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.

Le sujet de thèse est une étude théorique des modes normaux d’oscillations plasma dans des chaînes de jonctions Josephson supra-conductrices. Les propriétés de ces modes normaux peuvent être contrôlés en choisissant une modulation spatiale appropriée de paramètres des jonctions le long de la chaîne et/ou un couplage approprié à l'environnement extérieur. Le travail théorique au sein du LPMMC se fait en étroite collaboration avec l'équipe expérimentale"Cohérence Quantique" à l'Institut Néel. Les problèmes spécifiques étudiés dans la thèse sont : modélisation détaillée du couplage des modes normaux à l'environnement pour leur caractérisation dans une expérience de transmission de micro-ondes, dissipation intrinsèque des oscillations du plasma à cause de quasi-particules hors équilibre, l'optimisation de la structure spatiale de la chaîne de jonctions Josephson pour son utilisation en tant qu'une super-inductance. / The subject of thesis is a theorerical study of normal modes of plasma oscillations in superconducting Josephson junction chains. The properties of these normal modes can be controlled by choosing an appropriate spatial modulation of the junction parameters along the chain and/or an appropriate coupling to the external environment. The theoretical work at LPMMC is performed in a close collaboration with the experimental Quantum Coherence group at Néel Institute. The specific problems studied in this thesis are : detailed modeling of the normal mode coupling to the environment for probing them in a microwave transmission experiment, intrinsic dissipation of plasma oscillations due to the presence of non-equilibrium quasi-particles, optimization of the spatial structure of the Josephson junction chain for its use as a super-inductance.

Ce manuscrit présente l’étude de transitions de phase quantiques dans l’oxyde BaCo2V2O8, un système antiferromagnétique quasi-unidimensionnel constitué de chaînes d’ions cobalt portant un spin effectif S = 1/2 caractérisé par une forte anisotropie de type Ising. Lors de ce travail, nous avons étudié les propriétés statiques et dynamiques de BaCo2V2O8 sous l’effet de différents paramètres physiques.Notre première étude a porté sur l’effet d’un champ magnétique transverse, i.e. appliqué perpendiculairement à l’axe Ising. Il a été proposé que lors de l’application d’un tel champ, un champ magnétique alterné effectif est induit perpendiculairement à l’axe d’anisotropie et au champ uniforme appliqué. La comparaison d'expériences de diffusion (élastique et inélastique) de neutrons et de calculs numériques nous a permis de montrer que ce champ alterné entre en compétition avec l’anisotropie. Ceci aboutit à une transition de phase originale, dite topologique, que l'on peut modéliser par une théorie quantique des champs nommée « modèle de double sine-Gordon » qui décrit la compétition entre deux excitations topologiques duales. Nous avons pu montrer que BaCo2V2O8 sous champ magnétique transverse était la première réalisation d'une telle théorie.La seconde étude était consacrée à BaCo2V2O8 sous champ magnétique longitudinal, i.e. un champ appliqué parallèlement à l’axe Ising. La dynamique de spins a été sondée grâce à la diffusion inélastique de neutrons et nous avons montré qu’au-dessus d’un champ critique de 4 T, celle-ci semble en accord avec le spectre des fluctuations de spin attendu pour un liquide de Tomonaga Luttinger (TLL). De plus, les calculs numériques ont confirmé que, du fait de l’anisotropie de type Ising dans ce système, la majorité du poids spectral du spectre en énergie est porté par les fluctuations de spins de type longitudinales. Ce résultat est la signature d'un comportement quantique sans analogue classique avec des fluctuations de basses énergies essentiellement longitudinales pilotant la physique du système. Enfin, c’est la première fois que la dynamique de spin dans des chaînes de type Ising a pu être sondée dans cette phase TLL.Les deux dernières études sont préliminaires. Le diagramme de phase de BaCo2V2O8 a été sondé par des mesures calorimétriques sous l’application d’une pression hydrostatique et d’un champ magnétique longitudinal. Afin