N-vicinities method for 3D Ising Model

TL;DR

This paper adapts the n-vicinities method to analyze the 3D Ising model, deriving the state equation for arbitrary dimensions and comparing results with simulations and exact solutions.

Contribution

It extends the n-vicinities method to the D-dimensional Ising model and analyzes the free energy behavior across different dimensions.

Findings

Good agreement with simulations for D > 2

Noticeable discrepancies for D < 3

Provides a generalized state equation for arbitrary D

Abstract

The n-vicinities method for approximate calculations of the partition function of a spin system was proposed previously. The equation of state was obtained in the most general form. In the present publication these results are adapted to the Ising model on the D-dimensional cubic lattice. The state equation is solved for an arbitrary dimension D and the behavior of the free energy is analyzed. For large values of D (D > 2) the obtained results are in good agreement with the ones obtained by means of computer simulations. For small values of D (D < 3), there are noticeable discrepancies with the exact results.