Test of renormalization predictions for universal finite-size scaling functions

TL;DR

This paper tests theoretical predictions of universal finite-size scaling functions for systems with algebraically decaying interactions, finding significant discrepancies between predictions and numerical simulations.

Contribution

It provides the first numerical verification of renormalization predictions for systems with long-range interactions and explores the epsilon-expansion in this context.

Findings

Numerical results disagree with the predicted singular epsilon-expansion.

The study demonstrates the feasibility of varying epsilon continuously in simulations.

Results challenge existing theoretical predictions for long-range interaction systems.

Abstract

We calculate universal finite-size scaling functions for systems with an n-component order parameter and algebraically decaying interactions. Just as previously has been found for short-range interactions, this leads to a singular epsilon-expansion, where epsilon is the distance to the upper critical dimension. Subsequently, we check the results by numerical simulations of spin models in the same universality class. Our systems offer the essential advantage that epsilon can be varied continuously, allowing an accurate examination of the region where epsilon is small. The numerical calculations turn out to be in striking disagreement with the predicted singularity.