could be generalized for both classes studied. The second problem is on the n photon interaction in a space with (1+1) dimensions at hard thermal loops. Using the imaginary time formalism and the Schwinger\'s model, we have shown that all terms of the retarded causal amplitudes with one or more loops have null contribution. We have got a physical interpretation of this result and we have done a parallel of how it relates with the CPT invariance of this theory. The last one is related with quantum gravitation in (3+1) dimensions. We have discussed the possibility to obtain the 1PI n graviton functions in static and long-wavelength limits from polynomials which could be written and related in a simple manner. To this end, we used the Ward identities and the Weyl invariance to relate the n and (n+1) graviton functions. Then, we used the Boltzmann transport equation formalism to get a better understanding of the results.

Orientador: Marcus Aloizio Martinez de Aguiar
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin
Made available in DSpace on 2018-08-29T15:51:25Z (GMT). No. of bitstreams: 1 Veronez_Matheus_D.pdf: 15544434 bytes, checksum: 608af036b8db9a50a1b8ed957b506d84 (MD5) Previous issue date: 2015
Resumo: Representações no espaço de fase são ferramentas bastante difundidas no estudo e na simulação de sistemas quânticos, principalmente devido aos seus apelos clássicos. Tanto na mecânica quântica quanto na clássica, elementos similares, tal como densidades de probabilidade, podem ser definidos e usados para comparar ambos regimes. Neste trabalho construímos a partir de primeiros princípios uma corrente quântica no espaço de fase na representação de estados coerentes canônicos. Determinamos a corrente quântica para sistemas sob evolução de uma hamiltoniana genérica e mostramos que ela pode ser expandida numa série de potências em $hbar$ cujo termo de ordem mais baixa é a corrente clássica. Calculamos analiticamente a corrente para alguns sistemas uni-dimensionais simples. A corrente quântica apresenta propriedades não-clássicas, por exemplo, inversão de momento e surgimento de novos pontos de estagnação aos pares durante a dinâmica. Mostramos que estes pares são compostos por um ponto de sela, que é um zero da densidade de probabilidade e possui uma carga topológica de -1, e por um vórtice, que possui carga +1. Ambos pontos constituem o que denominamos dipolo topológico. Analisamos o papel destes dipolos no espalhamento de uma partícula por uma barreira gaussiana e mostramos que suas localizações em relação às superfícies de energia clássicas e em relação aos máximos da densidade de probabilidade são assinaturas de tunelamento
Abstract: Phase space representations are widely used tools to study and simulate the quantum dynamics of systems, mainly due to its natural classical appeal. In both classical and quantum mechanics, corresponding but not equivalent structures, such as probability densities, can be defined and explored to compare both dynamical regimes. In this work, we constructed from first principles the quantum phase space current for a quantum system in the canonical coherent states representation. We determined the quantum current for systems evolving under a general Hamiltonian, and we showed that the current can be expanded as a power series in $hbar$, whose lowest order term is the classsical current. We also calculated analytically the quantum current for simple one-dimensional systems. The quantum current presents non-classical features, such as momentum inversion and emergence of new stagnation points which appear in pairs during the dynamics. We showed that the pairs are composed by a saddle point, which is a zero of the phase space probability density and bears a topological charge -1, and a vortex, with charge +1. Both points constitute what we named a topological dipole. We analysed the role the dipoles play in the scattering of a particle by a gaussian barrier, and we showed that the location of the dipoles in relation to the classical energy surfaces and the quantum probability density maxima is a fingerprint of quantum tunneling
Doutorado
Física
Doutor em Ciências
2013/02248-0
157615/2011-1
FAPESP
CNPQ