d’obtenir des pressions allant jusqu’à 10 GPa, nous avons utilisé une cellule à enclumes de diamant. Nous avons effectué des mesures de chaleur spécifique qui nous ont permis de sonder l'effet de la pression sur le Hamiltonien de BaCo2V2O8 au travers de son diagramme de phase $(H, P, T)$. Enfin, nous avons étudié l’effet de la substitution des ions magnétiques Co2+ par des impuretés non-magnétiques Mg2+. Les expériences de diffraction neutronique sous champ longitudinal ont montré que la température et le champ critiques diminuent proportionnellement à la concentration en impuretés. La dynamique de spins à champ magnétique nul a aussi été sondée et révèle l’apparition de modes non-dispersifs, provenant possiblement de l’effet de segmentation des chaînes par les impuretés.En conclusion, nos études expérimentales couplées à des calculs numériques nous ont permis de dévoiler une physique extrêmement riche dans ce composé modèle pour l'étude du magnétisme quantique et des transitions de phase quantiques. / This manuscript is devoted to the study of quantum phase transitions in the BaCo2V2O8 oxide, a quasi-one dimensional antiferromagnet consisting of spin chains of cobalt magnetic ions carrying an effective spin S = 1/2 showing a strong Ising-like anisotropy. To achieve this, we have studied BaCo2V2O8 under the effect of different physical parameters.Our first study concerned the effect of a transverse magnetic field, i.e. applied perpendicularly to the Ising axis. It has been shown that when BaCo2V2O8 is subjected to such a field, an effective staggered magnetic field is induced perpendicularly to both the Ising-axis and the uniform applied field. Using neutron scattering experiments (both elastic and inelastic) compared to numerical calculations, we have proved that this staggered field competes with the Ising-like anisotropy. This leads to a very original quantum phase transition. Our system can actually be mapped onto a quantum field theory called “double sine-Gordon model”, describing the competition between two dual topological excitations. We have thus shown that BaCo2V2O8 under a transverse magnetic field is the first experimental realization of such a theory.The second study was devoted to the effect of a longitudinal magnetic field, i.e. a field applied parallel to the Ising-axis. The spin-dynamics have been investigated by means of inelastic neutron scattering experiments and it has been shown that above a critical field of 4 T, it corresponds to the one expected for a Tomonaga Luttinger liquid phase (TLL). Moreover, numerical calculations have shown that, because of the Ising-like anisotropy in this system, the majority of the spectral weight in the energy spectrum is carried by longitudinal spin fluctuations. This result is the signature of a quantum behavior without classical analogous with low energy longitudinal fluctuations driving the physics of the system. Finally, this is the first time that the dispersion spectrum for an Ising-like spin 1/2 chain could be probed in this TLL phase.The last two studies are preliminary work. The phase diagram of BaCo2V2O8 has been probed by calorimetric measurements under pressure and under a longitudinal magnetic field. Pressures up to 10 GPa have been obtained using a diamond anvil cell. We have then performed specific heat measurements allowing us to investigate the effect of pressure on the Hamiltonian of BaCo2V2O8 through its (H, P, T) phase diagram. Finally, we have also started to study the effect of the substitution of magnetic ions Co2+ by non-magnetic impurities Mg2+. The neutron diffraction experiments under a longitudinal magnetic field have shown that the critical temperature and critical field decrease proportionally to the concentration of impurities. The spin-dynamics at zero-field has also been investigated and reveals the appearance of non-dispersive magnetic modes, which possibly comes from the finite size effect of the spin chains segmented by the non-magnetic impurities.In conclusion, our experimental studies associated to numerical calculations allowed us to unveil a very rich physics in this model compound for the study of quantum magnetism and quantum phase transitions.

