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Condensed Matter Physics and Exactly Soluble Models

Selecta of Elliott H. Lieb

  • Book
  • © 2004

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Editors:
  1. Bruno Nachtergaele

    1. Department of Mathematics, University of California, Davis, USA

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  2. Jan Philip Solovej

    1. Department of Mathematics, University of Copenhagen, Copenhagen, Denmark

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  3. Jakob Yngvason

    1. Institut für Theoretische Physik, Universität Wien, Wien, Austria

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  • This is a collection of groundbreaking papers by E. Lieb, one of the founding fathers of mathematical solid state physics

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Table of contents (40 chapters)

  1. Front Matter

    Pages I-IX
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  2. Commentaries

    1. Front Matter

      Pages 1-1
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    2. A Survey by the Editors

      • Bruno Nachtergaele, Jan Philip Solovej, Jakob Yngvason
      Pages 3-16

  3. Condensed Matter Physics

    1. Front Matter

      Pages 17-17
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    2. Correlated Electrons

      1. Violation of the Noncrossing Rule: The Hubbard Hamiltonian for Benzene
        • Ole J. Heilmann, Elliott H. Liebt
        Pages 21-54
      2. Two Theorems on the Hubbard Model
        • Elliott H. Lieb
        Pages 55-58
      3. The Hubbard model: Some Rigorous Results and Open Problems
        • Elliott H. Lieb
        Pages 59-77
      4. Flux Phase of the Half-Filled Band
        • Elliott H. Lieb
        Pages 79-82

    3. Peierls Instability

      1. Proof of the Peierls Instability in One Dimension
        • Tom Kennedy, Elliott H. Lieb
        Pages 85-88
      2. Stability of the Peierls instability for ring-shaped molecules
        • Elliott H. Lieb, Bruno Nachtergaele
        Pages 89-103

    4. Falicov—Kimball Mode

      1. A Model for Crystallization: A Variation on the Hubbard Model
        • Elliott H. Lieb
        Pages 107-117
      2. Phase Separation due to Quantum Mechanical Correlations
        • James K. Freericks, Elliott H. Lieb, Daniel Ueltschi
        Pages 119-122

    5. Ordering of Energy Levels

      1. Theory of Ferromagnetism and the Ordering of Electronic Energy Levels
        • Elliott Lieb, Daniel Mattis
        Pages 125-133
      2. Ordering Energy Levels of Interacting Spin Systems
        • Elliott Lieb, Daniel Matris
        Pages 135-137
      3. Magnetic Properties of Some Itinerant-Electron Systems at <i>T</i> &gt; 0
        • Michael Aizenman, Elliott H. Lieb
        Pages 139-142

    7. Quantum Spin Systems

      1. A Proof of Part of Haldane’s Conjecture on Spin Chains
        • Ian Affleck, Elliott H. Lieb
        Pages 235-247
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Keywords

About this book

This is the third Selecta of publications of Elliott Lieb, the first two being Stabil­ ity of Matter: From Atoms to Stars, edited by Walter Thirring, and Inequalities, edited by Michael Loss and Mary Beth Ruskai. A companion fourth Selecta on Statistical Mechanics is also edited by us. Elliott Lieb has been a pioneer of the discipline of mathematical physics as it is nowadays understood and continues to lead several of its most active directions today. For the first part of this selecta we have made a selection of Lieb's works on Condensed Matter Physics. The impact of Lieb's work in mathematical con­ densed matter physics is unrivaled. It is fair to say that if one were to name a founding father of the field, Elliott Lieb would be the only candidate to claim this singular position. While in related fields, such as Statistical Mechanics and Atomic Physics, many key problems are readily formulated in unambiguous mathematical form, this is less so in Condensed Matter Physics, where some say that rigor is "probably impossible and certainly unnecessary". By carefully select­ ing the most important questions and formulating them as well-defined mathemat­ ical problems, and then solving a good number of them, Lieb has demonstrated the quoted opinion to be erroneous on both counts. What is true, however, is that many of these problems turn out to be very hard. It is not unusual that they take a decade (even several decades) to solve.

Editors and Affiliations

  • Department of Mathematics, University of California, Davis, USA

    Bruno Nachtergaele

  • Department of Mathematics, University of Copenhagen, Copenhagen, Denmark

    Jan Philip Solovej

  • Institut für Theoretische Physik, Universität Wien, Wien, Austria

    Jakob Yngvason

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