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Kühn Editor for The Americas Institut für Theoretische Teilchenphysik Department of Physics Universität Karlsruhe University of California Postfach 69 80 Riverside, CA 92521 76128 Karlsruhe, Germany Phone: +1 (951) 827-5331 Phone: +49 (7 21) 6 08 33 72 Fax: +1 (951) 827-4529 Fax: +49 (7 21) 37 07 26 Email: chandra.edu Email: johann.edu www-ttp.de/∼jk Peter Wölfle Thomas Müller Institut für Theorie der Kondensierten Materie Institut für Experimentelle Kernphysik Universität Karlsruhe Fakultät für Physik Postfach 69 80 Universität Karlsruhe 76128 Karlsruhe, Germany Postfach 69 80 Phone: +49 (7 21) 6 08 35 90 76128 Karlsruhe, Germany Fax: +49 (7 21) 69 81 50 Phone: +49 (7 21) 6 08 35 24 Email: woelfle@tkm.de Fax: +49 (7 21) 6 07 26 21 www-tkm.de Email: thomas.de www-ekp.de Complex Systems, Editor Fundamental Astrophysics, Editor Frank Steiner Abteilung Theoretische Physik Joachim Trümper Universität Ulm Max-Planck-Institut für Extraterrestrische Physik Albert-Einstein-Allee 11 Postfach 13 12 89069 Ulm, Germany 85741 Garching, Germany Phone: +49 (7 31) 5 02 29 10 Phone: +49 (89) 30 00 35 59 Fax: +49 (7 31) 5 02 29 24 Fax: +49 (89) 30 00 33 15 Email: frank.steiner@uni-ulm.de Email: jtrumper@mpe.de/theo/qc/group.com Stefan Kehrein The Flow Equation Approach to Many-Particle Systems With 24 Figures ABC www.com Stefan Kehrein Ludwig-Maximilians-Universität München Fakultät für Physik Theresienstr. 37 80333 München Germany E-mail: stefan.de Library of Congress Control Number: 2006925894 Physics and Astronomy Classification Scheme (PACS): 01.-w ISSN print edition: 0081-3869 ISSN electronic edition: 1615-0430 ISBN-10 3-540-34067-X Springer Berlin Heidelberg New York ISBN-13 978-3-540-34067-6 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer.
Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the author using a Springer LATEX macro package Cover concept: eStudio Calamar Steinen Cover production: design &production GmbH, Heidelberg Printed on acid-free paper SPIN: 10985205 56/techbooks 543210 www.com To Michelle www.com Preface Over the past decade, the flow equation method has developed into a new ver- satile theoretical approach to quantum many-body physics.
Its basic concept was conceived independently by Wegner [1] and by Glazek and Wilson [2, 3]: the derivation of a unitary flow that makes a many-particle Hamiltonian in- creasingly energy-diagonal. This concept can be seen as a generalization of the conventional scaling approaches in many-body physics, where some ultra- violet energy scale is lowered down to the experimentally relevant low-energy scale [4]. The main difference between the conventional scaling approach and the flow equation approach can then be traced back to the fact that the flow equation approach retains all degrees of freedom, i. the full Hilbert space, while the conventional scaling approach focusses on some low-energy subspace.
One useful feature of the flow equation approach is therefore that it allows the calculation of dynamical quantities on all energy scales in one unified framework. Since its introduction, a substantial body of work using the flow equa- tion approach has accumulated. It was used to study a number of very dif- ferent quantum many-body problems from dissipative quantum systems to correlated electron physics. Recently, it also became apparent that the flow equation approach is very suitable for studying quantum many-body non- equilibrium problems, which form one of the current frontiers of modern theoretical physics.
Therefore the time seems ready to compile the research literature on flow equations in a consistent and accessible way, which was my goal in writing this book. The choice of material presented here is necessarily subjective and moti- vated by my own research interests. Still, I believe that the work compiled in this book provides a pedagogical introduction to the flow equation method from simple to complex models while remaining faithful to its nonpertur- bative character. Most of the models and examples in this book come from condensed matter theory, and a certain familiarity with modern condensed matter theory will be helpful in working through this book.1 Purposely, this book is focussed on the method and not on the physical background and moti- vation of the models discussed.
By working through it, a student or researcher 1 An excellent and highly recommended introduction is, for example, P. An- derson’s classic textbook [4].com VIII Preface should become well equipped to investigate models of one’s own interest using the flow equation approach. Most of the derivations are worked out in con- siderable detail, and I recommend to study them thoroughly to learn about the application and potential pitfalls of the flow equation approach. The flow equation approach is under active development and many issues still need to be addressed and answered.
