Phase transitions above the upper critical dimension

TL;DR

This paper reviews the renormalization group approach to critical phenomena above the upper critical dimension, clarifying how hyperscaling and finite-size scaling are valid in all dimensions through dangerous irrelevance.

Contribution

It extends the understanding of RG scaling above the upper critical dimension by incorporating dangerous irrelevance into the correlation sector, reconciling theory with analytical and numerical results.

Findings

Hyperscaling and finite-size scaling are valid above the upper critical dimension.

Dangerous irrelevance in the correlation sector is crucial for RG scaling.

Current theory includes new insights and results on critical phenomena above $d_{uc}$.

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 $d_{\rm uc}$. 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 $d_{\rm uc}$. 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 $d_{\rm uc}$. 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.