Dynamical Mean Field Theory

This book is a short introduction to the Dynamical Mean-Field Theory for strongly correlated electrons. Its purpose is to focus on various local decoupling schemes in order to derive a self-consistent approximation and to map the lattice problem onto an impurity problem. Hubbard, Holstein, and Falicov-Kimball models are mainly used to provide examples of calculation. Numerous basic c/c++ programs are given along the book to develop confidence in computing actual numerical results.

​​This is the first book that provides a detailed summary of one of the most successful new condensed matter theories - dynamical mean-field theory (DMFT) - in both static and dynamical cases of systems of different sizes. DMFT is one of the most successful approaches to describe the physical properties of systems with strong electron-electron correlations such as bulk materials, multi-layers, surfaces, 2D materials and nanostructures in both metallic and insulating phases. Strongly correlated materials usually include partially-filled localized d- or f-orbitals, and DMFT takes into account crucial for these systems time-resolved interaction between electrons when they “meet” on one atom and occupy one of these orbitals. The First Part of the book covers the general formalism of DMFT as a many-body theory, followed by generalizations of the approach on the cases of finite systems and out-of-equilibrium regime. In the last Chapter of the First Part we discuss generalizations of the approach on the case when the non-local interactions are taken into account. The Second Part of the book covers methodologies of merging DMFT with ab initio static Density Functional Theory (DFT) and Time-Dependent DFT (TDDFT) approaches. Such combined DFT+DMFT and DMFT+TDDFT computational techniques allow one to include the effects of strong electron-electron correlations at the accurate ab initio level. These tools can be applied to complex multi-atom multi-orbital systems currently not accessible to DMFT. The book helps broad audiences of students and researchers from the theoretical and computational communities of condensed matter physics, material science, and chemistry to become familiar with this state-of-art approach and to use it for reaching a deeper understanding of the properties of strongly correlated systems and for synthesis of new technologically-important materials.

We show how to set up a Cellular Dynamical Mean Field Theory for the Holstein's polaron problem using the exact solution of a cluster of n sites embedded in a Weiss's field. We show that a restricted basis, that allows excitations of phonons only for n sites at a time, reproduces exactly the equations of the n-site Dynamical Mean Field Theory, and enables to check the proposed decoupling scheme of the Green's functions via Exact Numerical Diagonalizations. We introduce a real space formulation of the Cellular Dynamical Mean Field Theory that applies to any lattice with or without periodic boundary conditions and that allows to partition the lattice into different kinds of clusters.

This book presents a self-contained introduction to techniques from field theory applied to stochastic and collective dynamics in neuronal networks. These powerful analytical techniques, which are well established in other fields of physics, are the basis of current developments and offer solutions to pressing open problems in theoretical neuroscience and also machine learning. They enable a systematic and quantitative understanding of the dynamics in recurrent and stochastic neuronal networks. This book is intended for physicists, mathematicians, and computer scientists and it is designed for self-study by researchers who want to enter the field or as the main text for a one semester course at advanced undergraduate or graduate level. The theoretical concepts presented in this book are systematically developed from the very beginning, which only requires basic knowledge of analysis and linear algebra.

This volume presents, for the very first time, an exhaustive collection of those modern numerical methods specifically tailored for the analysis of Strongly Correlated Systems. Many novel materials, with functional properties emerging from macroscopic quantum behaviors at the frontier of modern research in physics, chemistry and material science, belong to this class of systems. Any technique is presented in great detail by its own inventor or by one of the world-wide recognized main contributors. The exposition has a clear pedagogical cut and fully reports on the most relevant case study where the specific technique showed to be very successful in describing and enlightening the puzzling physics of a particular strongly correlated system. The book is intended for advanced graduate students and post-docs in the field as textbook and/or main reference, but also for other researchers in the field who appreciate consulting a single, but comprehensive, source or wishes to get acquainted, in a as painless as possible way, with the working details of a specific technique.

This novel book is the first comprehensive text on dynamical mean-field theory (DMFT), which has emerged over the past two decades as one of the most powerful new developments in many-body physics. Written by one of the key researchers in the field, the volume develops the formalism of many-body Green's functions using the equation of motion approach, which requires an undergraduate solid state physics course and a graduate quantum mechanics course as prerequisites. The DMFT is applied to study transport in multilayered nanostructures, which is likely to be one of the most prominent applications of nanotechnology in the coming years. The text is modern in scope focusing on exact numerical methods rather than the perturbation theory. Formalism is developed first for the bulk and then for the inhomogeneous multilayered systems. The science behind the metal-insulator transition, electronic charge reconstruction, and superconductivity are thoroughly described. The book covers complete derivations and emphasizes how to carry out numerical calculations, including discussions of parallel programing algorithms. Detailed descriptions of the crossover from tunneling to thermally activated transport, of the properties of Josephson junctions with barriers tuned near the metal-insulator transition, and of thermoelectric coolers and power generators are provided as applications of the theory./a

This book is a wide-ranging survey of the physics of out-of-equilibrium systems of correlated electrons, ranging from the theoretical, to the numerical, computational and experimental aspects. It starts from basic approaches to non-equilibrium physics, such as the mean-field approach, then proceeds to more advanced methods, such as dynamical mean-field theory and master equation approaches. Lastly, it offers a comprehensive overview of the latest advances in experimental investigations of complex quantum materials by means of ultrafast spectroscopy.