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arXiv:cond-mat/0612665 · Physical Review B

Extended scaling for ferromagnets

Ian A. Campbell, Koji Hukushima, Hajime Takayama

Published Oct 29, 2007• 18 citations
arXivHTMLPDF

Abstract

A simple systematic rule, inspired by high-temperature series expansion (HTSE) results, is proposed for optimizing the expression for thermodynamic observables of ferromagnets exhibiting critical behavior at ${T}_{c}$. This ``extended scaling'' scheme leads to a protocol for the choice of scaling variables, $\ensuremath{\tau}=(T\ensuremath{-}{T}_{c})∕T$ or $({T}^{2}\ensuremath{-}{T}_{c}^{2})∕{T}^{2}$ depending on the observable instead of $(T\ensuremath{-}{T}_{c})∕{T}_{c}$, and more importantly to temperature dependent noncritical prefactors for each observable. The rule corresponds to scaling of the leading term of the reduced susceptibility above ${T}_{c}$ as ${\ensuremath{\chi}}_{c}^{*}(T)\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\gamma}}$ in agreement with standard practice with scaling variable $\ensuremath{\tau}$ and for the leading term of the second-moment correlation length as ${\ensuremath{\xi}}_{c}^{*}(T)\ensuremath{\sim}{T}^{\ensuremath{-}1∕2}{\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\nu}}$. For the specific heat in bipartite lattices, the rule gives ${C}_{c}^{*}(T)\ensuremath{\sim}{T}^{\ensuremath{-}2}{[({T}^{2}\ensuremath{-}{T}_{c}^{2})∕{T}^{2}]}^{\ensuremath{-}\ensuremath{\alpha}}$. The latter two expressions are not standard. The scheme can allow for confluent and noncritical correction terms. A stringent test of the extended scaling is made through analyses of high-precision numerical and HTSE data, or real data, on the three-dimensional canonical Ising, $XY$, and Heisenberg ferromagnets. For the susceptibility $\ensuremath{\chi}(T)$ and the correlation length $\ensuremath{\xi}(T)$ of the three ferromagnets, their optimized expression, which consists of the leading terms [respectively, ${\ensuremath{\chi}}_{c}^{*}(T)$ and ${\ensuremath{\xi}}_{c}^{*}(T)$] and a quite limited number of confluent and noncritical correction terms, represents real data to surprisingly good approximations over the entire temperature range from ${T}_{c}$ to infinity. The temperature dependent prefactors introduced are of crucial importance not only in fixing the optimized expression at relatively high temperatures but also in determining appropriately the small amplitude correction terms. For the specific heat of the Ising ferromagnet, ${C}_{c}^{*}(T)$ combined with two noncritical correction terms which are calculated with no free parameters once the correlation length critical parameters are known reproduces real data nicely also over the whole temperature range.

