The enigmatic exponent koppa and the story of finite-size scaling above the upper critical dimension

TL;DR

This paper investigates the peculiar behavior of finite-size scaling and critical exponents, especially the enigmatic exponent koppa, in high-dimensional systems above the upper critical dimension, revealing new insights into critical phenomena.

Contribution

It introduces a novel understanding of the exponent koppa and clarifies the limitations of traditional finite-size scaling theories in high dimensions.

Findings

Finite-size scaling breaks down above the upper critical dimension.

The exponent koppa plays a crucial role in high-dimensional critical phenomena.

Traditional scaling relations need modification for high-dimensional systems.

Abstract

Scaling, hyperscaling and finite-size scaling were long considered problematic in theories of critical phenomena in high dimensions. The scaling relations themselves form a model-independent structure that any model-specific theory must adhere to, and they are accounted for by the simple principle of homogeneity. Finite-size scaling is similarly founded on the fundamental idea that only two length scales enter the game -- namely system length and correlation length. While all scaling relations are quite satisfactory for multitudes of physical systems in low dimensions, one fails in high dimensions...