The Green's function method has applications in several fields in Physics, from classical differential equations to quantum many-body problems. In the quantum context, Green's functions are correlation functions, from which it is possible to extract information from the system under study, such as the density of states, relaxation times and response functions. Despite its power and versatility, it is known as a laborious and sometimes cumbersome method. Here we introduce the equilibrium Green's functions and the equation-of-motion technique, exemplifying the method in discrete lattices of non-interacting electrons. We start with simple models, such as the two-site molecule, the infinite and semi-infinite one-dimensional chains, and the two-dimensional ladder. Numerical implementations are developed via the recursive Green's function, implemented in Julia, an open-source, efficient and easy-to-learn scientific language. We also present a new variation of the surface recursive Green's function method, which can be of interest when simulating simultaneously the properties of surface and bulk.
The Green's function method has applications in several fields in Physics, from classical differential equations to quantum many-body problems. In the quantum context, Green's functions are correlation functions, from which it is possible to extract information from the system under study, such as the density of states, relaxation times and response functions. Despite its power and versatility, it is known as a laborious and sometimes cumbersome method. Here we introduce the equilibrium Green's functions and the equation-of-motion technique, exemplifying the method in discrete lattices of non-interacting electrons. We start with simple models, such as the two-site molecule, the infinite and semi-infinite one-dimensional chains, and the two-dimensional ladder. Numerical implementations are developed via the recursive Green's function, implemented in Julia, an open-source, efficient and easy-to-learn scientific language. We also present a new variation of the surface recursive Green's function method, which can be of interest when simulating simultaneously the properties of surface and bulk.
The Green's function method has applications in several fields in Physics, from classical differential equations to quantum many-body problems. In the quantum context, Green's functions are correlation functions, from which it is possible to extract information from the system under study, such as the density of states, relaxation times and response functions. Despite its power and versatility, it is known as a laborious and sometimes cumbersome method. Here we introduce the equilibrium Green's functions and the equation-of-motion technique, exemplifying the method in discrete lattices of non-interacting electrons. We start with simple models, such as the two-site molecule, the infinite and semi-infinite one-dimensional chains, and the two-dimensional ladder. Numerical implementations are developed via the recursive Green's function, implemented in Julia, an open-source, efficient and easy-to-learn scientific language. We also present a new variation of the surface recursive Green's function method, which can be of interest when simulating simultaneously the properties of surface and bulk.
The Green’s function method is among the most powerful and versatile formalisms in physics, and its nonequilibrium version has proved invaluable in many research fields. With entirely new chapters and updated example problems, the second edition of this popular text continues to provide an ideal introduction to nonequilibrium many-body quantum systems and ultrafast phenomena in modern science. Retaining the unique and self-contained style of the original, this new edition has been thoroughly revised to address interacting systems of fermions and bosons, simplified many-body approaches like the GKBA, the Bloch equations, and the Boltzmann equations, and the connection between Green’s functions and newly developed time-resolved spectroscopy techniques. Small gaps in the theory have been filled, and frequently overlooked subtleties have been systematically highlighted and clarified. With an abundance of illustrative examples, insightful discussions, and modern applications, this book remains the definitive guide for students and researchers alike.
The Green's function method is one of the most powerful and versatile formalisms in physics, and its nonequilibrium version has proved invaluable in many research fields. This book provides a unique, self-contained introduction to nonequilibrium many-body theory. Starting with basic quantum mechanics, the authors introduce the equilibrium and nonequilibrium Green's function formalisms within a unified framework called the contour formalism. The physical content of the contour Green's functions and the diagrammatic expansions are explained with a focus on the time-dependent aspect. Every result is derived step-by-step, critically discussed and then applied to different physical systems, ranging from molecules and nanostructures to metals and insulators. With an abundance of illustrative examples, this accessible book is ideal for graduate students and researchers who are interested in excited state properties of matter and nonequilibrium physics.
A review of electronic dynamics of single-impurity and many-impurity Anderson models is contained in this report. Those models are used widely for many of the applications in diverse fields of interest, such as surface physics, theory of chemisorption and adsorbate reactions on metal surfaces, physics of intermediate valence systems, theory of heavy fermions, physics of quantum dots and other nanostructures. While standard treatments are generally based on perturbation methods, our approach is based on the non-perturbative technique for the thermodynamic Green functions. The method of the irreducible Green functions is used as the basic tool. This irreducible Green functions method allows one to describe the quasiparticle spectra with damping of the strongly correlated electron systems in a very general and natural way and to construct the relevant dynamical solution in a self-consistent way on the level of Dyson equation without decoupling the chain of the equations of motion for the Green functions. The subject matter includes the improved interpolating solution of the Anderson model. It was shown that an interpolating approximation, which simultaneously reproduces the weak-coupling limit up to second order in the interaction strength U and the strong-coupling limit up to second order in the hybridization V (and thus also fulfils the atomic limit) can be formulated self-consistently. This approach offers a new way for the systematic construction of approximate interpolation dynamical solutions of strongly correlated electron systems.
