Spectral density and calculation of free energy

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.Key words: Ising model, free energy, spectral density, n-vicinity method.In the both formulas, J is the interaction constant between neighboring spins.1 The factor N in the exponent means that we calculate the energy () E s per one spin (see Eq. (1.4).

Approximation of the n-vicinity method

“…Более того, к настоящему времени вид распределения энергий известен только для одномерной модели Изинга. Это обстоятельство инициировало ряд работ, авторы которых пытаются получить выражения для распределения энергий в рамках различных приближений и численными методами [8][9][10][11][12][13].…”

n-vicinity Method and 1D Ising Model

“…In Ref. [6] we showed that if 0 E is the energy of the ground state and the total density is an exponent…”