Universality in nontrivial continuum limits: a model calculation

TL;DR

This paper numerically investigates the continuum limit at a non-trivial fixed point of Dyson's hierarchical model, revealing universal behavior in renormalized couplings and amplitude ratios, with implications for understanding critical phenomena.

Contribution

It provides the first detailed numerical analysis of universal amplitude ratios and renormalized couplings at a non-trivial fixed point in Dyson's hierarchical model.

Findings

Dimensionless renormalized couplings are approximately universal.

Ratios of subleading amplitudes are also universal.

Renormalized couplings grow factorially with 2l, while amplitude ratios grow linearly.

Abstract

We study numerically the continuum limit corresponding to the non-trivial fixed point of Dyson's hierarchical model. We discuss the possibility of using the critical amplitudes as input parameters. We determine numerically the leading and subleading critical amplitudes of the zero-momentum $2l$-point functions in the symmetric phase up to l=10 for randomly chosen local measures. In the infinite cutoff limit, the dimensionless renormalized coupling constants are in very good approximation universal (independent of the choice of the local measure). In addition, ratios of subleading amplitudes also appear to be universal. If we neglect very small log-periodic corrections, the non-universal features of the 2l-point functions appear to depend only the non-universal features of the 2-point function. We infer that when 2l becomes large, the dimensionless renormalized couplings grow as (2l)! despite the non-perturbative nature of our calculation, while the universal ratios of subleading amplitudes grow linearly.