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Physical review. B, Condensed matter

Nonlinear-renormalization-group study of a measurable quantity in a large range of temperature above Tc for Ising-like systems

C. Bagnuls, Claude Bervillier

Published Jun 1, 1983• 13 citations
Publisher

Abstract

We perform a numerical study, in the framework of the massive ${\ensuremath{\varphi}}^{4}$ field theory in dimension $d=3$ (${\ensuremath{\varphi}}_{3}^{4}$) for Ising-like systems above ${T}_{c}$, of the variation in $t$ of the dimensionless combination $R(t)={{\ensuremath{-}{t}^{3}[\frac{\ensuremath{\partial}C(t)}{\ensuremath{\partial}t}]{\ensuremath{\xi}}^{3}(t)}}^{\frac{1}{3}}$; $C$ being the specific heat, $\ensuremath{\xi}$ the correlation length, and $t$ a critical variable which vanishes at ${T}_{c}$. $R(0)$ is the universal combination of critical amplitudes ${R}_{\ensuremath{\xi}}^{+}$. The range in $t$ studied goes from the classical region to the true critical region. Together with estimations of $R(t)$ for discrete values of $t$, we provide an analytical form of fit for the central values. This will allow experimentalists to improve the analysis of confluent corrections to scaling.

DOI
10.1103/physrevb.27.6995

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We perform a numerical study, in the framework of the massive \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 field theory in dimension d=3d=3d=3 (\ensuremathφ34{\ensuremath{\varphi}}_{3}^{4}\ensuremathφ34​) for Ising-like systems above Tc{T}_{c}Tc​, of the variation in ttt of the dimensionless combination R(t)=\ensuremath−t3[\ensuremath∂C(t)\ensuremath∂t]\ensuremathξ3(t)13R(t)={{\ensuremath{-}{t}^{3}[\frac{\ensuremath{\partial}C(t)}{\ensuremath{\partial}t}]{\ensuremath{\xi}}^{3}(t)}}^{\frac{1}{3}}R(t)=\ensuremath−t3[\ensuremath∂t\ensuremath∂C(t)​]\ensuremathξ3(t)31​; CCC being the specific heat, \ensuremathξ\ensuremath{\xi}\ensuremathξ the correlation length, and ttt a critical variable which vanishes at Tc{T}_{c}Tc​. R(0)R(0)R(0) is the universal combination of critical amplitudes R\ensuremathξ+{R}_{\ensuremath{\xi}}^{+}R\ensuremathξ+​. The range in ttt studied goes from the classical region to the true critical region. Together with estimations of R(t)R(t)R(t) for discrete values of ttt, we provide an analytical form of fit for the central values. This will allow experimentalists to improve the analysis of confluent corrections to scaling.
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We estimate, at d=3d=3d=3, from ${\ensuremath{\varphi}}^{4}$ field theory the universal ratios of confluent critical correction amplitudes relative to the correlation length, the susceptibility, and the specific heat above Tc{T}_{c}Tc​. Our central values are comparable to these available from analysis of high-temperature series and the $\ensuremath{\epsilon}$ expansion. Furthermore, experiments in progress on a binary mixture are in good agreement with one of our results, the ratio of the confluent corrections for the specific heat and susceptibility which have not been previously estimated. We also present, in detail, a derivation of the universality of these quantities in the framework of field theory.
We use a high-precision Monte Carlo simulation to determine the universal specific-heat amplitude ratio A + / A − in the three-dimensional Ising model via the impact angle ϕ of complex temperature zeros. We also measure the correlation-length critical exponent ν from finite-size scaling and the specific-heat exponent α through hyperscaling. Extrapolations to the thermodynamic limit yield ϕ = 59.2(1.0)°, A + / A − = 0.56(3), ν = 0.630 48(32) and α = 0.1086(10). These results are compatible with some previous estimates from a variety of sources and rule out recently conjectured exact values.
The microscopic approach to calculating and investigating the three-dimensional (3D) Ising-like system thermodynamics above and below the phase transition temperature Tc is developed using the collective variables method.Mathematical description is performed within the framework of the higher non-Gaussian approximation ( 6 model) taking into account the corrections to scaling.A nonuniversal factor depending on microscopic parameters of the system is singled out in the expressions for leading critical amplitudes and correction-to-scaling amplitudes.Numerical estimates of the critical region size, phase transition temperature as well as the graphs of temperature dependences of entropy and other thermodynamic characteristics near Tc are given for various values of eective radius of the exponentially decreasing interaction potential.The variation of the free energy of the system at the phase transition point, average spin moment, and speci c heat with increasing ratio of the potential eective radius to the simple cubic lattice constant is traced.The results of calculations and their comparison with the other authors' data show that the 6 model provides a more adequate quantitative description of the critical behaviour of a 3D Ising-like system than the 4 model.