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_{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.

Monte Carlo Simulation of Long Hard-Sphere Polymer Chains in Two to Five Dimensions

“…Above the upper critical dimension D > 4 "the puzzles of finite-size scaling are still not fully resolved" [55]. Although there is only the one Gaussian fixed point it has been argued [56] that the way excluded volume impacts scaling should still be quantifiable by the crossover exponent φ = (4 − D)/2 and that the leading-order correction to scaling for D = 5 should, therefore, be of order L −1/2 .…”

On the new universality class in structurally disordered $n$-vector model with long-range interactions

“…It is said that they belong to the short-range universality class. For 𝑑 larger than the upper critical dimension 𝑑 𝑢𝑐 = 4 the model is governed by the mean-field exponents, see [9] for more details. Introducing the long-range interaction (2) drastically changes the picture of the critical behaviour of the 𝑛-vector model (1).…”

“…An analytical continuation of the Borel transform is achieved by representing it in the form of the diagonal Padé approximant [1/1] 𝐵 (𝜖), where subscript 𝐵 is used to distinguish from the Padé approximant of the original series (9). Finally, the resummed function is obtained via an inverse Borel These data were obtained from interpolation of the NPRG results for critical exponents of the long-range 𝑛-vector model [15].…”

Finite-size scaling of the majority-voter model above the upper critical dimension

“…Above 𝑑 𝑐 , the situation is much simpler and the critical exponents take the values predicted by the mean-field theory. However, the correct finite-size scaling of thermodynamic averages above the upper critical dimension has been clarified only recently [2].…”

“…This theory has recently been extended to quantum phase transitions [11]. Despite some indications that the same finite-size scaling holds with free boundary conditions, the problem is not completely settled [2,12].…”