Ising Model with a Magnetic Field

Abstract

The paper presents the low temperature expansion of the 2D Ising model in the presence of the magnetic field in powers of x=exp (-J/(kT)) and z=exp (B/(kT)) with full polynomials in z up to x^{88} and full polynomials in x^4 up to z^{-60}, in the latter case the polynomials are explicitly given. The new result presented in the paper is an expansion not in inverse powers of z but in (z^2+x^8)^{-k} where the subsequent coefficients (polynomials in x^4) turn out to be divisible by increasing powers of (1-x^4). This result gives a hint about the intriguing ‘off-diagonal’ correlations in the Ising model mixing the Bne 0 contributions with the usual low temperature B=0 expansion what may be useful on the road to find the full analytic expression for the partition function of the Ising model with non-vanishing magnetic field. The paper describes both the analytic expansions of the partition function and the efficient combinatorial methods to get the coefficients of the expansion.