Oscillatory Behavior of Critical Amplitudes of the Gaussian Model on a Hierarchical Structure

TL;DR

This paper investigates the oscillatory critical behavior of the Gaussian model on a hierarchical lattice, revealing non-standard singularities and challenging traditional finite-size scaling assumptions.

Contribution

It demonstrates that the critical amplitudes exhibit oscillations modulated by a logarithmic periodic function, and shows finite-size scaling does not hold in this non-standard critical context.

Findings

Critical amplitudes oscillate periodically in logarithmic scale.

Finite-size scaling hypothesis is invalid for this model.

The singular behavior of correlation length differs from standard predictions.

Abstract

We studied oscillatory behavior of critical amplitudes for the Gaussian model on a hierarchical structure presented by a modified Sierpinski gasket lattice. This model is known to display non-standard critical behavior on the lattice under study. The leading singular behavior of the correlation length $\xi$ near the critical coupling $K=K_c$ is modulated by a function which is periodic in $\ln|\ln(K_c-K)|$. We have also shown that the common finite-size scaling hypothesis, according to which for a finite system at criticality $\xi$ should be of the order of the size of system, is not applicable in this case. As a consequence of this, the exact form of the leading singular behavior of $\xi$ differs from the one described earlier (which was based on the finite-size scaling assumption).