DOI
10.1103/physrevb.76.134421

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A simple systematic rule, inspired by high-temperature series expansion (HTSE) results, is proposed for optimizing the expression for thermodynamic observables of ferromagnets exhibiting critical behavior at Tc{T}_{c}Tc​. This ``extended scaling'' scheme leads to a protocol for the choice of scaling variables, \ensuremathτ=(T\ensuremath−Tc)∕T\ensuremath{\tau}=(T\ensuremath{-}{T}_{c})∕T\ensuremathτ=(T\ensuremath−Tc​)∕T or (T2\ensuremath−Tc2)∕T2({T}^{2}\ensuremath{-}{T}_{c}^{2})∕{T}^{2}(T2\ensuremath−Tc2​)∕T2 depending on the observable instead of (T\ensuremath−Tc)∕Tc(T\ensuremath{-}{T}_{c})∕{T}_{c}(T\ensuremath−Tc​)∕Tc​, and more importantly to temperature dependent noncritical prefactors for each observable. The rule corresponds to scaling of the leading term of the reduced susceptibility above Tc{T}_{c}Tc​ as \ensuremathχc∗(T)\ensuremath∼\ensuremathτ\ensuremath−\ensuremathγ{\ensuremath{\chi}}_{c}^{*}(T)\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\gamma}}\ensuremathχc∗​(T)\ensuremath∼\ensuremathτ\ensuremath−\ensuremathγ in agreement with standard practice with scaling variable \ensuremathτ\ensuremath{\tau}\ensuremathτ and for the leading term of the second-moment correlation length as \ensuremathξc∗(T)\ensuremath∼T\ensuremath−1∕2\ensuremathτ\ensuremath−\ensuremathν{\ensuremath{\xi}}_{c}^{*}(T)\ensuremath{\sim}{T}^{\ensuremath{-}1∕2}{\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\nu}}\ensuremathξc∗​(T)\ensuremath∼T\ensuremath−1∕2\ensuremathτ\ensuremath−\ensuremathν. For the specific heat in bipartite lattices, the rule gives Cc∗(T)\ensuremath∼T\ensuremath−2[(T2\ensuremath−Tc2)∕T2]\ensuremath−\ensuremathα{C}_{c}^{*}(T)\ensuremath{\sim}{T}^{\ensuremath{-}2}{[({T}^{2}\ensuremath{-}{T}_{c}^{2})∕{T}^{2}]}^{\ensuremath{-}\ensuremath{\alpha}}Cc∗​(T)\ensuremath∼T\ensuremath−2[(T2\ensuremath−Tc2​)∕T2]\ensuremath−\ensuremathα. The latter two expressions are not standard. The scheme can allow for confluent and noncritical correction terms. A stringent test of the extended scaling is made through analyses of high-precision numerical and HTSE data, or real data, on the three-dimensional canonical Ising, XYXYXY, and Heisenberg ferromagnets. For the susceptibility \ensuremathχ(T)\ensuremath{\chi}(T)\ensuremathχ(T) and the correlation length \ensuremathξ(T)\ensuremath{\xi}(T)\ensuremathξ(T) of the three ferromagnets, their optimized expression, which consists of the leading terms [respectively, \ensuremathχc∗(T){\ensuremath{\chi}}_{c}^{*}(T)\ensuremathχc∗​(T) and \ensuremathξc∗(T){\ensuremath{\xi}}_{c}^{*}(T)\ensuremathξc∗​(T)] and a quite limited number of confluent and noncritical correction terms, represents real data to surprisingly good approximations over the entire temperature range from Tc{T}_{c}Tc​ to infinity. The temperature dependent prefactors introduced are of crucial importance not only in fixing the optimized expression at relatively high temperatures but also in determining appropriately the small amplitude correction terms. For the specific heat of the Ising ferromagnet, Cc∗(T){C}_{c}^{*}(T)Cc∗​(T) combined with two noncritical correction terms which are calculated with no free parameters once the correlation length critical parameters are known reproduces real data nicely also over the whole temperature range.
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 apply and test the recently proposed 'extended scaling' scheme in an analysis of the magnetic susceptibility of Ising systems above the upper critical dimension. The data are obtained by Monte Carlo simulations using both the conventional Wolff cluster algorithm and the Prokof'ev–Svistunov worm algorithm. As already observed for other models, extended scaling is shown to extend the high-temperature critical scaling regime over a range of temperatures much wider than that achieved conventionally. It allows for an accurate determination of leading and sub-leading scaling indices, critical temperatures and amplitudes of the confluent corrections.
Using a renormalized linked-cluster-expansion method, we have extended to order ${\ensuremath{\beta}}^{23}$ the high-temperature series for the susceptibility $\ensuremath{\chi}$ and the second-moment correlation length $\ensuremath{\xi}$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}.$ Our best estimates for the inverse critical temperatures are ${\ensuremath{\beta}}_{c}^{\mathrm{sc}}=0.221654(1)$ and ${\ensuremath{\beta}}_{c}^{\mathrm{bcc}}=0.1573725(6).$ For the susceptibility exponent we get $\ensuremath{\gamma}=1.2375(6)$ and for the correlation length exponent $\ensuremath{\nu}=0.6302(4).$ The ratio of the critical amplitudes of $\ensuremath{\chi}$ above and below the critical temperature is estimated to be ${C}_{+}{/C}_{\ensuremath{-}}=4.762(8).$ The analogous ratio for $\ensuremath{\xi}$ is estimated to be ${f}_{+}{/f}_{\ensuremath{-}}=1.963(8).$ For the correction-to-scaling amplitude ratio we obtain ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$