Estudamos o amortecimento radiativo e o ruído de spins de um material magnético acoplado a um circuito ressonante. O amortecimento radiativo em ressonância magnética é um fenômeno de dissipação, na qual a magnetização preparada após um pulso de Rabi sofre um decaimento até seu estado de equilíbrio. O material magnético perde energia através do seu acoplamento com o circuito ressonante, que deve estar sintonizado na freqüência de Larmor dos spins do material. Apesar deste fenômeno ter sido estudado há vários anos, nenhuma descrição quântica completa lhe foi dada. Apresentamos um modelo hamiltoniano quântico que descreve o amortecimento radiativo. Para isto usamos o método de equações de Langevin quânticas. Mostramos que além do amortecimento radiativo do material magnético, se o circuito está em um estado inicial coerente, a magnetização adquire um movimento complicado não-trivial. Usando as mesmas equações de Langevin, estudamos a influência da amostra no ruído do circuito ressonante. Calculamos a densidade espectral da corrente no caso em que todo o sistema está em equilíbrio térmico. Pudemos verifcar a efcácia do método comparando-o com estudos anteriores. Além disso, estudamos as alterações do ruído do circuito quando uma tensão oscilante externa é aplicada. Nesta situação surgem dois outros picos laterais ao pico central do espectro de absorção da amostra magnética. Isso leva a três depressões no espectro da corrente do circuito. Este efeito deve-se à separação dupla dos estados de energia dos spins. Comentamos sobre a analogia deste fenômeno com a fluorescência ressonante observada na Óptica Quântica.
We study the radiation damping and the spin noise of a magnetic material coupled with a resonant circuit. Radiation damping in magnetic resonance is a dissipation phenomenon, where magnetization prepared after a Rabi pulse decays toward its equilibrium state. The magnetic sample loses its energy by the coupling with resonant circuit, that must be tuned in Larmor frequency of the sample spins. Even though this phenomenon had been studied many years ago, no full quantum description was done. We present a quantum Hamiltonian model, that explains the radiation damping. We use quantum Langevin equation method for this task. Beyond radiation damping, we show the magnetization acquires an unusual intrincate motion, if the circuit initial state is coherent. Using the same Langevin equation, we study the sample influence on the resonant circuit noise. We calculate the current spectral density in the case of thermal equilibrium of whole system. We can verify the method efectiveness, comparing former papers. Moreover we study modifcations in the circuit noise, if an external oscillating tension is applied. In this situation, other two peaks emerge in the central peak sidebands of the sample absorption spectrum. It leads to appear three dips in circuit current spectrum. This efect is due to the splitting of the spin energy states. We comment about the analogy between this phenomenon and the resonance fluorescence in Quantum Optics.

Ce mémoire présente mes activités de recherche dans le domaine de la mécanique statistique des systèmes désordonnés, en particulier sur les modèles de champ moyen à connectivité finie. Ces modèles présentent de nombreuses transitions de phase dans la limite thermodynamique, avec des applications tant pour la physique des verres que pour leurs liens avec des problèmes d'optimisation de l'informatique théorique. Leur comportement sous l'effet de fluctuations quantiques est aussi discuté, en lien avec des perspectives de calcul quantique.

Utilizamos um formalismo de operadores e a técnica de grupo de renormalizacao de Dasgupta, Ma e Hu para analisar o efeito de distribuições inomogêneas dos parâmetros sobre o comportamento crítico de um modelo estocástico simples. O processo de contato em uma dimensão constitui talvez o modelo mais simples que apresenta uma transição de fase para um estado absorvente. Nós usamos as seqüências de Fibonacci, duplicação de período e triplicação de período para introduzir inomogeneidades aperiódicas no processo de contato unidimensional e em uma cadeia quântica de spin. Usando procedimento de grupo de renormalização de desordem forte, estabelecemos algumas relações entre propriedades dos operadores renormalizados e grandezas termodinâmicas ou médias. Fomos capazes de testar o critério de relevância de flutuações geométricas de Harris-Luck, de obter vários expoentes críticos, e de observar aspectos característicos de dinâmica lenta e oscilações log-periódicas. A sequência de triplicação de período nos leva aos expoentes = ln (7/9)/ ln (4/9), = ln (9/7)/ ln 4, = ln 3/ ln (3/2) e k = ln 2/ ln (3/2). Usamos técnicas de Monte Carlo para confirmar os resultados de grupo de renormalização. As simulações numéricas indicam a validade do critério de relevância de Harris-Luck, e corroboram o caráter universal do comportamento crítico desses sistemas aperiódicos.
We use an operator formalism and the renormalization-group technique of Dasgupta, Ma and Hu to analyze the effects of a nonhomogeneous distribution of parameters on the critical behavior of simple stochastic model system. The contact process in one dimension is perhaps the simplest model to display a phase transition to an absorbing stationary state. We use the Fibonacci, period-doubling and period-tripling sequences for introducing aperiodic inhomogeneities in the one dimensional contact process and in a quantum Ising chain. Using strong-disorder renormalization-group procedures, we establish some relations between properties of renormalized operator and of thermodynamic or mean quantities. We were able to test a well-known criterion of relevance of geometric fluctuations, to obtain a number of critical exponents, and to point out features of slow-dynamics and log-periodic oscillations. The period-tripling sequence leads to the critical exponents = ln (7/9)/ ln (4/9), = ln (9/7)/ ln 4, = ln 3/ ln (3/2) and k = ln 2/ ln (3/2). We then used Monte Carlo techniques to check renormalization-group results. The numerical simulations indicate the validity of the Harris-Luck criterion of relevance of the geometric fluctuations, and generally support the universal character of the critical behavior of these aperiodic systems.

