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Advanced Condensed Matter Theory

Andreas Mielke

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

Strong correlations are important for the understanding of many phenomena in modern condensed matter physics. Examples are high temperature superconductivity (Nobel prize 1987 http://www.nobel.se/physics/ laureates/1987/index.html) and the fractional quantum Hall effect (Nobel prize 1998 http://www.nobel. se/physics/laureates/1998/index.html). The theory for strong correlations needs new concepts and methods to describe such phenomena and to solve the corresponding models. The aim of this course is to provide these methods, to introduce the models, and to help understanding the phenomena in strongly correlated systems. Not the only one but the most important model to describe strongly correlated Fermions is the Hubbard model. In the modern theory of strong correlations it plays the same role as the Ising model in statistical physics: It serves as a standard model to describe and understand most of the phenomena in strongly correlated systems. The Hubbard model was initially introduced to describe the physical behaviour of transition metals, to understand magnetic phenomena in itinerant electron systems (ferromagnetism, anti-ferromagnetism, ferrimagnetism), to describe the Mott transition, and to describe π-electron systems in quantum chemistry. In one space dimension it describes a Luttinger liquid. Although the model has a very simple structure, the behaviour depends strongly on the parameters, the interaction strength, the density, and the underlying lattice. In this course we try to give an overview over the physics of strongly correlated electrons and the Hubbard model. The students are expected to know quantum mechanics and statistical physics. Some knowledge in condensed matter physics is helpful as well. Acknowledgement I am grateful to Johannes Holck. He carefully read the manuscript, found lots of errors, and had several ideas for improvements. About this text The manuscript is available as pdf. Links for citations are coloured in dark red, links within the text in dark blue, links to external material in dark green. Copyright c ©2015ff Andreas Mielke

