Phase transitions above the upper critical dimension

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 $d_{\mathrm{uc}}$. 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 $d_{\mathrm{uc}}$. 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 $d_{\mathrm{uc}}$. 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.

References in corpus (17)

• Classical critical behavior of spin models with long-range interactions
• Finite-size Scaling and Universality above the Upper Critical Dimensionality
• Scaling Relations for Logarithmic Corrections
• Hyperscaling above the upper critical dimension
• Self-Consistent Scaling Theory for Logarithmic Correction Exponents
• Finite-size scaling above the upper critical dimension
• Finite size scaling of the correlation length above the upper critical dimension
• On the role of Fourier modes in finite-size scaling above the upper critical dimension
• Finite size scaling of the 5D Ising model with free boundary conditions
• A new critical exponent koppa and its logarithmic counterpart koppa-hat
• Monte Carlo investigations of phase transitions: status and perspectives
• Lee-Yang theory of the Curie-Weiss model and its rare fluctuations
• Universal Finite-Size Scaling for Percolation Theory in High Dimensions
• Professor C. N. Yang and Statistical Mechanics
• Universal scaling relations for logarithmic-correction exponents
• The two-dimensional 4-state Potts model in a magnetic field
• Boundary effects on finite-size scaling for the 5-dimensional Ising model