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Cambridge University Press eBooks

Series Expansion Methods for Strongly Interacting Lattice Models

J. Oitmaa, Chris J. Hamer, Weihong Zheng

Published Apr 6, 2006• 285 citations
Publisher

Abstract

Perturbation series expansion methods are sophisticated numerical tools used to provide quantitative calculations in many areas of theoretical physics. This book gives a comprehensive guide to the use of series expansion methods for investigating phase transitions and critical phenomena, and lattice models of quantum magnetism, strongly correlated electron systems and elementary particles. Early chapters cover the classical treatment of critical phenomena through high-temperature expansions, and introduce graph theoretical and combinatorial algorithms. The book then discusses high-order linked-cluster perturbation expansions for quantum lattice models, finite temperature expansions, and lattice gauge models. Also included are numerous detailed examples and case studies, and an accompanying resources website, www.cambridge.org/9780521842426, contains programs for implementing these powerful numerical techniques. A valuable resource for graduate students and postdoctoral researchers working in condensed matter and particle physics, this book will also be useful as a reference for specialized graduate courses on series expansion methods.

DOI
10.1017/cbo9780511584398

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We review recent advances in the calculations of high-order convergent expansions for quantum many-particle systems. Calculations for ground state properties, including correlation functions and static susceptibilities, for spin models as well as for models of many fermions, such as Hubbard and Kondo models, are discussed. A historical perspective to the subject is provided. Recently important technical advances have been made in perturbative calculations of the excitation spectra of quantum many-particle systems, which enable the calculation of these spectra to high orders. The method, along with its applications, are explained. Fairly comprehensive, though simplified, algorithms for generating lists of relevant clusters, their lattice embeddings and subclusters are presented. The perturbative recursion relations and their computer implementation are also discussed in detail. A compilation is made of various series expansion studies that have been carried out for condensed matter problems. The scope and limitations of these methods are explained, and several open problems are noted.
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The Ising model provides a detailed mathematical description of ferromagnetism and is widely used in statistical physics and condensed matter physics. In this Student's Guide, the author demystifies the mathematical framework of the Ising model and provides students with a clear understanding of both its physical significance, and how to apply it successfully in their calculations. Key topics related to the Ising model are covered, including exact solutions of both finite and infinite systems, series expansions about high and low temperatures, mean-field approximation methods, and renormalization-group calculations. The book also incorporates plots, figures, and tables to highlight the significance of the results. Designed as a supplementary resource for undergraduate and graduate students, each chapter includes a selection of exercises intended to reinforce and extend important concepts, and solutions are also available for all exercises.