In connection with the contribution “Quantum Condensates in Nuclear Matter” some problems are given to become more familiar with the techniques of many-particle physics.
The quantum theory of many-particle systems is deeply connected to the mathematical formalism and techniques of quantum field theory, which are crucial in order describe the emergent phenomena that arise in condensed matter physics. The main goal of this thesis is to provide a unified description of quantum field theory methods in the context of many-particle physics in order to extend and develop these techniques to better understand the different quantum properties of matter. This both furthers the conceptual understanding of quantum many-body correlation and also allows for the development of novel theoretical techniques that combine the different formalisms employed in condensed matter theory, quantum chemistry and nuclear physics. To achieve this aim, the thesis is composed of two parts. Part I deals with many-body fermion-fermion interactions and is physically motivated by the non-relativistic electronic structure problem. Here, we introduce the Keldysh contour formulation of quantum field theory to provide a unified presentation of quantum systems both in and out-ofequilibrium. Then, we extend the conventional quantum theory of many-particle systems, based on hermitian quantum mechanics, to describe systems governed by non-hermitan Hamiltonians. This allows us to present a natural unification of two major pillars in the ab initio quantum theory of many-particle systems: coupled-cluster theory and the Green’s function formalism. This unification is further demonstrated through the emergence of the coupled-cluster amplitude equations from the Green’s function formalism. In particular, this analysis has direct bearing on the development of novel theoretical approaches to the electronic structure and nuclear many-body problems. Our findings are also relevant to the study of general non-hermitian as well as open quantum systems. Part II focuses on the quantum field theory of electron-nuclear interactions in infinite solids. Here, we use the Keldysh contour representation presented in Part I to understand excited state dynamics that include electron-electron and electron-lattice interactions in materials. We also derive the analytic continuation of the Green’s function for systems that have continuous eigenspectra. This analysis demonstrates the relationship between the Green’s function formalism and scattering theory, making the role of the continuum and causality conditions explicitly clear in the context of the pole condition of the Dyson equation. Using the analytic continuation of the propagator, we provide an in-depth analysis of electron-phonon and exciton-phonon scattering channels, with a particular focus on phonon-driven exciton dissociation scattering. Our theoretical analysis then firmly demonstrates the connection between the extension of the Bethe-Salpeter equation approach to include electron-phonon interactions and effective field theories based on composite exciton-phonon coupling. We subsequently combine the techniques presented in Part I, based on the Dyson supermatrix formulation, with the analysis presented in Part II to calculate, from first principles, the effects of including electron-phonon interactions on the excited states of heterogeneous semiconductors. Finally, we provide a summary of the fundamental contributions presented in this thesis and the potential further research directions to be pursued.
The influence of Franco Iachello in Physics during the last 50 years and, in particular, in the use of algebraic methods applied to very different physical problems has been broad, extense and have permeated most branches of Physics, from Nuclear and Molecular to Particle and Condensed Matter physics. Apart of many other contributions, at the beginning of the 2000’s he introduced the concept of critical point symmetry and triggered the study of the many faces of quantum phase transitions in nuclei and other mesoscopic systems. In this contribution, we present the analysis of the scaling properties of the Lipkin model spectrum at the phase transition point and we focus on the differences between first and second order quantum phase transitions. Moreover, we explain how the obtained results can be also of application for the interacting boson model.