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arXiv:1010.6244 · Physical Review B

Extended scaling analysis of the S=12 Ising ferromagnet on the simple cubic lattice

Ian A. Campbell, P. H. Lundow

Published Jan 13, 2011• 13 citations
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Abstract

It is often assumed that for treating numerical (or experimental) data on continuous transitions the formal analysis derived from the renormalization-group theory can only be applied over a narrow temperature range, the ``critical region''; outside this region correction terms proliferate rendering attempts to apply the formalism hopeless. This pessimistic conclusion follows largely from a choice of scaling variables and scaling expressions, which is traditional but very inefficient for data covering wide temperature ranges. An alternative ``extended scaling'' approach can be made where the choice of scaling variables and scaling expressions is rationalized in the light of well established high-temperature series expansion developments. We present the extended scaling approach in detail, and outline the numerical technique used to study the three-dimensional ($3$D) Ising model. After a discussion of the exact expressions for the historic $1$D Ising spin chain model as an illustration, an exhaustive analysis of high quality numerical data on the canonical simple cubic lattice $3$D Ising model is given. It is shown that in both models, with appropriate scaling variables and scaling expressions (in which leading correction terms are taken into account where necessary), critical behavior extends from ${T}_{c}$ up to infinite temperature.

DOI
10.1103/physrevb.83.014411

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It is often assumed that for treating numerical (or experimental) data on continuous transitions the formal analysis derived from the renormalization-group theory can only be applied over a narrow temperature range, the critical region''; outside this region correction terms proliferate rendering attempts to apply the formalism hopeless. This pessimistic conclusion follows largely from a choice of scaling variables and scaling expressions, which is traditional but very inefficient for data covering wide temperature ranges. An alternative extended scaling'' approach can be made where the choice of scaling variables and scaling expressions is rationalized in the light of well established high-temperature series expansion developments. We present the extended scaling approach in detail, and outline the numerical technique used to study the three-dimensional (333D) Ising model. After a discussion of the exact expressions for the historic 111D Ising spin chain model as an illustration, an exhaustive analysis of high quality numerical data on the canonical simple cubic lattice 333D Ising model is given. It is shown that in both models, with appropriate scaling variables and scaling expressions (in which leading correction terms are taken into account where necessary), critical behavior extends from Tc{T}_{c}Tc​ up to infinite temperature.
Three-dimensional Ising model ferromagnets on different lattices with nearest-neighbor interactions, and on simple-cubic lattices with equivalent interactions out to further neighbors, are studied numerically. The susceptibility data for all these systems are analyzed using the critical renormalization-group theory formalism over the entire temperature range above Tc{T}_{c}Tc​ with an appropriate choice of scaling variable and scaling expressions. Representative experimental data on a metallic ferromagnet (Ni) and an elementary fluid (Xe) are interpreted in the same manner so as to estimate effective coordination numbers.
If scaling variables and scaling expressions are chosen judiciously, ``critical'' scaling analyses do not need to be restricted to a narrow range of temperatures near the critical point. The standard scaling variable $(T\ensuremath{-}{T}_{c})/{T}_{c}$ is inadapted to wide ranges of temperature because it diverges at high temperatures. With the variable $\ensuremath{\tau}=(T\ensuremath{-}{T}_{c})/T$, in the high-dimension (mean-field) limit, the reduced susceptibility and the second-moment correlation length of the Ising ferromagnet depend on temperature as $\ensuremath{\chi}(T)={\ensuremath{\tau}}^{\ensuremath{-}1}$ and $\ensuremath{\xi}(T)={T}^{\ensuremath{-}1/2}{\ensuremath{\tau}}^{\ensuremath{-}1/2}$ exactly over the entire temperature range above the critical temperature Tc{T}_{c}Tc​. For the canonical two-dimensional square lattice near-neighbor Ising ferromagnet, it is shown that compact ``extended scaling'' expressions analogous to the high-dimensional limit form but with the appropriate exponents give accurate approximations to the true temperature dependencies, again over the entire temperature range from Tc{T}_{c}Tc​ to infinity. Within this approach, for near-neighbor interaction systems, there is no crossover temperature above which mean-field-like behavior sets in.
We study the second-moment correlation length and the reduced susceptibility of two ferromagnetic Ising models with zero-temperature ordering. By introducing a scaling variable motivated by high-temperature series expansions, we are able to scale data for the one-dimensional Ising ferromagnet rigorously over the entire temperature range. Analogous scaling expressions are then applied to the two-dimensional fully frustrated Villain model where excellent finite-size scaling over the entire temperature range is achieved. Thus we broaden the applicability of the extended scaling method to Ising systems having a zero-temperature critical point.
While the three-dimensional Ising model has defied analytic solution, various numerical methods like Monte Carlo, Monte Carlo renormalization group, and series expansion have provided precise information about the phase transition. Using Monte Carlo simulation that employs the Wolff cluster flipping algorithm with both 32-bit and 53-bit random number generators and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising Model, with lattice sizes ranging from 16^{3} to 1024^{3}. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, e.g., logarithmic derivatives of magnetization and derivatives of magnetization cumulants, we have obtained the critical inverse temperature K_{c}=0.221654626(5) and the critical exponent of the correlation length ν=0.629912(86) with precision that exceeds all previous Monte Carlo estimates.
From a consideration of high temperature series expansions in ferromagnets and in spin glasses, we propose an extended scaling scheme involving a set of scaling formulas which expresses to leading order the temperature (T) and the system size (L) dependences of thermodynamic observables over a much wider range of T than the corresponding one in the conventional scaling scheme. The extended scaling, illustrated by data on the canonical 2d ferromagnet and on the 3d bimodal Ising spin glass, leads to consistency in the estimates of critical parameters obtained from scaling analyses for different observables.
We have substantially extended the high-temperature and low-magnetic-field (and the related low-temperature and high-magnetic-field) bivariate expansions of the free energy for the conventional three-dimensional Ising model and for a variety of other spin systems generally assumed to belong to the same critical universality class. In particular, we have also derived the analogous expansions for the Ising models with spin s=1,3/2,…s=1,3/2,\dots{}s=1,3/2,… and for the lattice Euclidean scalar-field theory with quartic self-interaction, on the simple-cubic and the body-centered-cubic lattices. Our bivariate high-temperature expansions, which extend through 24th order, enable us to compute, through the same order, all higher derivatives of the free energy with respect to the field, namely, all higher susceptibilities. These data make more accurate checks possible, in critical conditions, both of the scaling and the universality properties with respect to the lattice and the interaction structure and also help to improve an approximate parametric representation of the critical equation of state for the three-dimensional Ising model universality class.