Phase transitions above the upper critical dimension

Bertrand Berche, Tim Ellis, Yurij Holovatch, Ralph Kenna

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14 Citations (Scopus)
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Abstract

These lecture notes provide an overview of the renormalization group (RG) as a successful framework to understand critical phenomena above the upper critical dimension d_{uc}duc. After an introduction to the scaling picture of continuous phase transitions, we discuss the apparent failure of the Gaussian fixed point to capture scaling for Landau mean-field theory, which should hold in the thermodynamic limit above d_{uc}duc. We recount how Fisher’s dangerous-irrelevant-variable formalism applied to thermodynamic functions partially repairs the situation but at the expense of hyperscaling and finite-size scaling, both of which were, until recently, believed not to apply above d_{uc}duc. We recall limitations of various attempts to match the RG with analytical and numerical results for Ising systems. We explain how the extension of dangerous irrelevancy to the correlation sector is key to marrying the above concepts into a comprehensive RG scaling picture that renders hyperscaling and finite-size scaling valid in all dimensions. We collect what we believe is the current status of the theory, including some new insights and results. This paper is in grateful memory of Michael Fisher who introduced many of the concepts discussed and who, half a century later, contributed to their advancement.
Original languageEnglish
Number of pages44
JournalSciPost Physics Lecture Notes
Volume60
DOIs
Publication statusPublished - 12 Aug 2022

Bibliographical note

Copyright B. Berche et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation. (CC BY)

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