Triviality problem and the high-temperature expansions of the higher susceptibilities for the Ising and the scalar field models on four-, five- and six-dimensional lattices

TL;DR

This paper extends high-temperature series expansions up to order 24 for Ising and scalar field models on high-dimensional lattices, providing numerical evidence supporting the triviality of their continuum limits in dimensions four and above.

Contribution

It significantly advances high-temperature expansion data for high-dimensional Ising and scalar models, enabling detailed analysis of their triviality properties.

Findings

Results support the triviality hypothesis in four, five, and six dimensions.

High-temperature expansions are validated as effective for studying high-dimensional models.

Numerical data align with renormalization group predictions.

Abstract

High-temperature expansions are presently the only viable approach to the numerical calculation of the higher susceptibilities for the spin and the scalar-field models on high-dimensional lattices. The critical amplitudes of these quantities enter into a sequence of universal amplitude-ratios which determine the critical equation of state. We have obtained a substantial extension through order 24, of the high-temperature expansions of the free energy (in presence of a magnetic field) for the Ising models with spin s >= 1/2 and for the lattice scalar field theory with quartic self-interaction, on the simple-cubic and the body-centered-cubic lattices in four, five and six spatial dimensions. A numerical analysis of the higher susceptibilities obtained from these expansions, yields results consistent with the widely accepted ideas, based on the renormalization group and the constructive approach to Euclidean quantum field theory, concerning the no-interaction ("triviality") property of the continuum (scaling) limit of spin-s Ising and lattice scalar-field models at and above the upper critical dimensionality.