Ising Model with a Magnetic Field

Abstract

AbstractThe paper presents the low temperature expansion of the 2D Ising model in the presence of the magnetic field in powers of $$x=\exp (-J/(kT))$$ x = exp ( - J / ( k T ) ) and $$z=\exp (B/(kT))$$ z = exp ( B / ( k T ) ) with full polynomials in z up to $$x^{88}$$ x 88 and full polynomials in $$x^4$$ x 4 up to $$z^{-60}$$ 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 $$(z^2+x^8)^{-k}$$ ( z 2 + x 8 ) - k where the subsequent coefficients (polynomials in $$x^4$$ x 4 ) turn out to be divisible by increasing powers of $$(1-x^4)$$ ( 1 - x 4 ) . This result gives a hint about the intriguing ‘off-diagonal’ correlations in the Ising model mixing the $$B\ne 0$$ B ≠ 0 contributions with the usual low temperature $$B=0$$ 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.