Spectral density and calculation of free energy

TL;DR

This paper compares two approximation methods for spectral density in Ising models, analyzing their impact on free energy calculations and critical behavior, revealing limitations and strengths of each approach.

Contribution

It introduces and evaluates the n-vicinity method as an alternative to polynomial approximation for spectral density in Ising models.

Findings

Free energy is nearly unaffected by the approximation method.

Derivatives of free energy vary significantly near the critical temperature.

Polynomial approximation can incorrectly predict the system reaching the ground state at finite temperature.

Abstract

For planar and cubic Ising models, we examined two ways of approximation of a spectral density that describes a degeneracy of energy levels. We approximated the exponent of the spectral density by polynomials of even degrees and using our n-vicinity method [8, 9]. According our analysis, the free energy is almost independent of the chosen method of approximation. However, its derivatives depend on the way of approximation and substantially differ in the neighborhood of a critical temperature. Our calculations showed that when approximating by polynomials the system necessarily finds itself in the ground state at a finite temperature, which is forbidden. The n-vicinity method approximates the derivatives of the free energy correctly for the cubic Ising model and it works poorly in the planar case.