Orientador: Marcos Cesar de Oliveira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-09-24T17:07:35Z (GMT). No. of bitstreams: 1 Oliveira_ThiagoRodriguesde_D.pdf: 2410853 bytes, checksum: 48f52d2d48ef1be2ecb155881b8e16df (MD5) Previous issue date: 2008 / Resumo: Esta dissertação tenta contribuir ao entendimento das possíveis interconexões entre a Teoria de Informação Quântica e Matéria Condensada, um novo campo de pesquisa em amplo desenvolvimento. Mais especificamente, investigamos o papel do emaranhamento, ou correlações quânticas, em transições de fase quânticas contínuas. Enquanto o papel do primeiro na Teoria de Informação dispensa apresentação, as últimas são de grande interesse por exibir um comportamento universal, o qual se origina na divergência de um comprimento de correlação. É esta origem mútua em correlações de ambos os fenômenos que cria uma expectativa de uma possível relação entre estes. Nosso trabalho, embasado no estudo do modelo XY unidimensional em um campo transverso, aponta evidências de um favorecimento do emaranhamento multipartite em detrimento do bipartite na transição, e assim da importância do primeiro no estabelecimento de correlações de longo alcance. Nessa tarefa, acabamos por definir uma classe de medidas de emaranhamento multipartite, generalizando o Emaranhamento Global introduzido por Meyer e Wallach em2002. Mostramos que algumas destas classes provêem informações adicionais à do Emaranhamento Global, além de serem escritas de forma simples em termos de funções de correlação. Tal simplicidade permite o estabelecimento de uma relação formal entre uma dessas classes e transições de fase sinalizadas por divergências na energia. Ao final estudamos o papel da quebra de simetria no emaranhamento bipartite e multipartite, evidenciando, uma vez mais, a maior importância do último em relação ao primeiro. Em uma segunda parte, examinamos o uso de estados de produtos de matrizes na aproximação de estados fundamentais de sistemas críticos. Estes estados podem ser vistos como o ansatz utilizado no Grupo de Renormalização de Matriz Densidade (DMRG), quando este é encarado como um método variacional. Analisando o poder de aproximação de tais estados, agora no modelo de Ising, descobrimos que a "dimensão" do ansatz (ou número de graus de liberdade renormalizados) é uma variável relevante do grupo de renormalização de maneira análoga ao tamanho finito do sistema. Isto possibilita uma análise de escala em relação a essa "dimensão" dos estados de produto de matrizes, com uma possível obtenção de propriedades críticas a baixo custo computacional / Abstract: This thesis attempts to contribute to the understanding of possible connections between Quantum Information and Condensed Matter theories, a new field of research in broad development. Specifically, we investigated the role of entanglement, or quantumcorrelations, in continuous quantum phase transitions. While the importance of the first in the theory of Quantum Information is well known dispense presentation, the latter are of great interest as they exhibit a universal behavior, which descent fromthe divergence of the correlation length. This mutual origin of both in correlations is what creates an expectation of a possible link between them. Our work, based on the study of XY dimensional model in a transverse field, brings evidence of multipartite entanglement being favored, in detriment of bipartite in the transition, and thus in the importance of the first in the establishment of long-range correlations. During our journey, we define a class of measures of multipartite entanglement, generalising the Global Entanglement introduced by Meyer and Wallach in 2002. We show that some of these classes provide additional information to the Global Entanglement, as well as being written in a simple way in terms of correlation functions . This simplicity allows the establishment of a formal relationship between those classes and phases transitions marked by non-analycities in the energy. At the end, we studied the role of spontaneous symmetry breaking in the bipartite and multipartite entanglement, demonstrating once again a major role of the last over the first. In a second part, we examine the use of Matrix Product States to approximate ground states of critical systems. This class of states can be seen as the ansatz used in the Density Matrix Renormalization Group (DMRG), when this one is understood as a variational method. Analyzing the power of approximation of these states, now in Ising model, we found that the "dimension" of the ansatz (or number of renormalized degrees of freedom) is a relevant variable in the renormalization group, in a analogous way to the finite size of the system. This enables an analysis of scaling regarding the "size" of Matrix Product States, with a possible acquisition of critical properties at low computation cost / Doutorado / Física da Matéria Condensada / Doutor em Ciências

