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Estimates of Critical Quantities from an Expansion in Mass: Ising Model on the Simple Cubic Lattice

  • Condensed Matter
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Abstract

In the Ising model on the simple cubic lattice, we describe the inverse temperature β and other quantities relevant for the computation of critical quantities in terms of a dimensionless squared mass M. The critical behaviors of those quantities are represented by the linear differential equations with constant coefficients which are related to critical exponents. We estimate the critical temperature and exponents via an expansion in the inverse powers of the mass under the use of δ-expansion. The critical inverse temperature β c is estimated first in unbiased manner and then critical exponents are also estimated in biased and unbiased self-contained way including ω, the correction-to-scaling exponent, ν, η, and γ.

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Notes

  1. For example, when the site j is on the diagonal direction with j = J(1,1,1), it is found at the leading order that \(<s_{0}s_{j}>\sim \frac {(3J)!}{(J!)^{3}}(\tanh \beta )^{3J}\). At large enough J, \(\lim _{J\to \infty }\left (\frac {(3J)!}{(J!)^{3}}\right )^{1/3J}=3\) and < s 0 s j > ∼ exp[3J log3β]. Thus we obtain \(\xi \sim \sqrt {3}\log (3M)\).

  2. In addition to the group of contributions τ −γ(1 + c o n s t×τ 𝜃+⋯ ), Aharony and Fisher found the presence of the group labelled by τ 1−α = τ −γ⋅τ γ + 1−α where α stands for the exponent of specific heat. See A. Aharony and M. E. Fisher, Phys. Rev. B 27, 4394 (1983). It is however considered just as the correction high enough to be omitted in this analysis (α is positive but small).

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  1. Division of Mathematics and Science, Chiba Institute of Technology, Shibazono 2-1-1, Narashino, Chiba, 275-0023, Japan

    Hirofumi Yamada

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Yamada, H. Estimates of Critical Quantities from an Expansion in Mass: Ising Model on the Simple Cubic Lattice. Braz J Phys 45, 584–603 (2015). https://doi.org/10.1007/s13538-015-0353-8

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