Spectral density and calculation of free energy

We examine a spectral density, which characterizes the degeneracy of each energy, for 2D and 3D Ising models. We compare two approximations for the spectral density, the polynomial approximation (Dixon et al., 2005) and the nv-approximation suggested by us in Kryzhanovsky and Litinskii (2017). Our analysis shows that while the free energy is almost independent of the approximation method, its derivatives depend on the approximation, and they differ significantly near the critical temperature. We show that for the polynomial approximation, the ground state of the system is reached at a finite temperature, in contradiction with fundamental physical principles. For the 2D Ising system, the nv-approximation is also incorrect. However, it works rather well when the dimension of the system is equal to three or higher.