The LiHo_xY_{1-x}F_4 magnetic material in a transverse magnetic field B_x perpendicular to the Ising spin direction has long been used to study tunable quantum phase transitions in pure and random disordered systems. We first present analytical and numerical evidences for the validity of an effective spin-1/2 approach to the description of a general dipolar spin glass model with strong uniaxial Ising anisotropy and subject to weak B_x. We relate this toy model to the LiHo_xY_{1-x}F_4 transverse field Ising material. We show that an effective spin-1/2 model is able to capture both the qualitative and quantitative aspects of the physics at small B_x. After confirming the validity of the effective spin-1/2 approach, we show that the field-induced magnetization along the x direction, combined with the local random dilution-induced destruction of crystalline mirror symmetries generates, via the predominant dipolar interactions between Ho^{3+} ions, random fields along the Ising z direction. This identifies LiHo_xY_{1-x}F_4 in B_x as a new random field Ising system. We show that the random fields explain the smearing of the nonlinear susceptibility at the spin glass transition with increasing B_x. In this thesis, we also investigate the phase diagram of non-diluted LiHoF_4 in the presence of B_x, by performing Monte-Carlo simulations. A previous quantum Monte Carlo (QMC) simulation found that even for small B_x where quantum fluctuations are small, close to the classical critical point, there is a discrepancy between experiment and the QMC results. We revisit this problem, focusing on weak B_x close to the classical T_c, using an alternative approach. For small B_x, by applying a so-called cumulant expansion, the quantum fluctuations around the classical T_c are taken into account perturbatively. We derived an effective perturbative classical Hamiltonian, on which MC simulations are performed. With this method we investigate different proposed sources of uncertainty which can affect the numerical results. We fully reproduce the previous QMC results at small B_x. Unfortunately, we find that none of the modifications to the microscopic Hamiltonian that we explore are able to provide a B_x-T phase diagram compatible with the experiments in the small semi-classical B_x regime.

The LiHo_xY_{1-x}F_4 magnetic material in a transverse magnetic field B_x perpendicular to the Ising spin direction has long been used to study tunable quantum phase transitions in pure and random disordered systems. We first present analytical and numerical evidences for the validity of an effective spin-1/2 approach to the description of a general dipolar spin glass model with strong uniaxial Ising anisotropy and subject to weak B_x. We relate this toy model to the LiHo_xY_{1-x}F_4 transverse field Ising material. We show that an effective spin-1/2 model is able to capture both the qualitative and quantitative aspects of the physics at small B_x. After confirming the validity of the effective spin-1/2 approach, we show that the field-induced magnetization along the x direction, combined with the local random dilution-induced destruction of crystalline mirror symmetries generates, via the predominant dipolar interactions between Ho^{3+} ions, random fields along the Ising z direction. This identifies LiHo_xY_{1-x}F_4 in B_x as a new random field Ising system. We show that the random fields explain the smearing of the nonlinear susceptibility at the spin glass transition with increasing B_x. In this thesis, we also investigate the phase diagram of non-diluted LiHoF_4 in the presence of B_x, by performing Monte-Carlo simulations. A previous quantum Monte Carlo (QMC) simulation found that even for small B_x where quantum fluctuations are small, close to the classical critical point, there is a discrepancy between experiment and the QMC results. We revisit this problem, focusing on weak B_x close to the classical T_c, using an alternative approach. For small B_x, by applying a so-called cumulant expansion, the quantum fluctuations around the classical T_c are taken into account perturbatively. We derived an effective perturbative classical Hamiltonian, on which MC simulations are performed. With this method we investigate different proposed sources of uncertainty which can affect the numerical results. We fully reproduce the previous QMC results at small B_x. Unfortunately, we find that none of the modifications to the microscopic Hamiltonian that we explore are able to provide a B_x-T phase diagram compatible with the experiments in the small semi-classical B_x regime.

The vacuum of quantum field theory is not empty space but filled with quantum vacuum fluctuations, which give rise to many intriguing effects. The first part of this Thesis addresses cosmic inflation, where the quantum fluctuations of the inflaton field freeze and get amplified in the expanding universe. Afterwards, we turn our attention towards Bose-Einstein condensates, a laboratory system. Since most of our calculations are performed using a mean-field expansion, we will study the accuracy of a finite-range interaction potential onto such an expansion. Exploiting the universality of quantum fluctuations, several aspects of cosmic inflation will be identified in ballistically expanding Bose-Einstein condensates. The effective action technique for calculating the quantum backreaction will be scrutinized. Finally, we consider dynamic quantum phase transitions in the last part of this Thesis. To this end two specific scenarios will be investigated: firstly, the structure formation during the superfluid to Mott-insulator transition in the Bose-Hubbard model; and secondly, the formation of spin domains as a two-dimensional spin-one Bose gas is quenched from the (polar) paramagnetic to the (planar) ferromagnetic phase. During this quench, the symmetry of the ground state is spontaneously broken and vortices (topological defects) form.