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Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
In this thesis, we are dedicated to study the time evolution generated by the hamiltonian of a classical Yang-Mills-Higgs theory with gauge symmetry SU(2) on a spatial lattice. In particular, we study energy transfer and equilibration processes among the gauge and Higgs sectors, calculate the maximal Liapunov exponents regarding to random initial conditions in the regime of weak coupling, where one expects them to be related to the high-temperature static plasmon damping rate, and investigate their energy and Higgs self-coupling parameter dependence. We further examine finite-time and finite-size errors, value the impact of the Higgs fields on the instabilty of constant non-abelian magnetic fields and comment on the implications of our obtained results for the thermalization properties of gauge fields at finite temperature in the presence of matter.

Efeitos de solventes tem sido um tema de grande interesse científco. Em particular, o estudo dos efeitos de solventes no espectro eletrônico de absorção tem sua própria motivação e complexidade. Neste trabalho, nós estudamos os efeitos da solução aquosa na estabilidade conformacional e no espectro eletrônico de absorção da molécula Óxido Mesitil (OM). Essa molécula pertence a família das cetonas ,-insaturadas e, semelhantemente aos outros membros da família, ela apresenta transições eletrônicas sensíveis ao solvente. Inicialmente, estudamos os isômeros syn e anti do OM isoladamente usando cálculos quânticos para determinar a energia livre relativa, a barreira de rotação, os momentos de dipolo e as transições eletrônicas de absorção. Nosso melhor resultado mostra que o isômero syn do OM é a conformação mais estável, por cerca de 1.3 kcal/- mol calculado com nível MP2/aug-cc-pVDZ. Com o mesmo nível de cálculo, obtivemos os momentos de dipolo de 2.80 e 3.97 D para os isômeros syn e anti respectivamente, que estão em boa concordância com os valores experimentais de 2.8 e 3.7 D. Para o espectro eletrônico de absorção, analisamos a banda mais intensa, -*, com diferentes funcionais de densidade e funções base. Obtivemos o comprimento de transição de 229 nm calculado com nível TD-B3LYP/6-311++G** para o isômero syn em muito boa concordância com o valor experimental de 231 nm medido em solução de iso-octano (solvente de baixa polaridade). Para realizar os estudos em solução, geramos estruturas supermoleculares dos isômeros do OM em solução aquosa usando simulações computacionais com o método Monte Carlo. Usamos os potenciais Lennard-Jones e Coulomb para descrever as interações intermoleculares com os parâmetros do campo de força OPLS. Verifcamos que as cargas atômicas OPLS não descrevem bem o potencial eletrostático do OM. Portanto, realizamos um processo iterativo para incluir a polarização do soluto na presença do solvente para descrever melhor as interações entre o OM e as moléculas de água. Assim, obtivemos um aumento de cerca de 80% nos momentos de dipolo dos isômeros isolados. Adicionalmente, calculamos a energia livre relativa entre os isômeros em solução aquosa usando teoria de perturbação termodinâmica. Obtivemos que o isômero anti do OM é a conformação mais estável, por cerca de 2.8 kcal/mol. Examinando os efeitos de solvente no espectro eletrônico de absorção do OM, identificamos que existem duas contribuições competindo para o deslocamento da banda -*. Uma contribuição vem da mudança conformacional syn anti do OM devido a mudança de polaridade, baixa alta, do solvente. Essa mudança conformacional provoca um deslocamento para o azul de 1210 cm-1 na transição -*. A outra contribuição vem do efeito do solvente na estrutura eletrônica do OM, que provoca um deslocamento para o vermelho de - 4460 cm-1 nessa transição. Adicionando essas duas contribuições, temos o efeito do solvente total no espectro eletrônico de absorção do OM em solução aquosa. Nosso melhor resultado é um valor médio de 248 nm obtido com 75 cálculos TD-B3LYP/6-311++G** de estruturas supermoleculares estatisticamente descorrelacionadas compostas por um anti-OM rodeado por 14 moléculas de água explícitas embebidas no campo eletrostático de 236 moléculas de água tratadas como cargas pontuais simples. Esse resultado está em muito boa concordância com o resultado experimental de 243 nm em solução aquosa. Sendo assim, este trabalho demonstra que a mudança conformacional syn anti é essencial para entender o deslocamento espectral da transição -* do OM em água.
Solvent effects have been the subject of considerable scientifc interest. In particular, the study of solvent effects in electronic absorption spectroscopy has its own motivation and complexities. In this work we study the effects of the aqueous solution in the conformational stability and the electronic absorption spectrum of the Mesityl Oxide (OM) molecule. This molecule belongs to the family of the ,-unsaturated ketones and, like other members of the family, presents sensitivity to solvent in the absorption transitions. Initially we studied the isolated syn and anti isomers of OM by performing quantum mechanical calculations to obtain the relative free energy, the rotational barrier, the dipole moments and the electronic absorption transitions. Our best result showed that the OM syn isomer is the most stable conformer, by approximately 1.3 kcal/mol calculated with the MP2/aug-cc-pVDZ level. With the same level of calculation, we obtained the dipole moments of 2.80 and 3.97 D for the syn and anti isomers respectively, which are in good agreement with the experimental values of 2.8 and 3.7 D. For the electronic absorption spectrum, we analyzed the most intense band, -*, with different density functional and basis sets. We obtained a transition wavelength of 229 nm calculated with TD-B3LYP/6-311++G** level for the syn isomer in good agreement with the experimental value of 231 nm measured in iso-octane (solvent of low polarity). For performing the in-solution studies, we generated supermolecular structures of the OM isomers in aqueous solution using computer simulations with the Monte Carlo method. We used the Lennard-Jones and Coulomb potentials to describe the intermolecular interactions with the OPLS force field parameters. We found that the OPLS atomic charges do not describe well the electrostatic potential of OM. Therefore we performed an iterative process for including the solute polarization in the presence of the solvent to better describe the interactions between the OM and the water molecules. We obtained an increase of about 80% in the dipole moments of the isolated isomers. Additionally, we calculated the relative free energy between the isomers in aqueous solution using thermodynamic perturbation theory. We found that the anti isomer is the most stable conformer in aqueous solution, by about 2.8 kcal/mol. Examining the solvent effects in the electronic absorption spectrum of OM, we found that there are two competing contributions to the -* band shift. One contribution is due to the syn anti conformational change of OM caused by the low high polarity change of the solvent. This conformational change led to a blue shift of 1210 cm-1 in the * band. The remaining contribution is due to the solvent effect in the electronic structure of OM, which led to a red shift of -4460 cm-1. Adding these two contributions, we obtained the total solvent effect in the electronic absorption spectrum of OM in aqueous solution. Our best result was an average wavelength transition of 248 nm obtained using 75 TD-B3LYP/6-311++G** quantum calculations on statistically uncorrelated supermolecular structures composed by one anti-OM surrounded by 14 explicit water molecules in the electrostatic embedding composed of 236 water molecules described as simple point charges. This result is in very good agreement with the experimental result of 243 nm in aqueous solution. Thus, this work demonstrates that the syn anti conformational change is the essential ingredient to understand the spectral shift of the - * absorption transition of OM in water.