In this thesis we present methodologies that have been developed to efficiently solve for the non-equilibrium Green’s Function (NEGF) of interacting electron systems, and demonstrate their applications to several different condensed matter systems. These solvers are based on the numerical integration of the equations of motion for the NEGF, which are known as the Kadanoff-Baym Equations (KBE). The KBE are a general set of equations which allow for the study of a wide range of systems that are of interest to condensed matter physicists, including superconductivity, nuclear matter, molecules and atoms, and qubits, among many others. They are also useful for solving impurity models and systems in the dynamical mean-field theory (DMFT) approximation through their ability to incorporate dynamical hybridization functions. The generality of these equations comes from their connection to time-dependent Many-Body Perturbation Theory (MBPT), which introduces non-Markovian dynamical correlations through the self-energy functional. MBPT ensures important conservation laws, such as those for density and energy, are obeyed no matter the type of time-dependent perturbations that are applied to the Hamiltonian. The non-Markovian correlations manifest in a kernel that couples the dynamics to the entire history of the system. This coupling leads to computational costs that become prohibitively expensive as longer-time dynamics are probed, this is especially true for systems which decohere slowly, leading to undamped dynamics and kernels. Due to the large amount of interest in non-equilibrium quantum dynamics, the availability of efficient solvers for the KBE are of great importance. This thesis will focus on the implementation and analysis of integration methods for the KBE, including compression algorithms which lower computational complexity and adaptive integration techniques which allow for resolution of dynamics which vary over wide ranges of timescales. Chapter 1 gives an introduction to quantum systems and the problems we will be studying. Chapter 2 provides a derivation of the NEGF and its representation on the Keldysh contour. Chapter 3 derives the KBE from MBPT and introduces the self- energy and its conservation laws. Chapter 4 introduces the numerical techniques used to integrate the KBE, as well as an analysis of its causality structure. In Chapter 5, we use compression techniques to lower the complexity of the KBE, and provide details on its implementation. In Chapter 6 we apply the compression algorithm to study the Higgs mode in a superconducting system. In Chapter 7, we couple our solver to an adaptive integrator and analyze its performance against fixed-timestep integrators. Finally, Chapter 8 provides a summary of the results and an outlook for future work on efficient solvers.
Introduction Mathematically, given a linear differential operator L x , one encounters the solution to the inhomogeneous differential equation Here ρ ( x ) is a given source function. For a given boundary condition, we assume a solution to exist. The solution can be reduced to a simpler problem. Let G ( x, y ) is the Green's function. This is a function of x with y a parameter. Take G ( x, y ) to satisfy the same boundary conditions as φ ( x ). Then since An example of L x is, of course, the Schrödinger operator We take ρ ( x ) = V ( x ) ψ ( x , t ), ψ ( x , t ) being the wave function and V ( x ) the potential operator. We are interested in Green's functions taken over from the techniques of quantum field theory (Schweber, 1961; Lifshitz and Petaevskii, 1981). We will concern ourselves particularly with one- and two-time Green's functions, since our principal interest is to show a connection to the calculations of linear response theory (Chapter 15) as well as to quantum kinetic equations. Then we wish to compare the methods with those described in Chapter 4. In this we will follow the work of L. P. Kadanoff and G. Baym (1962) and of L. V. Keldysh (1965) and also Zubarev (1974). We will not discuss equilibrium statistical mechanics utilizing Green's function techniques for many-body problems. The literature is exhaustive (see Abrikosov et al. , 1963; Fetter and Walecka, 1971). A good general introduction is the book by G. D. Mahan (2000). One- and two-time quantum Green's functions and their properties Let us introduce the creation operator ψ † ( r, t ) and annihilation operator ψ ( r, t ) of the second quantization formalism (see Schweber, 1961). They have the equal time commutation rules for Bose and Fermi particles: The Hamiltonian operator for the particles is and the number density of particles at r t is the operator Note that r = ( r 1 … r N ) for (1, 2, 3, …, N ) and V (| r − r ′|) is the pair potential depending, as in Chapter 4, on the scalar distance between the particles.