Effective-Dimension Theory of Critical Phenomena above Upper Critical Dimensions

TL;DR

This paper introduces an effective-dimension theory to resolve longstanding inconsistencies in the renormalization-group predictions of critical phenomena above the upper critical dimension, unifying various results and deriving new scaling laws.

Contribution

It presents a novel perspective on dangerous irrelevant variables, clarifies the origin of multiple critical exponents, and develops an effective-dimension framework that consistently explains existing and new results.

Findings

Unveiled the origin of two sets of critical exponents.

Developed an effective-dimension theory accounting for critical fluctuations.

Derived a new finite-size scaling law for correlation functions.

Abstract

Phase transitions and critical phenomena are among the most intriguing phenomena in nature and their renormalization-group theory is one of the greatest achievements of theoretical physics. However, the predictions of the theory above an upper critical dimension $d_c$ seriously disagree with reality. In addition to its fundamental significance, the problem is also of practical importance because both complex systems with spatial or temporal long-range interactions and quantum phase transitions can substantially lower $d_c$. The extant scenarios built on a dangerous irrelevant variable (DIV) to resolve the problem introduce two sets of critical exponents and even two sets of scaling laws whose origin is obscure. Here, we consider the DIV from a different perspective and clearly unveil the origin of the two sets of exponents and hence the intrinsic inconsistency in those scenarios. We then develop an effective-dimension theory in which critical fluctuations and system volume are fixed at an effective dimension by the DIV. This enables us to account for all the extant results consistently. A novel asymptotic finite-size scaling behavior for a correlation function together with a new anomalous dimension and its associated scaling law is also derived.