Investigamos as propriedades de baixa temperatura do modelo de Heisenberg unidimensional de spin 1 desordenado, com flutuações geométricas nos acoplamentos induzidas por sequências aperiódicas e determinísticas. Escolhemos duas sequências com propriedades distintas, a sequência de Fibonacci e a sequência 6-3. Nosso objetivo é entender como essas flutuações geométricas modificam a física da fase de Haldane, que corresponde ao estado fundamental da cadeia de spin 1 uniforme. Introduzimos deferentes adaptações do grupo de renormalização de desordem forte (SDRG) de Ma, Dasgupta e Hu, que tem sido amplamente usado no estudo de cadeias de spin aleatórias. Usamos ainda simulações numéricas de Monte Carlo quântico e do grupo de renormalização da matriz densidade para confirmar as previsões do SDRG, assim como estudar as propriedades do estado fundamental, conforme a modulação se torna mais forte. Não encontramos uma transição de fase para a cadeia modulada pela sequência de Fibonacci, enquanto para a cadeia modulada pela sequência 6-3 encontramos uma transição para uma fase sem gap, dominada pela aperiodicidade, similar àquela encontrada na cadeia XXZ de spin 1/2. Mostramos que abordagens que preveem o mesmo comportamento qualitativo na cadeia aleatória de spin 1 podem levar a previsões qualitativamente incompatíveis na cadeia aperiódica.
We investigate the low-temperature properties of the one-dimensional spin-1 Heisenberg model with geometric fluctuations induced by aperiodic but deterministic coupling distributions, involving two parameters. We focus on two aperiodic sequences, the Fibonacci sequence and the 6-3 sequence. Our goal is to understand how these geometric fluctuations modify the physics of the (gapped) Haldane phase, which corresponds to the ground state of the uniform spin-1 chain. We utilize different adaptations of the strong-disorder renormalization-group (SDRG) scheme of Ma, Dasgupta and Hu, widely employed in the study of random spin chains, supplemented by quantum Monte Carlo and density-matrix renormalization-group numerical calculations, to study the nature of the ground state as the coupling modulation is increased. We find no phase transition for the Fibonacci chain, while we show that the 6-3 chain exhibits a phase transition to a gapless, aperiodicity-dominated phase similar to that found for the aperiodic spin-1/2 XXZ chain. We show that approaches providing the same qualitative behaviour in the random spin-1 chain may lead to qualitatively incompatible predictions in aperiodic chain.