High-temperature series expansions are used to examine the dependence of critical-point exponents upon lattice anisotropy (different interaction strengths in different directions in the lattice). The two-spin correlation function ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ is calculated to tenth order in (1kBT\frac{1}{{k}_{B}T}kB​T1​) for the Ising Hamiltonian ${\mathcal{H}}_{l \mathrm{anis}}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}\stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}\stackrel{z}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}$ for a wide range of anisotropy parameters $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}$ and for both the sc and fcc lattices; here the first summation is over all pairs of nearest-neighbor sites whose relative displacement vector ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{\mathrm{ij}}$ has no zzz component, while the second summation is over all other pairs of nearest-neighbor sites. Hence for R=0R=0R=0, both the sc and fcc lattices reduce to the two-dimensional square lattice, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc becomes a one-dimensional linear chain and the fcc becomes two noninteracting three-dimensional bcc lattices. The series for ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ are then used to obtain series of corresponding lengths for the specific heat, susceptibility, and second moment. Analysis of these series yields results consistent with the universality hypothesis of critical-point exponents. Specifically, it is found that when lattice anisotrophy is introduced, the critical-point exponents studied (the susceptibility exponent $\ensuremath{\gamma}$ and the correlation length exponent $\ensuremath{\nu}$) do not appear to change from their values for an isotropic lattice. The problem of next-nearest-neighbor interactions is treated using similar methods in Paper II of this series (and briefly discussed in this paper).
We have extended through ${\ensuremath{\beta}}^{23}$ the high-temperature expansion of the second field derivative of the susceptibility for Ising models of general spin, with nearest-neighbor interactions, on the simple cubic and the body-centered cubic lattices. Moreover, the expansions for the nearest-neighbor correlation function, the susceptibility, and the second correlation moment have been extended up to ${\ensuremath{\beta}}^{25}.$ Taking advantage of these new data, we can improve the accuracy of direct estimates of critical exponents and of hyperuniversal combinations of critical amplitudes such as the renormalized four-point coupling gr{g}_{r}gr​ or the quantity usually denoted by ${R}_{\ensuremath{\xi}}^{+}.$ In particular, we obtain $\ensuremath{\gamma}=1.2371(1),$ $\ensuremath{\nu}=0.6299(2),$ ${\ensuremath{\gamma}}_{4}=4.3647(20),$ gr=1.404(3),{g}_{r}=1.404(3),gr​=1.404(3), and ${R}_{\ensuremath{\xi}}^{+}=0.2668(5).$ We have used a variety of series extrapolation procedures and, in some of the analyses, we have assumed that the leading correction-to-scaling exponent $\ensuremath{\theta}$ is universal and roughly known. We have also verified, to high precision, the validity of the hyperscaling relation and of the universality property both with regard to the lattice structure and to the value of the spin.
Thermodynamic functions in the vicinity of an ordinary critical point are expected to obey asymptotic scaling laws as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and the ordering field, hhh, approach zero. However, the optimal scaling variables are the nonlinear scaling fields, ${g}_{t}=t+{b}_{t}{h}^{2}+{c}_{t}{t}^{2}+\ensuremath{\cdots}$ and ${g}_{h}=h(1+{c}_{h}t+\ensuremath{\cdots})$. The nonlinearities yield correction factors to the leading power-law (and scaling) variation of thermodynamic quantities, LLL, of the form ($1+{a}_{L}t+{b}_{L}{t}^{2}+\ensuremath{\cdots}$), etc., where the correction amplitudes aL,bL,…{a}_{L},{b}_{L},\dots{}aL​,bL​,…, are uniquely determined by the nonlinear scaling-field coefficients. It follows that "analytic" corrections to, e.g., the susceptibility, are directly related to those for the free energy and magnetization (in zero field). The term bth2{b}_{t}{h}^{2}bt​h2 also generates nonanalytic contributions such as an additive, energylike term, varying as ${|t|}^{1\ensuremath{-}\ensuremath{\alpha}}$ in the zero-field susceptibility, and factors like ($1+{c}_{L}{|h|}^{2\ensuremath{-}\frac{1}{\ensuremath{\Delta}}}$) on the critical isotherm, t=0t=0t=0. Irrelevant scaling fields yield further, in general distinct, nonanalytic corrections, and cause shifts in Tc{T}_{c}Tc​ and various amplitudes although "universal" ratios remain constant.