Fixed point stability and decay of correlations

TL;DR

This paper investigates the stability of fixed points in the renormalization-group flow of $ ext{Phi}^4$ theories, proposing and supporting the conjecture that the stable fixed point corresponds to the fastest decay of correlations, characterized by the largest $ ext{eta}$ exponent.

Contribution

The paper provides a proof of the conjecture within the $ ext{epsilon}$-expansion framework and discusses its validity in lower-dimensional cases, supporting the idea that the stable fixed point maximizes the decay rate of correlations.

Findings

The stable fixed point has the largest $ ext{eta}$ exponent.

The conjecture is proven in the $ ext{epsilon}$-expansion.

Numerical cases support the conjecture beyond the $ ext{epsilon}$-expansion.

Abstract

In the framework of the renormalization-group theory of critical phenomena, a quantitative description of many continuous phase transitions can be obtained by considering an effective $\Phi^4$ theories, having an N-component fundamental field $\Phi_i$ and containing up to fourth-order powers of the field components. Their renormalization-group flow is usually characterized by several fixed points. We give here strong arguments in favour of the following conjecture: the stable fixed point corresponds to the fastest decay of correlations, that is, is the one with the largest values of the critical exponent $\eta$ describing the power-law decay of the two-point function at criticality. We prove this conjecture in the framework of the $\epsilon$-expansion. Then, we discuss its validity beyond the $\epsilon$-expansion. We present several lower-dimensional cases, mostly three-dimensional, which support the conjecture. We have been unable to find a counterexample.