Construímos um arranjo experimental para o estudo de ressonadores acústicos, que tem sido considerados como análogos clássicos de bilhares quânticos. O equipamento mantém estabilidade durante vários dias, o que é uma condição necessária para a obtenção de espectros de autofreqüências com a resolução requerida para a caracterização precisa destes sistemas. Caracterizamos 7 amostras, que são placas de alumínio com espessura < 2 mm e que possuem as seguintes geometrias: dois estádios de Sinai, com e sem dessimetrização planar; três triângulos sendo um equilátero, um retângulo e outro escaleno, este com todos os ângulos agudos e irracionais em unidades de ; além de duas amostras circulares, com e sem dessimetrização planar. Observamos que três amostras apresentam estatísticas GOE, uma 2GOE, uma semi-Poisson, uma Poisson com perda de níveis, e outra aparentemente intermediária entre a GOE e a 2GOE, que nao foi possível classificar. A qualidade dos dados também permitiu a obtenção das energias dos espectros, onde obtivemos resultados coerentes com a classificação a
We have built an experimental apparatus to study acoustic resonators which have been considered as classical analogs of quantum billiards. The equipment was able to keep the stability during several days, which is a requirement to the precise eigenfrequency measurements allowing a characterization of the systems. We have characterized 7 samples made of aluminum plates with thickness smaller than 2 mm having the following geometries: two Sinai\'s stadiums (with and without planar symmetry), an equilateral triangle, a rectangle triangle, and a scalene triangle with three acute and irrational angles, and two circular shaped samples, with and without planar symmetry. We observed that three of the samples followed the GOE statistics (the asymmetrical Sinai stadium, the rectangle triangle and the scalene one). The asymmetrical Sinai stadium was described by 2GOE statistics, the equilateral triangle by the semi-Poisson, the symmetrical circle by a Poisson with missing levels and the asymmetrical circle has statistics apparently between 1GOE and 2GOE which was not possible to classify. The high quality of data allowed us to calculate the spectra energies and we found these results compatible with the previous one.

Neste trabalho realizo uma adaptação do método de Ma, Dasgupta e Hu para o estudo e caracterização das transições de fase quânticas, induzidas por um campo transverso, em cadeias XY de spins 1/2, unidimensionais e aperiódicas, no espírito da adaptação correspondente para cadeias XXZ. O presente trabalho determina de forma analítica uma série de expoentes críticos associados às transições ferro-paramagnéticas do sistema, e dá pistas quanto à natureza das estruturas presentes no estado fundamental. Os resultados são então testados pelo emprego da técnica de férmions livres, da análise de nite size scaling e, no limite de Ising, de resultados extraídos do mapeamento do problema em uma caminhada aleatória.
We employ an adaptation of the Ma, Dasgupta, Hu method in order to analyze the quantum phase transition, induced by a transversal magnetic eld, at spin-1/2 aperiodic XY chains, in analogy to the corresponding adaptation for XXZ chains. We derive analytical expressions for some cri tical exponents related with the ferro-paramagnetic transitions, and shed light onto the nature of the ground state structures. The main results obtained by this approach were tested by the free-fermion method, nite-size scaling analyses and, at the Ising limit of the model, by using results derived from a mapping to a random-walk problem.

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Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior
Considering a quantum gas, the foundations of standard thermostatistics are investigated in the context of non-Gaussian statistical mechanics introduced by Tsallis and Kaniadakis. The new formalism is based on the following generalizations: i) Maxwell- Boltzmann-Gibbs entropy and ii) deduction of H-theorem. Based on this investigation, we calculate a new entropy using a generalization of combinatorial analysis based on two different methods of counting. The basic ingredients used in the H-theorem were: a generalized quantum entropy and a generalization of collisional term of Boltzmann equation. The power law distributions are parameterized by parameters q;, measuring the degree of non-Gaussianity of quantum gas. In the limit q ?1; ?0, the gaussian thermostatistics is recovered. A complementary study is related to a perfect gas in the context of general relativity. Using the non-Gaussian effects on the concept of entropy flux, and on the collisional term of the Boltzmann equation, we generalize the H-theorem within the Tsallis and Kaniadakis frameworks. In the first one, the nonextensive parameter is constrained to the interval [0,2]
Considerando um g?s qu?ntico, os fundamentos da termoestat?stica padr?o s?o investigados no contexto da mec?nica estat?stica n?o-gaussiana introduzida por Tsallis e Kaniadakis. O novo formalismo ? baseado nas seguintes generaliza??es: i) entropia de Maxwell-Boltzmann-Gibbs e ii) dedu??o do Teorema-H. Com base neste estudo, calculamos uma nova entropia usando a generaliza??o da an?lise combinat?ria baseadas em dois diferentes m?todos de contagem. Os ingredientes b?sicos usados no teorema-H foram: uma entropia qu?ntica generalizada e uma generaliza??o do termo colisional da equa??o de Boltzmann. As distribui??es lei de pot?ncia calculadas s?o parametrizadas pelos par?metros q; , medindo o grau de n?o-gaussianidade do sistema. No limite q ?1; ?0, a termoestat?stica gaussiana ? recuperada. Um estudo complementar est? relacionado com um g?s perfeito no contexto da relatividade geral. Utilizando os efeitos n?o-gaussiano no conceito de fluxo de entropia, e no termo colisional da equa??o de transporte de Boltzmann, n?s generalizamos o teorema- H nos formalismos de Tsallis e Kaniadakis. No formalismo de Tsallis, o par?metro n?o extensivo est? restrito ao intervalo [0,2]