I hope that this book will motivate its readers to contribute to these developments. I will try to keep track of such developments on my Internet homepage, and hope for e-mail feedback from the readers of this book. In particular, I am grateful for mentioning typos, which will be compiled on my homepage. Both in my research on flow equations and in writing the present book, I owe debts of gratitude to numerous colleagues.
First of all, I am deeply indebted to my Ph. advisor Franz Wegner, whose presentation of his new “flow equation scheme” in our Heidelberg group seminar in 1992 started both this whole line of research and my involvement in it. I also owe a very special acknowledgment to Andreas Mielke, with whom I have started my work on flow equations back in 1994. Our joint work has set the foundations of many of the developments presented in this book.
During my work on flow equations, I have also profited greatly from many discussions with Dieter Vollhardt. I am particularly grateful to him for his continued interest and encouragement. I also thank the participants of my flow equation lecture in Augsburg during the summer term 2005, which gave me the opportunity to test my presentation of the material that is compiled in this book. Among them I am especially thankful to Peter Fritsch, Lars Fritz, Andreas Hackl, Verena Körting, and Michael Möckel for proofreading parts of this manuscript.
The original idea to write this book is due to a suggestion by Peter Wölfle, and I am very grateful to him for starting me on this project and for his continued interest in the flow equation approach in general. This book project and a lot of the research compiled in it has only been possible due to a Heisenberg fellowship of the Deutsche Forschungsgemeinschaft (DFG). This gave me the necessary free time to pursue this project, and it is pleasure to acknowledge the DFG for this generous and unbureaucratic support through the Heisenberg program. Finally, I thank my colleagues at the University of Augsburg for many valuable discussions, and everyone else not mentioned here by name with whom I have worked on flow equations in the past decade.
For everything else and much more, I thank Michelle. Augsburg Stefan Kehrein February 2006 www.com References IX References 1. Anderson: Basic Notions of Condensed Matter Physics, 6th edn (Addison- Wesley, Reading Mass.com Contents 1 Introduction .2 Flow Equations: Basic Ideas .3 Outline and Scope of this Book. 9 2 Transformation of the Hamiltonian .1 Energy Scale Separation .1 Potential Scattering Model .2 Flow Equation Approach .2 Infinitesimal Unitary Transformations .3 Choice of Generator .3 Example: Potential Scattering Model .1 Setting up the Flow Equations .2 Methods of Solution .3 Strong-Coupling Case.
40 3 Evaluation of Observables .3 Fluctuation–Dissipation Theorem .1 Potential Scattering Model .2 Resonant Level Model .com XII Contents 4 Interacting Many-Body Systems .4 Normal-Ordered Expansions .5 Normal-Ordering with Respect to Which State? .1 Expansion in 1st Order (1-Loop Results) .2 Expansion in 2nd Order (2-Loop Results) .4 Transformation of the Spin Operator .5 Spin Correlation Function and Dynamical Susceptibility .6 Pseudogap Kondo Model .3 Spin–Boson Model .1 Flow of the Hamiltonian .2 Low-Energy Observables .4 Interacting Fermions in d > 1 Dimensions .1 Flow Equations and Fermi Liquid Theory .2 Flow Equations and Molecular-Field Type Hamiltonians .1 Construction of Effective Hamiltonians: The Fröhlich Transformation Re-examined .2 Block-Diagonal Hamiltonians .1 Strong-Coupling Behavior: Sine–Gordon Model .1 Sine–Gordon Model .2 Flow Equation Analysis .3 Conventional Scaling vs.2 Steady Non-Equilibrium: Kondo Model with Voltage Bias .1 Kondo Model in Non-Equilibrium .2 Flow Equation Analysis .3 Correlation Functions in Non-Equilibrium: Spin Dynamics .3 Real Time Evolution: Spin–Boson Model .4 Outlook and Open Questions .com 1 Introduction This introductory chapter provides a brief overview of the flow equation method and its relation to other methods in condensed matter theory. The aim of this chapter is to define the framework of the method, which will be filled out in more detail in the following chapters of this book.1 Motivation The fundamental challenge of condensed matter theory can be summed up by the observation that while we know all the relevant laws of nature for describing condensed matter systems, the number of degrees of freedom in such systems is typically much too large to allow a direct solution based on these laws. This observation is reflected in the multitude of phenomena that can be observed in condensed matter systems, from different kinds of ordering to phase transitions and novel states of matter like superconductiv- ity and fractional Quantum Hall liquids. In order to arrive at a theoretical understanding of such complex phenomena, various stages of simplifications and suitable modeling are necessary.
The resulting many-particle model then needs to be solved with a reliable theoretical method. Theoretical methods for solving quantum many-particle problems can be broadly classified in three main categories: 1. Perturbative analytical expansions 2. Exact analytical solutions 3.
Numerical solutions using computers All these approaches have their specific advantages and shortcomings.