A fundamental challenge in quantum many-body physics is to understand the universal properties of strongly correlated systems. In this work, we establish a universal and exact identity from the Dyson-Schwinger equations within the Keldysh-Schwinger field theory for systems with generic bilinear interactions. Our derivation demonstrates the emergence of composite particles as "elementary excitations", whose Green's functions definitively determine the original single-particle Green's function. This universal relation uniquely identifies the composite particles governing correlations and rigorously connects their spectra to the observable single-particle spectrum. Thus, our exact identity reveals a new pathway toward a paradigm for understanding many-body correlated systems.
A fundamental challenge in quantum many-body physics is to understand the universal properties of strongly correlated systems. In this work, we establish a universal and exact identity from the Dyson-Schwinger equations within the Keldysh-Schwinger field theory for systems with generic bilinear interactions. Our derivation demonstrates the emergence of composite particles as "elementary excitations", whose Green's functions definitively determine the original single-particle Green's function. This universal relation uniquely identifies the composite particles governing correlations and rigorously connects their spectra to the observable single-particle spectrum. Thus, our exact identity reveals a new pathway toward a paradigm for understanding many-body correlated systems.
Schwinger Keldysh field theory is a widely used paradigm to study nonequilibrium dynamics of quantum many-body systems starting from a thermal state. We extend this formalism to describe nonequilibrium dynamics of quantum systems starting from arbitrary initial many-body density matrices. We show how this can be done for both bosons and fermions, and for both closed and open quantum systems, using additional sources coupled to bilinears of the fields at the initial time, calculating Green's functions in a theory with these sources, and then taking appropriate set of derivatives of these Green's functions with respect to initial sources to obtain physical observables. The set of derivatives depends on the initial density matrix. The physical correlators in a dynamics with arbitrary initial conditions do not satisfy Wick's theorem, even for noninteracting systems. However, our formalism constructs intermediate ``$n\ensuremath{-}\mathrm{particle}$ Green's functions'' that obey Wick's theorem and provide a prescription to obtain physical correlation functions from them. This allows us to obtain analytic answers for all physical many-body correlation functions of a noninteracting system even when it is initialized to an arbitrary density matrix. We use these exact expressions to obtain an estimate of the violation of Wick's theorem and relate it to presence of connected multiparticle initial correlations in the system. We illustrate this new formalism by calculating density and current profiles in many-body fermionic and bosonic open quantum systems initialized to nontrivial density matrices. We have also shown how this formalism can be extended to interacting many-body systems.
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.
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.
Abstract Strong connection between the single-particle excitation and the collective excitation stands out as one of the features of Bose–Einstein condensates (BECs). We discuss theoretically these single-particle and density excitations of BECs focusing on the exact properties of the one-body and two-body Green’s functions developed by Gavoret and Nozières. We also investigate these excitations by using the many-body approximation theory at nonzero temperatures. First, we revisited the earlier study presented by Gavoret and Nozières, involving the subsequent results given by Nepomnyashchii and Nepomnyashchii, in terms of the matrix formalism representation. This matrix formalism is an extension of the Nambu representation for the single-particle Green’s function of BECs to discuss the density and current response functions efficiently. We describe the exact low-energy properties of the correlation functions and the vertex functions, and discuss the correspondence of the spectra between the single-particle excitation and the density excitation in the low-energy and low-momentum limits at T = 0. After deriving the exact low-energy structures of the one-body and two-body Green’s functions, we develop a many-body approximation theory of BECs with making the use of the matrix formalism for describing the single-particle Green’s function and the density response function at nonzero temperatures. We show how the peaks of the single-particle spectral function and the density response function behave with an increasing temperature. Many-body effect on the single-particle spectral function and the density response function is included within a random phase approximation, where satellite structures emerge because of beyond-mean-field effects. Criticisms are also made on recent theories casting doubt upon the conventional wisdom of the BEC: the equivalence of the dispersion relations between the single-particle excitation and the collective excitation in the low-energy and low-momentum regime.
Develops the many-body one-particle Green function, and explains its physical interpretation in terms of the spectral function, self-energy, and quasiparticle. lifetime. Its application in angle-resolved photoemission spectroscopy is presented in detail. The time-evolution operator in the interaction picture is derived, and time-ordering and adiabatic switching-on are introduced as precursor tools to construct the Feynman–Dyson many-body perturbation theory. A detailed account of Wick’s theorem, normal ordering, and contractions is outlined. Feynman diagrams are constructed, and the emergence of the infinite Dyson series from irreducible diagrams is outlined. Two-particle Green function and the particle-hole excitation spectra are developed. Diagramatic application of RPA for interacting systems is described. The finite-temperature Matsubara Green function is introduced and developed, together with its Fourier series representation and the evaluation of Matsubara sums.
Exactly solved models provide rigorous understanding of many-body phenomena in strongly correlated systems. In this article, we report a breakthrough in uncovering universal many-body correlated properties of the quantum integrable Lieb-Liniger model. We exactly calculate the dynamical correlation functions by computing the form factors through a newly developed method, by which we are capable of calculating all possible 'relative excitations' over the ground state or a finite temperature state to high precision. Consequently, full spectral functions obtained for the model manifest the unique power-law singularity behaviour at the spectral threshold, confirming the validity of nonlinear Luttinger liquid theory. Our method advances the theory of dynamical correlation functions with high precision towards the thermodynamic limit, and is capable of benchmarking experimental observation of such novel correlated properties.