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
We investigate here the low-energy properties of two strongly correlated electronic models in two spatial dimensions. The first one consists in a version of the Hubbard model in which are considered just the degrees of freedom of the system in the neighborhood of the so-called hot spots, which are defined as the intersection of the Fermi surface of the model with the antiferromagnetic zone. Initially, we set our theory up by linearizing the dispersion model in hot spots and consider all the interacting processes between these regions that conserve momentum within a reciprocal-lattice wave vector. In order to access the physics of the model, we then turn to the renormalization group method of quantum field theory and derive the flow equations for the couplings in the two-loop approximation. As a result, we obtain that the Fermi surface is strongly renormalized in hot spots as the renormalized couplings flow to a non-trivial fixed point in the low-energy limit. Then we suggest that this system can be viewed as an example of a non-Fermi liquid in two spatial dimensions, due to the lack of well defined quasiparticle fermionic excitations in the region close to hot spots. Moreover, we solve the Callan-Symanzik equation for the oneparticle Green function up to two-loop order, calculate the density of states in the hot spots, and derive the renormalization group equations for the order parameters of the potential instabilities which may eventually occur in the system at lower energies. We verify that the system can be characterized, in this regime, in terms of an emergent pseudospin symmetry [SU(2)]4, which leads to the appearance of entangled orders in the region close to the non-trivial fixed point of the model. We also show that the fermionic excitations in the adjacent regions to the hot spots get a gap in both charge a spin excitation spectra. Because of this, we argue that the Fermi surface of the model can be reconstructed, leading therefore to the formation of either Fermi arcs or electronic pockets. The second model analyzed in this thesis was the three-band Emery model, which describes all the interacting processes between fermionic excitations localized in both copper (Cu) and oxygen (O) orbitals in the CuO2 unit cell. By making use of a Hubbard-Stratonovich transformation, we introduce two order parameters in the system: one for the so-called ΘII-loop-current order, which violates Z2 time-reversal symmetry, and another one for the entangled phase with dx 2 -y 2 symmetry involving the singlet superconducting instability and the quadrupole density wave order, whose wave vector points in the direction of the Brillouin zone diagonal. Minimizing the free energy of the model, we derive the self-consistent mean-field equations for these order parameters. The solution of these equations for the zero temperature regime shows that the two phases compete with themselves for the same region of the phase space and, consequently, the system tends not to display coexistence between them. We argue that this effect could be the main reason for the fact that the quadrupole density wave order has never been observed in experiments performed on the cuprate superconductors. Next, we analyze the competition between the ΘII-loop-current order, which is experimentally observed, and charge order with dx 2 -y 2 symmetry and wave vectors in the direction of the main axes of the Brillouin zone. As a result, we obtain that the system only exhibits coexistence between the ΘII-loop-current phase and the bidirectional charge order. Due to the existence of a pseudospin symmetry in this model, we also confirm that the ΘII-loop-current phase coexists with the bidirectional pair density wave order. Finally, we discuss the implications of these results for the pseudogap phase of the cuprate superconductors, which appears in the underdoped regime in these systems.
Investigamos aqui as propriedades de baixa energia de dois modelos eletrônicos fortemente correlacionados em duas dimensões espaciais. O primeiro deles consiste em uma versão do modelo de Hubbard em que são considerados apenas os graus de liberdade do sistema na vizinhança dos chamados hot spots, que são definidos como a intersecção da superfície de Fermi do modelo com a zona antiferromagnética. Inicialmente, definimos a nossa teoria linearizando a dispersão do modelo nos hot spots e consideramos todos os processos de interação entre essas regiões que conservam momento a menos de um vetor da rede recíproca. Para acessar a física do modelo, recorremos então ao método de grupo de renormalização de teoria de campos e derivamos as equações de fluxo para os acoplamentos na aproximação de dois loops. Como resultado, obtemos que a superfície de Fermi do modelo sofre forte renormalização nos hot spots, ao mesmo tempo que os acoplamentos renormalizados fluem para um ponto fixo não trivial no limite de baixa energia. Sugerimos então que esse sistema pode ser visto como um exemplo de um líquido de não-Fermi em duas dimensões espaciais, devido à ausência de excitações fermiônicas do tipo quasipartícula bem definidas na região próxima aos hot spots. Além disso, resolvemos a equação de Callan- Symanzik para a função de Green de uma partícula na aproximação de dois loops, calculamos a densidade de estados nos hot spots, e derivamos as equações de grupo de renormalização para os parâmetros de ordem das possíveis instabilidades que podem, eventualmente, ocorrer no sistema em baixas energias. Verificamos que o sistema pode ser caracterizado, nesse regime, em termos de uma simetria emergente de pseudospin [SU(2)]4, que leva ao aparecimento de ordens emaranhadas na região próxima ao ponto fixo não trivial do modelo. Mostramos também que as excitações fermiônicas nas regiões adjacentes aos hot spots adquirem um gap nos espectros de excitação de carga e spin. Devido a isso, argumentamos que a superfície de Fermi do modelo pode ser reconstruída, levando assim à formação de arcos de Fermi ou pockets eletrônicos. O segundo modelo analisado nesta tese foi o modelo de três bandas de Emery, que descreve todos processos de interação entre as excitações fermiônicas localizadas nos orbitais do cobre (Cu) e do oxigênio (O) na célula unitária de CuO2. Através de uma transformada de Hubbard-Stratonovich, introduzimos dois parâmetros de ordem no sistema: um para a chamada fase de corrente de loop do tipo ΘII, que viola a simetria de reversão temporal Z2, e outro para a fase emaranhada com simetria dx 2 -y 2 envolvendo a instabilidade supercondutora do tipo singleto e a ordem de densidade de carga quadrupolar, cujo vetor de onda aponta na direção da diagonal da zona de Brillouin. Minimizando a energia livre do modelo, derivamos as equações auto-consistentes de campo médio para esses parâmetros de ordem. A solução dessas equações para o regime de temperatura nula mostra que as duas fases competem entre si pela mesma região do espaço de fase e, consequentemente, o sistema tende a não exibir coexistência entre as mesmas. Argumentamos que esse efeito pode ser a principal razão para o fato de a fase onda de densidade quadrupolar nunca ter sido observada em experimentos realizados nos cupratos supercondutores. Em seguida, analisamos a competição entre as fases de corrente de loop do tipo ΘII, observada experimentalmente, e ordem de carga com simetria dx2-y2 e vetores de onda na direção dos eixos principais da zona de Brillouin. Como resultado, obtemos que o sistema exibe coexistência apenas entre as fases de corrente de loop do tipo ΘII e ordem de carga bidirecional. Devido à existência de uma simetria de pseudospin nesse modelo, confirmamos também que a fase de corrente de loop do tipo ΘII coexiste com a fase onda de densidade de pares bidirecional. Por fim, discutimos as implicações dos nossos resultados para a fase de pseudogap dos cupratos supercondutores, que emerge no chamado regime subdopado nesses sistemas.

