The study of the correlated motion of electrons in solids is of increasing importance in condensed matter physics. In the past few years, the discovery of high-temperature superconductors has stimulated an enormous theoretical effort in this area, building on earlier theories of heavy-fermion and organic superconductors, and magnetic insulators. In a separate development, the discovery of the fractional quantum Hall effect stimulated research into the behaviour of the two-dimensional electron gas in a strong transverse magnetic field. The lectures give a systematic presentation of the current status of the theory in these areas. They cover the fractional quantum Hall effect and the many-body physics of the Hubbard model and its extension, paying particular attention to the properties of doped insulators which are relevant for high-temperature superconductivity. They detail discussions of situations for which controlled calculations may be carried out - specifically infinite dimensions, one dimension, and generalized models in which the fermions have N components and N to infinity.
The study of the correlated motion of electrons in solids is of increasing importance in condensed matter physics. In the past few years, the discovery of high-temperature superconductors has stimulated an enormous theoretical effort in this area, building on earlier theories of heavy-fermion and organic superconductors, and magnetic insulators. In a separate development, the discovery of the fractional quantum Hall effect stimulated research into the behaviour of the two-dimensional electron gas in a strong transverse magnetic field. The lectures give a systematic presentation of the current status of the theory in these areas. They cover the fractional quantum Hall effect and the many-body physics of the Hubbard model and its extension, paying particular attention to the properties of doped insulators which are relevant for high-temperature superconductivity. They detail discussions of situations for which controlled calculations may be carried out - specifically infinite dimensions, one dimension, and generalized models in which the fermions have N components and N to infinity.
Open questions related with heavy electrons, H.R. Ott electron spectroscopy of strongly correlated systems, J.W. Allen single hole in 1D and 2D hubbard model, S. Sorella surface acoustic wave studies of the FQHE, R.L. Willett Chern-Simons-Landau-Ginzburg theory of the FQHE and its experimental prediction, S.C. Zhang magnetic fluctuations in La2-xSrxCuO4, G. Aeppli single particle spectral function for the large U Hubbard model, P. Wolfe strongly correlated systems in infinite dimensions and their zero-dimensional counterpart, G. Kotliar stable hc-e vortices in a gauge theory of superconductivity in strongly correlated systems, S. Sachdev electronic mechanisms for superconductivity in doped C60, G. Baskaran weakly correlated electrons - many questions and few answers, H.J. Schulz normal state properties of the high Tc oxides - gauge theories, L. Loffe single particle and optical gaps in charge-transfer insulators, Y. Bang et al a renormalization group approach for many-Fermion problems, R. Shankar spin singlet quantum hall effect and nonabelian Landau-Ginzburg theory, A. Balatsky De Haas-Van Alphen and Shubnikov-De Haas oscillations in strongly correlated electron gas, P. Lederer et al strongly correlated Fermi liquid and the normal state of high Tc superconductors, Z. Wang determinant Monte Carlo for the Hubbard model with arbitrarily gauged auxiliary fields, A.M. Tremblay spin-charged coupling in doped Mott insulators, J. Wheatley.
Motivated by possible applications to cold atomic gases in a rotating trap, we give an overview of some important concepts in two-dimensional vortex physics. In the context of Josephson junction arrays, we first discuss the crucial role payed by vortices in both the thermal superconducting to normal metal transition, and the zero temperature superconducting to Mott insulator transition. Important notions on vortex dynamics in these arrays are then presented, with emphasis on experimental results indicating the quantum nature of vortex motion at low temperature. We move then to the Fractional Quantum Hall effect, as observed in electronic systems, and introduce the notions of fractional excitations, generalized statistics, and composite Fermions, which have played a key role in the theoretical description of this remarkable phenomenon. Very likely, cold atomic gases in a rotating trap will soon provide a way to explore the bosonic version of the Fractional Quantum Hall effect, and thus shedding new light on these subtle strongly correlated quantum fluids.
The main aim of this book is to give a self-contained and representative cross section through present-day research in solid-state physics. This covers metallic and mesoscopic transport, localization by disorder and superconductivity, including questions related to high-temperature superconductors and to heavy fermion systems. An important part of the book is devoted to itinerant-electron magnetism, discussing paramagnons, strong correlation, magnetization fluctuations and spin density waves. All the formal tools used in these chapters are developed in the first part of the book which contains a thorough discussion of second quantization, and of perturbation theory for an arbitrary complex time path, and also describes the functional approach to Feynman diagrams, including general ward identities. Each chapter contains an extensive list of the relevant literature and a series of problems with detailed solutions which complement the main text. The book is meant as both a course and a research tool.
The structure of matter is intimately related to the fundamental role played by many-fermion systems. The development in the last few decades of the microscopic many-body theory of correlated fermion systems has been a fertile ground not only for spectacular achievements in basic science in various areas of research, but also for technological applications. Among the numerous phenomena discovered and studied in systems of many fermions, one can cite the superconductivity in metals, the superfluidity in 3He, nuclear matter and nuclei, the quantum Hall effect, the giant resonances in nuclei, the Anderson location and the metal-insulator transition, hole and electron diffusion in doped semiconductors, etc. All these phenomena can be understood only in terms of correlations occurring in many-fermion systems, and the formulation of the correct microscopic theory of each phenomenon has marked a milestone in pure science as well as the starting point for the exploitation of its potential technological applications. It is likely that in the future further developments will take place in this field of basic science.The Hispalensis International School is a summer school aimed mainly at young physicists, both theoreticians and experimentalists, engaged in research work at the predoctoral or recent postdoctoral level. The objective of the School provide an opportunity for participants to come into contact with experienced researchers and hear their clear account of the state of the art of many-body theories in nuclear physics, as well as in related fields, and the main future lines of development.