Spin Systems

Roland Bauerschmidt; David C. Brydges; Gordon Slade

doi:10.1007/978-981-32-9593-3_1

Spin Systems

Roland Bauerschmidt, David C. Brydges, Gordon Slade

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter

Abstract

We provide an introduction to the theory of critical phenomena and discusses several of the models which serve as guiding examples. The Ising and multi-component |φ|⁴ spin models are introduced and motivated, with emphasis on their critical behaviour. The theory of the mean-field model is developed in a self-contained manner. The Gaussian free field is introduced and its relation to simple random walk is explained. The notion of universality is discussed. Recent results for the critical behaviour of the |φ|⁴ model are summarised, including the existence of logarithmic corrections to mean-field critical exponents in dimension d = 4.

Original language	English (US)
Title of host publication	Lecture Notes in Mathematics
Publisher	Springer Verlag
Pages	3-28
Number of pages	26
DOIs	https://doi.org/10.1007/978-981-32-9593-3_1
State	Published - 2019

Publication series

Name	Lecture Notes in Mathematics
Volume	2242
ISSN (Print)	0075-8434
ISSN (Electronic)	1617-9692

Keywords

Critical exponent
Critical phenomena
Ising model
Mean-field model
|φ| model

ASJC Scopus subject areas

Algebra and Number Theory

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10.1007/978-981-32-9593-3_1

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AB - We provide an introduction to the theory of critical phenomena and discusses several of the models which serve as guiding examples. The Ising and multi-component |φ|4 spin models are introduced and motivated, with emphasis on their critical behaviour. The theory of the mean-field model is developed in a self-contained manner. The Gaussian free field is introduced and its relation to simple random walk is explained. The notion of universality is discussed. Recent results for the critical behaviour of the |φ|4 model are summarised, including the existence of logarithmic corrections to mean-field critical exponents in dimension d = 4.

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KW - Ising model

KW - Mean-field model

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Spin Systems

Abstract

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Keywords

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