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Strong correlations are important for the understanding of many phenomena in modern condensed matter physics. Examples are high temperature superconductivity (Nobel prize 1987 http://www.nobel.se/physics/ laureates/1987/index.html) and the fractional quantum Hall effect (Nobel prize 1998 http://www.nobel. se/physics/laureates/1998/index.html). The theory for strong correlations needs new concepts and methods to describe such phenomena and to solve the corresponding models. The aim of this course is to provide these methods, to introduce the models, and to help understanding the phenomena in strongly correlated systems. Not the only one but the most important model to describe strongly correlated Fermions is the Hubbard model. In the modern theory of strong correlations it plays the same role as the Ising model in statistical physics: It serves as a standard model to describe and understand most of the phenomena in strongly correlated systems. The Hubbard model was initially introduced to describe the physical behaviour of transition metals, to understand magnetic phenomena in itinerant electron systems (ferromagnetism, anti-ferromagnetism, ferrimagnetism), to describe the Mott transition, and to describe π-electron systems in quantum chemistry. In one space dimension it describes a Luttinger liquid. Although the model has a very simple structure, the behaviour depends strongly on the parameters, the interaction strength, the density, and the underlying lattice. In this course we try to give an overview over the physics of strongly correlated electrons and the Hubbard model. The students are expected to know quantum mechanics and statistical physics. Some knowledge in condensed matter physics is helpful as well. Acknowledgement I am grateful to Johannes Holck. He carefully read the manuscript, found lots of errors, and had several ideas for improvements. About this text The manuscript is available as pdf. Links for citations are coloured in dark red, links within the text in dark blue, links to external material in dark green. Copyright c ©2015ff Andreas Mielke
The concept of electronic correlations plays an important role in modern condensed matter physics. It refers to interaction effects which cannot be explained within a static mean‐field picture as provided by Hartree‐Fock theory. Electronic correlations can have a very strong influence on the properties of materials. For example, they may turn a metal into an insulator (Mott‐Hubbard metal‐insulator transition). In these lecture notes I (i) introduce basic notions of the physics of correlated electronic systems, (ii) discuss the construction of mean‐field theories by taking the limit of high lattice dimensions, (iii) explain the simplifications of the many‐body perturbation theory in this limit which provide the basis for the formulation of a comprehensive mean‐field theory for correlated fermions, the dynamical mean‐field theory (DMFT), (v) derive the DMFT self‐consistency equations, and (vi) apply the DMFT to investigate electronic correlations in models and materials.
A state-of-the-art review is given of the investigations of the Hubbard model, which is the main model used in the theory of strongly correlated electron systems. It is shown how the main correlation effects — the metal–insulator phase transition, the appearance of localised magnetic moments, and the breakdown of the Fermi-liquid behaviour — appear in the case of strong electron correlations when U ~ W, where U is the Coulomb repulsion parameter and W is the band width. Different approaches to the problem, not based on perturbation theory in terms of the parameter U/W or W/U, are presented. A new method in the theory of strongly correlated systems, in which the limit of infinite number of dimensions of space, d = ∞, is the main feature of the review. The physical results obtained by this approach are compared with those deduced by traditional methods. An overview of the behaviour of strongly correlated systems is given for wide ranges of the model parameters U and W, and of the electron density n.
Modern Condensed Matter Physics brings together the most important advances in the field of recent decades. It provides instructors teaching graduate-level condensed matter courses with a comprehensive and in-depth textbook that will prepare graduate students for research or further study as well as reading more advanced and specialized books and research literature in the field. This textbook covers the basics of crystalline solids as well as analogous optical lattices and photonic crystals, while discussing cutting-edge topics such as disordered systems, mesoscopic systems, many-body systems, quantum magnetism, Bose–Einstein condensates, quantum entanglement, and superconducting quantum bits. Students are provided with the appropriate mathematical background to understand the topological concepts that have been permeating the field, together with numerous physical examples ranging from the fractional quantum Hall effect to topological insulators, the toric code, and majorana fermions. Exercises, commentary boxes, and appendices afford guidance and feedback for beginners and experts alike.
The art of being wise is the art of knowing what to overlook. William James Summary This chapter is devoted to idealized models and theoretical concepts that underlie the topics in the rest of this book. Among the most dramatic effects are the Wigner and Mott transitions, exemplified by electrons in a homogeneous background of positive charge and by the Hubbard model of a crystal. Fermi liquid theory is the paradigm for understanding quasi-particles and collective excitations in metals, building on a continuous link between a non-interacting and an interacting system. The Luttinger theorem and Friedel sum rule are conservation laws for quantities that do not vary at all with the interaction. The Heisenberg and Ising models exemplify the properties of localized electronic states that act as spins. The Anderson impurity model is the paradigm for understanding local moment behavior and is used directly in dynamical mean-field theory. The previous chapters discuss examples of experimental observations where effects of interactions can be appreciated with only basic knowledge of physics and chemistry. The purpose of this chapter is to give a concise discussion of models that illustrate major characteristics of interacting electrons. These are prototypes that bring out features that occur in real problems, such as the examples in the previous chapter. They are also pedagogical examples for the theoretical methods developed later, with references to specific sections. The Wigner transition and the homogeneous electron system The simplest model of interacting electrons in condensed matter is the homogeneous electron system, also called homogeneous electron gas (HEG), an infinite system of electrons with a uniform compensating positive charge background. It was originally introduced as a model for alkali metals. Now the HEG is a standard model system for the development of density functionals and a widely used test system for the many-body perturbation methods in Chs. 10–15. It is also an important model for quantum Monte Carlo calculations, described in Chs. 23–25. To define the model, we take the hamiltonian in Eq. (1.1) and replace the nuclei by a rigid uniform positive charge with density equal to the electron charge density n .
Strongly interacting quantum systems exhibit same of the most interesting phenomena in condensed matter physi=. Until very recently the study of these systems has heen confined to either numerical simulations of rather small systems (which still have an enormous number of q“ant”m stat=) or approximate ~na. lytical methods which are uncontrolled. Mat of the effort concentrated on the fermionic systems. In the last few yearn the hmonic systems have attracted a lot of interest.l ,2,3,4,5 Example of the bmonic strongly correlated systems inc]ude ~h~rt coherence length superconductors, Jc6ephson junction arrays, the dynamics of flux lines in high temperature superconductors and the quantum hall effect.6 The ~ntial physics of all of these problems is the competition between the hopping matr~ elements and the short range strong repulsion. The minimum mdel which contains the bmicphysics of theproblem istheso called Bme Hubbard model:
Modern Condensed Matter Physics brings together the most important advances in the field of recent decades. It provides instructors teaching graduate-level condensed matter courses with a comprehensive and in-depth textbook that will prepare graduate students for research or further study as well as reading more advanced and specialized books and research literature in the field. This textbook covers the basics of crystalline solids as well as analogous optical lattices and photonic crystals, while discussing cutting-edge topics such as disordered systems, mesoscopic systems, many-body systems, quantum magnetism, Bose–Einstein condensates, quantum entanglement, and superconducting quantum bits. Students are provided with the appropriate mathematical background to understand the topological concepts that have been permeating the field, together with numerous physical examples ranging from the fractional quantum Hall effect to topological insulators, the toric code, and majorana fermions. Exercises, commentary boxes, and appendices afford guidance and feedback for beginners and experts alike.