Condensed Matter Physics and Exactly Soluble Models

1. Condensed Matter Physics

   1. Quantum Spin Systems

      1. Rigorous Results on Valence-Bond Ground States in Antiferromagnets

         • Ian Affleck, Tom Kennedy, Elliott H. Lieb, Hal Tasaki

         Pages 249-252

      2. Valence Bond Ground States in Isotropic Quantum Antiferromagnets

         • Ian Affleck, Tom Kennedy, Elliott H. Lieb, Hal Tasaki

         Pages 253-304

      3. Ground State Properties of a Fully Frustrated Quantum Spin System

         • Elliott H. Lieb, Peter Schupp

         Pages 305-308

   2. The Polaron

      1. Exact Ground State Energy of the Strong-Coupling Polaron

         • Elliott H. Lieb, Lawrence E. Thomas

         Pages 311-321

   3. Interacting Bosonss

      1. Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas

         • Elliott H. Lieb

         Pages 325-335

      2. Bose-Einstein quantum phase transition in an optical lattice model

         • Michael Aizenman, Elliott H. Lieb, Robert Seiringer, Jan Philip Solovej, Jakob Yngvason

         Pages 337-349

      3. The Quantum-Mechanical Many-Body Problem: The Bose Gas

         • Elliott H. Lieb, Robert Seiringer, Jan Philip Solovej, Jakob Yngvason

         Pages 351-435

2. Exactly Soluble Models

   1. Front Matter

      Pages 437-437

      PDF

   2. Some of the Early History of Exactly Soluble Models

      • Elliott H. Lieb

      Pages 439-446

   3. The Six-Vertex Ice Type Models

      1. Exact Solution of the Problem of the Entropy of Two-Dimensional Ice

         • Elliott H. Lieb

         Pages 449-451

      2. Exact Solution of the <i>F</i> Model of an Antiferroelectric

         • Elliott H. Lieb

         Pages 453-455

      3. Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric

         • Elliott H. Lieb

         Pages 457-459

      4. Residual Entropy of Square Ice

         • Elliott H. Lleb

         Pages 461-471

      5. Two-Dimensional Ferroelectric Models

         • Elliott H. Lieb

         Pages 473-474

      6. Relations between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem

         • H. N. V. Temperley, E. H. Lieb

         Pages 475-504

      7. Analytic Properties of the Free Energy for the “Ice” Models

         • M. L. Glasser, D. B. Abraham, E. H. Lieb

         Pages 505-518

   4. Ising and Dimer Models

      1. Two-Dimensional Ising Model as a Soluble Problem of Many Fermions

         • T. D. Schultz, D. C. Mattis, E. H. Lieb

         Pages 521-536

      2. Solution of the Dimer Problem by the Transfer Matrix Method

         • Elliott H. Lieb

         Pages 537-539

   5. Spin Chains

      1. Two Soluble Models of an Antiferromagnetic Chain

         • Elliott Lieb, Theodore Schultz, Daniel Mattis

         Pages 543-601

   6. Delta Function Bose Gas

      1. Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State

         • Elliott H. Lieb, Werner Liniger

         Pages 605-616

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.

• Bose Gas
• Bose-Einstein Condensation
• Exactly Soluble Models
• Helium-Atom-Streuung
• Hubbard Model
• Integrabe Models
• Ising Model
• Peirls Instability
• Polaron
• Quantum Spin Systems
• Spin Chain
• molecule

• 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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