Finite Size Scaling for O(N) phi^4-Theory at the Upper Critical Dimension

TL;DR

This paper develops a finite size scaling theory for O(N) phi^4-theory at the upper critical dimension, revealing mean-field behavior with logarithmic corrections linked to the theory's triviality, and proposes numerical tests using the Ising model.

Contribution

It derives a finite size scaling framework for O(N) phi^4-theory in four dimensions, highlighting N-dependent and independent logarithmic corrections related to triviality.

Findings

Mean-field like scaling with logarithmic corrections

Logarithmic corrections are N-independent for odd quantities

Numerical tests with the Ising model can verify triviality

Abstract

A finite size scaling theory for the partition function zeros and thermodynamic functions of O(N) phi^4-theory in four dimensions is derived from renormalization group methods. The leading scaling behaviour is mean-field like with multiplicative logarithmic corrections which are linked to the triviality of the theory. These logarithmic corrections are independent of N for odd thermodynamic quantities and associated zeros and are N dependent for the even ones. Thus a numerical study of finite size scaling in the Ising model serves as a non-perturbative test of triviality of phi^4_4-theories for all N.