Este trabalho consiste em um estudo teórico da coexistência de condensados de Bose-Einstein atômicos acoplados. Esta mistura de condensados pode consistir em átomos de um mesmo elemento em vários estados hiperfinos (estado interno). Outra situação em que esta mistura ocorre é quando um átomo em um único estado interno pode ocupar mais de um estados externo. As semelhanças e diferenças entre condensados de Bose-Einstein com vários estados internos e externos serão analisadas cuidadosamente. Somente na aproximação de Bose-Hubbard os estados externos serão tratados como internos. Condensados envolvendo duas ou mais espécies serão estudados com a aproximação de Thomas-Fermi com estados coerentes. Esta aproximação implica em desprezar o termo de energia cinética. Muitos resultados analíticos serão exibidos para a condição em que todos os comprimentos de espalhamento são iguais entre si. Esta condição é bastante próxima da observada em um condensado de vários estados internos. Além disso serão apresentados alguns resultados anaíticos onde o comprimento de espalhamento para espécies diferentes é zero. Esta condição é verificada para a aproximação de Bose-Hubbard para misturas de condensados com estados externos. O objetivo principal destes cálculos é estudar o papel da fase relativa entre as funções de onda nas soluções estacionárias e na evolução temporal destes sistemas.
This work consists of a theoretical study of the coexistance of coupled atomic Bose-Einstein condensates. These mixed condensates can consist of atoms of the same element in different hyperfine internal states. A different situation is that in which an atom in one internal state can ocupate different single particle external states. The similarities and differences between mixed Bose-Einstein condensates involving internal and external states will be analysed carefully. Only in the Bose-Hubbard approximation the external states can be treated like internal states. Condensates involving two or more species will be studied with in a Thomas- Fermi\'s approximation with coherent states. This approximation involves in discarding of the Hamiltonian the kinetic energy term. A number of analytical results are given for the case in which the different channel scatering lenghts are equal to each other. This condition is which well approximated in condensates involving many internal states. Beside some results will also be given for situations where the scatering lenghts for different species vanishes. This condiction is verified for the Bose-Hubbard approximation to externally mixed condensates. The principal goal of this calculations is to study the role of relative phase between the wave functions in the stationary solutions and in the time evolution of the systems.

