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Reflection Positivity and Phase Transitions in Lattice Spin Models

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Book cover Methods of Contemporary Mathematical Statistical Physics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1970))

Abstract

Phase transitions are one of the most fascinating, and also most perplexing, phenomena in equilibrium statistical mechanics. On the physics side, many approximate methods to explain or otherwise justify phase transitions are known but a complete mathematical understanding is available only in a handful of simplest of all cases. One set of tractable systems consists of the so called lattice spin models. Originally, these came to existence as simplified versions of (somewhat more realistic) models of crystalline materials in solid state physics but their versatile nature earned them a life of their own in many other disciplines where complex systems are of interest.

The present set of notes describes one successful mathematical approach to phase transitions in lattice spin models which is based on the technique of reflection positivity. This technique was developed in the late 1970s in the groundbreaking works of F. Dyson, J. Fr¨ohlich, R. Israel, E. Lieb, B. Simon and T. Spencer who used it to establish phase transitions in a host of physically-interesting classical and quantum lattice spin models; most notably, the classical Heisenberg ferromagnet and the quantum XY model and Heisenberg antiferromagnet. Other powerful techniques — e.g., Pirogov-Sinai theory, lace expansion or multiscale analysis in field theory — are available at present that can serve a similar purpose in related contexts, but we will leave their review to experts in those areas

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  1. Department of Mathematics, UCLA, 600 S. Michigan Avenue, 60605, Los Angeles, CA, USA

    Marek Biskup

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  1. Ctr. for Theoretical Study, Charles University, Jilska 1, Praha 1, 110 00, Czech Republic

    Roman Kotecký

  2. Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK

    Roman Kotecký

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Biskup, M. (2009). Reflection Positivity and Phase Transitions in Lattice Spin Models. In: Kotecký, R. (eds) Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Mathematics(), vol 1970. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92796-9_1

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