In der vorliegenden Arbeit wurde der elektrische Transport im Schwere-Fermionen-System YbRh2Si2 sowohl in seiner stöchiometrischen Form als auch mit teilweiser isoelektronischer Substitution von Ir oder Co auf dem Rh-Platz untersucht. In YbRh2Si2 liegt ein quantenkritischer Punkt vor, der zugänglich ist, indem der antiferromagnetische Phasenübergang mittels eines kleinen Magnetfelds zum absoluten Nullpunkt der Temperatur unterdrückt wird. Die zentralen Messungen des Hallkoeffizienten zeigen einen Übergang der in der Extrapolation zu T=0 zu einer Diskontinuität wird und somit auf eine Rekonstruktion der Fermifläche am quantenkritischen Punkt schließen lässt. Dies belegt die unkonventionelle Natur der Quantenkritikalität in YbRh2Si2. Unterstützt wird dies auf fundamentale Weise durch verknüpfungen mit unkonventionellem Skalierungsverhalten. In den Proben mit teilweiser Substitution wird der Einfluss einer Veränderung der Gitterparameter auf die Quantenkritikalität mit Hilfe von Widerstandsmessungen untersucht. Dabei zeigt sich, dass der magnetische Übergang von der Fermiflächenrekonstruktion separiert wird. Für Proben mit teilweiser Ir-Substitution, welche negativen Drücken entspricht, scheint im Zwischenbereich eine neuartige metallische Spinflüssigkeit hervorzutreten. / This work investigates the electrical transport of the heavy-fermion compound YbRh2Si2 in its stoichiometric form as well as with slight isoelectronic substitution of Ir or Co on the Rh site. A quantum critical point is present in YbRh2Si2 which is accessed by tuning the transition temperature of the antiferromagnetic order to absolute zero via the application of a small magnetic field. The central measurements of the Hall coefficient reveal a crossover which sharpens to a discontinuity in the extrapolation to zero temperature implying a reconstruction of the Fermi surface at the quantum critical point. This allows to rule out conventional descriptions of the quantum criticality in YbRh2Si2. A scaling analysis corroborates this on a fundamental basis. In the samples with partial substitution the effect of unit cell volume change on the quantum criticality was investigated by means of resistivity measurements. Surprisingly, the magnetic transition is separated from the Fermi surface reconstruction. For samples with Ir substitution corresponding to negative chemical pressure, a new metallic spin liquid seems to emerge in the intermediate regime.

