Improved high-temperature expansion and critical equation of state of three-dimensional Ising-like systems

TL;DR

This work advances the precision of critical parameters in 3D Ising-like systems by employing improved Hamiltonians that suppress leading scaling corrections, enabling high-accuracy high-temperature expansions up to order $\beta^{20}$ and robust extraction of universal quantities. It develops a global-stationarity parametric framework to reconstruct the full critical equation of state from small-field data, yielding tight universal ratios and connecting to $\epsilon$-expansion results. The combination of improved HT analysis, effective-potential couplings, and a stationary parametric representation produces exponents $\gamma=1.2371(4)$, $\nu=0.63002(23)$, $\eta=0.0364(4)$, $\alpha=0.1099(7)$, $\beta=0.32648(18)$, and $\delta=4.7893(22)$, with very consistent results across three models and good agreement with Monte Carlo and field-theoretic analyses. The methodology yields precise universal amplitude ratios (e.g., $g_4^*=23.49(4)$) and provides a coherent bridge between HT data, effective potentials, and the critical equation of state, with implications for comparing theory to experiments in Ising-class systems.

Abstract

High-temperature series are computed for a generalized $3d$ Ising model with arbitrary potential. Two specific ``improved'' potentials (suppressing leading scaling corrections) are selected by Monte Carlo computation. Critical exponents are extracted from high-temperature series specialized to improved potentials, achieving high accuracy; our best estimates are: $γ=1.2371(4)$, $ν=0.63002(23)$, $α=0.1099(7)$, $η=0.0364(4)$, $β=0.32648(18)$. By the same technique, the coefficients of the small-field expansion for the effective potential (Helmholtz free energy) are computed. These results are applied to the construction of parametric representations of the critical equation of state. A systematic approximation scheme, based on a global stationarity condition, is introduced (the lowest-order approximation reproduces the linear parametric model). This scheme is used for an accurate determination of universal ratios of amplitudes. A comparison with other theoretical and experimental determinations of universal quantities is presented.

Figures (1)

• Figure 1: Comparison of the determination of $g_4$ (plotted vs. $1/\lambda_4$) from HT series without (direct) and with the Roskies transform (RT) for the pure $\phi^4$ lattice model. The dashed line marks the more precise estimate (with its error) we derived from the analysis of the IHT expansion.