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We introduce a generalized Gross-Neveu (GN) model to describe the excitonic instabilities in two different systems: a small overlap semi-metal (SM) and a small gap semi-conductor (SMC), both in two (2d) and three-dimensions (3d). We identify the excitonic order parameter (EOP) and obtain the effective potential within the Large $N$ limit approach where the GN model can be exactly solved. We obtain the excitonic insulator (EI) phase diagrams as a function of temperature, chemical potential, overlap between bands and gaps of the system. We show that the EI may undergo first- or second-order thermal transitions depending on the regime whereupon this phase is approached. We also investigate the expected thermodynamic signatures for the specific heat above the fine-tuned excitonic quantum critical point (EQCP), in both 2d and 3d, in the SMC regime. We show that the EQCP is a different kind of critical point since although the EOP vanishes at the EQCP, there is always a finite gap in the SMC regime. We find that for high temperatures, the specific heat might exhibit a scaling behavior in the form $C_V/T \propto T^{(d-z)/z}$, where $d$ is the dimension of the system and $z$ is the dynamical critical exponent. The very low temperature behavior has a dominant exponential thermally activated term due to the presence of a gap that does not vanish at the excitonic transition., Comment: 12 pages and 17 figures

Conductivity and Charge Stiffness In this chapter, we will consider initially metal–insulator (MI) transitions which occur in pure systems without disorder. In the final section we consider a superfluid–insulator transition induced by some special type of disorder. We will show that the quantities which characterise the phase transitions in both problems obey similar scaling laws and are governed by similar exponents. Differently from the magnetic transitions treated before metal–insulator (MI) transitions have no obvious order parameter to distinguish the metallic from the insulating phase. This precludes a power expansion of the free energy in terms of a small quantity as for magnetic phase transitions. In spite of that, the concepts of phase transitions and critical phenomena turn out to be very useful to describe metal–insulator transitions. In case this problem is approached using the renormalisation group, the localisation transition is associated with an unstable fixed point and the flow of the RG equations to the different attractors is sufficient to distinguish the nature of the phases. Then, in general, we can associate a set of critical exponents with the MI transitions. Besides, these exponents are not independent but obey scaling relations. Since we are concerned with zero temperature instabilities, the quantum hyperscaling law, Eq. 1.18, plays a crucial role in this problem. We will distinguish here two different types of metal–insulator transitions. Those due to a competition between parameters of the relevant Hamiltonian and metal–insulator transitions arising by varying the number of electrons or chemical potential. An example of the former is the Mott transition due to the competition between kinetic energy and the local Coulomb repulsion as described by the half-filled band Hubbard model. The latter will be referred as density-driven metal– insulator transitions in analogy with the superfluid–insulator transition in bosonic systems with varying particle number (Fisher et al ., 1989). These two types of transitions may occur in the same system or Hamiltonian. They in general belong to different universality classes and consequently have different critical exponents.

Quantum phase transitions are strongly relevant in a number of fields, ranging from condensed matter to cold atom physics and quantum field theory. This book, now in its second edition, approaches the problem of quantum phase transitions from a new and unifying perspective. Topics addressed include the concepts of scale and time invariance and their significance for quantum criticality, as well as brand new chapters on superfluid and superconductor quantum critical points, and quantum first order transitions. The renormalisation group in real and momentum space is also established as the proper language to describe the behaviour of systems close to a quantum phase transition. These phenomena introduce a number of theoretical challenges which are of major importance for driving new experiments. Being strongly motivated and oriented towards understanding experimental results, this is an excellent text for graduates, as well as theorists, experimentalists and those with an interest in quantum criticality.

We study some important problems related to the transport properties of many body systems. It is divided in three parts, each one focusing in a specific topic. We obtain relevant results that improve our understanding of these systems. We investigate the effect of dissipation and time-dependent external sources, in the phase diagram of a many body system at zero and finite temperature. In the presence of time-dependent perturbations, dissipation is essential for the system to attain a steady, time independent state. In order to treat this time dependent problem, we use a Keldysh approach within an adiabatic approximation that allows us to study the phase diagram of this system as a function of the parameters of the system and temperature. We also discuss the nature of the quantum phase transitions of the system. Next, we study an important concept in the physics of metallic multi-band systems, that of hybridization, and how it affects the superconducting properties of a material. A constant or symmetric k-dependent hybridization in general act in detriment of superconductivity. We show here that when hybridization between orbitals in different sites assumes an anti-symmetric character having odd-parity it {it{enhances}} superconductivity. The antisymmetric hybridization in a problem study in this thesis (present in Chapter 3) allow us to propose a new system where it is possible to investigate Majorana fermions, even in absence of spin-orbit interactions. In the last part of this thesis we study the effect of spin-orbit coupling (SOC) on transport properties in magnetic nanostructures. In this system SOC plays an important role, because surfaces (or interfaces) introduce symmetry breaking which is a source of spin-orbit interaction. We study the role of Dzyaloshinshkii-Moriya (DM) interaction on spin-transport in a 3 layer system. We show that there is a DM interaction between magnetics ions in the layers and spin of conduction electrons. We study the influence of this DM interaction on transport within a simple model where each layer is represented by a point., Nous étudions plusieurs problémes reliés aux propriétés de transport dans les systèmes corrélés. La thèse contient 3 parties distinctes, chacune d'entre elles décrivant un aspect particulier. Nous avons obtenu dans chacun des cas des résultats qui permettent une meilleure compréhension du transport. Nous étudions l'effet de la dissipation et d'une perturbation extérieure dépendant du temps sur le diagramme de phases d'un systèmes à N corps à température nulle et à température finie. En présence de perturbation dépendant du temps, la dissipation joue un rôle important dans l'évolution vers un état stable indépendant du temps. Nous utilisons le formalisme de Keldysh dans l'approximation adiabatique qui permet d'étudier le diagramme de phases du système en fonction de parameter et de la température. Dans la 2ième partie, nous étudions un concept important pour la physique des systèmes métalliques à plusieurs bandes, le concept d'hybridation, et la façon dont l'hybridation affecte la supraconductivité du métal. De façon générale, une hybridation dépendante ou non du vecteur d'onde k a tendance à détruire la supraconductivité. Nous montrons dans ce chapitre qu'une hybridation antisymétrique a l'effet inverse et renforce la supraconductivité. Nous montrons que si l'hybridation est antisymétrique, la supraconductivité a des propriétés non-triviales. Nous proposons que dans un tel système, il puisse exister des fermions de Majorana, même en l'absence de couplage spin-orbite. Le dernier chapitre de la thèse porte sur les effets du couplage spin-orbite sur le transport dans les nanostructures magnétiques. Dans les nanostructures, le couplage spin-orbite joue un rôle important en raison de la brisure de symmétrie à la surface ou aux interfaces. En particulier, nous étudions l'effet de l'interaction Dzyaloshinskii-Moriya (DM) sur le transport de spin dans un système tri-couche. Nous montrons qu'il existe une interaction DM entre les moments des couches et les électrons de conduction, et l'influence de cette interaction sur le transport est étudiée dans un modèle simplifié ou chaque couche est représentée par un point.