Nous étudions le modèle O (N) relativiste, une généralisation quantique de la théorie φ⁴ utilisée en physique statistique pour étudier des transitions de phase. Ce modèle décrit certaines transitions de phase quantiques telles que la transition isolant de Mott-superfluide dans un gaz de bosons piégés dans un réseau optique ou la transition paramagnétique-antiferromagnétique dans un aimant. En deux dimensions d’espace, ces systèmes sont fortement corrélés près de la transition. Nous les étudions à l’aide du groupe de renormalisation non perturbatif, une formulation du groupe de renormalisation de Wilson. Nous nous intéressons aux propriétés universelles au voisinage de la transition de phase quantique à température nulle. Ainsi, nous déterminons les fonctions d’échelle universelles qui définissent la thermodynamique et démontrons que ces fonctions sont reliées à celles décrivant la force de Casimir critique dans un système classique tridimensionel. Ensuite, nous étudions le spectre d’excitation dans la phase ordonnée à température nulle. Pour N = 2 et 3, nous établissons l’existence d’un mode d’amplitude aussi appelé « mode de Higgs » par analogie avec le mécanisme de Higgs en physique des hautes énergies. Pour N = 1, nous montrons l’existence d’un état lié pour des dimensions proches de trois. Enfin, nous calculons la dépendance en fréquence de la conductivité à température nulle et confirmons son universalité, en particulier à la transition. Nous établissons que l’une des composantes du tenseur de conductivité dans la phase ordonnée est une quantité « superuniverselle », ne dépendant ni de la distance au point critique ni de N. / We study the relativistic O (N) model, a quantum generalization of the φ⁴ theory used in statistical physics to study some phase transitions. This model describes quantum phase transitions such as the Mott insulator-superluid transition in boson gases trapped in optical latices or the paramagnetic-antiferromagnetic transition in magnets. In two space dimensions, these systems exhibit strong correlations near the transition. We study them using the nonperturbative renormalization group, an implementation of Wilson’s renormalization group. We focus on the universal properties in the vicinity of the zero-temperature quantum phase transition. We determine the universal scaling functions which define the thermodynamics and we show that these functions are related to those describing the critical Casimir forces in a three-dimensional system. Then, we study the excitation spectrum in the zero-temperature ordered phase. For N = 2 and 3, we establish the existence of an amplitude mode, also called “Higgs mode” by analogy with the Higgs mechanism in high-energy physics. For N = 1, we show the existence of a bound state at dimensions close to three. Finally, we compute the frequency-dependent conductivity at zero temperature and confirm its universal character, in particular at the transition. We prove that one of the components of the conductivity tensor in the ordered phase is a “superuniversal” quantity depending neither on the distance to the critical point nor on N.

Made available in DSpace on 2016-06-02T20:15:26Z (GMT). No. of bitstreams: 1 4780.pdf: 9997745 bytes, checksum: 24a1f1c6cc3ce4ef4b6ac0cc5009255c (MD5) Previous issue date: 2013-01-11 / Universidade Federal de Sao Carlos / In this thesis, we study the behavior of Quantum Discord in the contexts of open quantum systems and spin chains. Furthermore, we investigate the thermalization process of a spin chain due to interaction with the environment. First, we present a review on the concept of quantum correlation, beginning with the first ideas on non-locality, and leading to the measure of quantum correlations called Quantum Discord. Afterwards, we study the dynamics of the quantum correlations between two non-interacting qubits coupled to Markovian and non- Markovian thermal reservoirs. In the context of spin chains, we investigate the behavior of quantum correlations at finite temperatures, starting with a system composed of two interacting spins, described by XYZ model, in order to generalize this study to the case of infinite unidimensional spin chains, described by XY and XXZ models. In this context, we investigate the relationship between quantum correlations and quantum phase transitions present in these two models. We conclude this thesis with a study of the thermalization process of two interacting spins weakly coupled to independent bosonic thermal reservoirs, or to a single collective reservoir, besides presenting some results for larger systems, composed of an arbitrary number of spins. / Nesta tese estudamos o comportamento da Discórdia Quântica nos contextos de sistemas quânticos abertos e cadeias de spins. Além disso, investigamos também o processo de termalização de uma cadeia de spins sujeita a interação com o meio-ambiente. Primeiramente, apresentamos uma revisão do conceito de correlação quântica, partindo das ideias iniciais sobre não-localidade e tendo como ponto final a medida de correlações quânticas denominada Discórdia Quântica. Posteriormente, estudamos a dinâmica das correlações quânticas entre dois qubits não-interagentes acoplados a reservatórios térmicos markovianos e não-markovianos. No contexto de cadeias de spins, investigamos o comportamento das correlações quânticas a temperaturas finitas, começando com um sistema formado por dois spins interagentes, descrito pelo modelo XYZ para, em seguida, generalizar este estudo para o caso de cadeias de spins unidimensionais infinitas, descritas pelos modelos XY e XXZ. Neste contexto, investigamos a relação entre as correlações quânticas e as transições de fase quânticas presentes nestes dois modelos. Finalizamos esta tese com um estudo sobre o processo de termalização de dois spins interagentes fracamente acoplados a reservatórios térmicos bosônicos independentes ou um único reservatório coletivo, além de apresentar alguns resultados referentes a sistemas maiores, formados por um número arbitrário de spins.