Sólitons são soluções clássicas de equações de campos não lineares, que possuem energia finita e densidade de energia localizada. Eles constituem pacotes de energia que se movem de maneira uniforme e não dispersiva, assemelhando-se a partículas estendidas. Quando se estuda um sistema à temperatura finita é possível tecer um paralelo entre a teoria quântica de campos e a mecânica estatística. Neste trabalho calculamos, na aproximação de um laço, a correção quântica à massa do kink do modelo 4 acoplado a um campo fermiônico. As contribuições bosônica e fermiônica são calculadas à temperatura zero e o comportamento das flutuações a temperatura finita também é analisado.
Solitons are classical solutions of non-linear field equations, that have finite energy and localised energy density. They constitute non-dispersive localised packages of energy moving uniformly, resembling extended particles. When studying a system at finite temperature one can make an analogy between quantum field theory and statistical mechanics. In this work we calculate, in one loop approximation, the quantum correction to the mass of the kink of the model 4 coupled to a fermionic field. The bosonic and fermionic contributions are calculated at zero temperature and the behavior of the finite temperature fluctuations are also analysed.

Condensed matter systems undergoing phase transitions rarely allow exact solutions. The presence of disorder renders the situation  even worse but collective Monte Carlo methods and parallel algorithms allow numerical descriptions. This thesis considers classical phase transitions in disordered spin systems in general and in effective models of superfluids with disorder and novel interactions in particular. Quantum phase transitions are considered via a quantum to classical mapping. Central questions are if the presence of defects changes universal properties and what qualitative implications follow for experiments. Common to the cases considered is that the disorder maps out correlated structures. All results are obtained using large-scale Monte Carlo simulations of effective models capturing the relevant degrees of freedom at the transition. Considering a model system for superflow aided by a defect network, we find that the onset properties are significantly altered compared to the $\lambda$-transition in $^{4}$He. This has qualitative implications on expected experimental signatures in a defect supersolid scenario. For the Bose glass to superfluid quantum phase transition in 2D we determine the quantum correlation time by an anisotropic finite size scaling approach. Without a priori assumptions on critical parameters, we find the critical exponent $z=1.8 \pm 0.05$ contradicting the long standing result $z=d$. Using a 3D effective model for multi-band type-1.5 superconductors we find that these systems possibly feature a strong first order vortex-driven phase transition. Despite its short-range nature details of the interaction are shown to play an important role. Phase transitions in disordered spin models exposed to correlated defect structures obtained via rapid quenches of critical loop and spin models are investigated. On long length scales the correlations are shown to decay algebraically. The decay exponents are expressed through known critical exponents of the disorder generating models. For cases where the disorder correlations imply the existence of a new long-range-disorder fixed point we determine the critical exponents of the disordered systems via finite size scaling methods of Monte Carlo data and find good agreement with theoretical expectations.

QC 20150306

As propriedades do estado fundamental do modelo de Heisenberg antiferroinagnético quântico de spin-1/2 na rede quadrada e na rede cúbica espacialmente anisotrópica são investigadas através de um novo método de Monte Carlo, baseado na estimativa do maior autovalor de uma matriz de elementos não negativos. A energia do estado fundamental e a magnetização \"staggered\" destes sistemas são calculadas em redes relativamente grandes com até 24 x 24 sítios para o caso de redes quadradas e 8 x 8 x 8 sítios para o caso de redes cúbicas. O método desenvolvido também pode ser usado como um novo algoritmo para a determinação direta da entropia de sistemas de spins de Ising através de simulações usuais de Monte Carlo. Usando este método, calculamos a entropia do antiferromagneto de Ising na presença de um campo magnético externo nas redes triangular e cúbica de face centrada.
The ground state properties of the antiferromagnetic quantum Heisenberg model with spin-112 defined on a square lattice and on a cubic lattice with spatial anisotropy are investigated through a new Monte Carlo method, based on the estimation of the largest eigenvalue of a matrix with nonnegative elements. The ground state energy and the staggered magnetization of these systems are calculated in relatively large lattices with up to 24 x 24 sites for the square lattices and 8 x 8 x 8 sites for cubic lattices. The method developped can also be used as a new algorithm for the direct determination of the entropy of Ising spin systems through ordinary Monte Car10 simulations. By using this method we calculate the entropy of the Ising antiferromagnetic in the presence of a magnetic field in the triangular and face centered cubic lattices.