25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple cubic lattice

TL;DR

This work computes a 25th-order high-temperature expansion for three-dimensional Ising-like systems on the simple cubic lattice and analyzes three improved Hamiltonians to suppress leading scaling corrections. By combining integral-approximant analyses, ratio-method checks, and asymptotic-coefficient matching, the authors obtain precise critical exponents: $oldsymbol{b3}=1.2373(2)$, $oldsymbol{}=0.63012(16)$, $oldsymbol{}=0.32653(10)$, $oldsymbol{}=0.03639(15)$, $oldsymbol{}=4.7893(8)$, and a correction-to-scaling exponent $oldsymbol{}=0.52(3)$; they also construct accurate parametric representations of the critical equation of state and report universal amplitude ratios consistent with prior theory and experiments. The study demonstrates that improved Hamiltonians significantly reduce systematic HT-analysis errors and yields a coherent, high-precision picture of critical behavior across multiple methodological cross-checks. The results advance quantitative understanding of the Ising universality class in three dimensions and provide robust benchmarks for theory and experiment alike.

Abstract

25th-order high-temperature series are computed for a general nearest-neighbor three-dimensional Ising model with arbitrary potential on the simple cubic lattice. In particular, we consider three improved potentials characterized by suppressed leading scaling corrections. Critical exponents are extracted from high-temperature series specialized to improved potentials, obtaining $γ=1.2373(2)$, $ν=0.63012(16)$, $α=0.1096(5)$, $η=0.03639(15)$, $β=0.32653(10)$, $δ=4.7893(8)$. Moreover, biased analyses of the 25th-order series of the standard Ising model provide the estimate $Δ=0.52(3)$ for the exponent associated with the leading scaling corrections. By the same technique, we study the small-magnetization expansion of the Helmholtz free energy. The results are then applied to the construction of parametric representations of the critical equation of state, using a systematic approach based on a global stationarity condition. Accurate estimates of several universal amplitude ratios are also presented.

Figures (15)

• Figure 1: Estimates of $\gamma$ as obtained by analyzing the HT series of $\chi$ for the $\phi^4$ model at $\lambda_4=1.10$ versus the order $n$ of the series considered in the analysis. Several approximants (defined in the text) are considered.
• Figure 2: Estimates of $\gamma$ as obtained by analyzing the HT series of $\chi$ for the improved models and for the spin-1/2 model, versus the order $n$ of the series considered in the analysis. IA2's are considered.
• Figure 3: Sequence $\beta_c^{(n)}$ for the $\phi^4$ model at $\lambda_4=1.10$ using the series for $\chi$. The dashed lines indicate the IA estimate of $\beta_c$.
• Figure 4: Sequence $\beta_c^{(n)}$ for the spin-1 model at $D=0.641$ using the series for $\chi$. The dashed lines indicate the IA estimate of $\beta_c$.
• Figure 5: Sequence $\beta_c^{(n)}$ for the spin-1/2 model using the series for $\chi$. The dashed lines indicate the MC estimate of $\beta_c$, while the dotted line corresponds to a $n^{-3/2}$ extrapolation of the four points with $n=22,23,24,25$.