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\begin{document}

\title{Critical scaling to infinite temperature}

\author{P. H.~Lundow}
\affiliation {Department of Theoretical Physics, KTH, SE-106 91
  Stockholm, Sweden}

\author{I. A.~Campbell}
\affiliation{Laboratoire des Collo\"ides, Verres et Nanomat\'eriaux,
  Universit\'e Montpellier II, 34095 Montpellier, France}



\begin{abstract}
Three dimensional Ising model ferromagnets on different lattices with near neighbor interactions and on simple cubic lattices with equivalent interactions out to further neighbors are studied numerically. With an appropriate choice of scaling variable and scaling expression the susceptibility data are analysed using the critical Renormalization Group Theory formalism over the entire temperature range above $T_c$. Experimental data on a metallic ferromagnet (Ni) and a fluid (Xe) are interpreted in the same manner to estimate effective interaction ranges.


\end{abstract}

\pacs{ 75.50.Lk, 05.50.+q, 64.60.Cn, 75.40.Cx}

\maketitle

In the very extensive studies which have been devoted to critical phenomena attention has understandably been mainly concentrated on the regime in the immediate neighborhood of the critical point; data are analysed using the leading terms in the Renormalization Group Theory (RGT) formalism. It is widely considered that as correction terms proliferate outside this narrow "critical region" they lead ultimately to a crossover towards Gaussian fixed point mean-field-like behavior above a temperature $T_g$ determined (at dimensions below the upper critical dimension) by the Ginzburg criterion \cite{ginzburg:60}. This criterion expresses the breakdown of a high temperature Landau regime due to critical fluctuations.

Sophisticated theoretical and numerical studies have been made of this crossover \cite{riedel:69,luijten:97,pelissetto:98,luijten:99,garrabos:08} in particular in the context of long range interactions. Experimental data on fluids, where effective interactions are expected to be long range, have been interpreted on this basis \cite{anisimov:95,sengers:09}.

Here we discuss from a different perspective numerical data on various 3d Ising systems, in particular models where the interactions extend beyond near neighbor. We conclude that with appropriate observables and choice of scaling variable, for this family of models at least there is no need to invoke a crossover or the Ginzburg criterion. The data can be convincingly interpreted using the critical RGT formalism over the whole temperature range from $T_c$ to infinity.

We then turn to experimental data on ferromagnets and on fluids, showing that they too can be analysed in a transparent manner using the same approach and without invoking crossovers. We obtain quantitative estimates of the effective interaction range for Ni and for Xe.

It was already established before the RGT (see e.g. \cite{gammel:63,fisher:67}) that in the magnetic case the response parameter to be scaled is the "reduced" susceptibility (see \cite{stanley:71} for definitions) :

%\begin{equation}
$\chi(\beta) = \langle m - \langle m \rangle\rangle^2= \chi_{T}(\beta)/\chi_{0}(\beta)$

%\label{chiscale}
%\end{equation}
where the thermodynamic susceptibility $\chi_{T}(\beta) \equiv [\partial{m}/\partial{H}]_{H \to 0}$
is normalized by the free spin susceptibility $\chi_{0}(\beta) \propto \beta$. (As usual we will set the interaction strength $J$ to $1$ and write $\beta \equiv 1/kT$).
The critical behavior is written \cite{stanley:71}
%\begin{equation}

$\chi(\beta) \equiv T\chi_{T}(\beta) \propto \epsilon^{-\gamma}[1 + \cdots]$

%\label{chidef}
%\end{equation}
where $\epsilon$ is an appropriately normalized scaling variable depending linearly on $(T - T_c)$ close to criticality.

The thermodynamic fluid analogue to $\chi_{T}$ is the isothermal compressibility on the critical isochore $K_{T}=(\partial{V}/\partial{p})/V$ so the parameter to be scaled \cite{stanley:71} is the compressibility normalized by the ideal gas compressibility $\langle(N-\langle N \rangle)^2 \rangle/\langle N \rangle$ where $N$ is the total number of particles, i.e.
%\begin{equation}

$TK_{T} \propto \epsilon^{-\gamma}[1 + \cdots]$

%\label{TKT}
%\end{equation}
Because the gas-liquid order parameter is a scalar the fluid transition should belong to the Ising universality class \cite{kadanoff:71}. Careful experimental measurements made over many years (see \cite{sengers:09}) have shown that the asymptotic fluid exponents at criticality are indeed those of the short range Ising universality class.

The critical scaling variable was initially taken to be $\epsilon = \tau = 1-\beta/\beta_c$ or $\epsilon=1-\tanh(\beta)/\tanh(\beta_c)$ \cite{gammel:63,fisher:67} but $\epsilon = t = (T/T_c)-1$ is more frequently employed. The latter convention has been used in a systematic way for analyses of high temperature experimental and numerical data of long range interaction models \cite{anisimov:95,luijten:97,garrabos:08,chamberlin:09}, defining an effective susceptibility exponent \cite{kouvel:64}
$\gamma_{t}(t) = -\partial{\log \chi_{T}}/\partial{\log t}$.
In the high temperature limit for any model $\chi_{T} \to \beta$ and $t \to T$ so $\gamma_{t}(t)$ must tend to $1$, with a necessary crossover from the critical $\gamma$ at some intermediate temperature.
%This is an automatic consequence of the choice of scaling expression and of scaling variable.

The full formal RGT Wegner scaling expression for $\chi$ (rather than $\chi_{T}$ ) in the thermodynamic limit including confluent and analytic correction terms is written rigorously in terms of $\tau$ as \cite{wegner:72,aharony:83,gartenhaus:88}
\begin{eqnarray}
\chi(\tau)= T\chi_{T}(\tau) = C_{\chi}\tau^{-\gamma}[1 + a_{\chi}\tau^{\theta}F_{a}(\tau)+ b_{\chi}F_{b}~~~~~~ \\  \nonumber
 + c_{\chi}\tau^{(1-\alpha)\gamma}F_{c}(\tau)
+ d_{\chi}\tau^{\gamma}F_{d} + a_{2,\chi}\tau^{\theta_{2}}F_{2}(\tau) +\cdots]
\label{chiwegner}
\end{eqnarray}
where $\gamma, \alpha$ and the confluent correction exponents $\theta_{i}$ are universal but the amplitudes are not universal; the $F_{i}$ are infinite analytic series in $\tau$ normalized to $1$ at $\tau=0$. Because $\tau \to 1$ as $T \to \infty$ these developments always remain well behaved, whereas because $t$ diverges as $T$ it is obviously very awkward to expand the equivalent expression written in terms of $t$ to high temperatures.


The temperature dependent effective exponent is defined in terms of $\tau$ and $\chi(\tau)$ by \cite{fahnle:84,orkoulas:00,butera:02,campbell:07,campbell:10}
\begin{equation}
\gamma_{eff}(\tau) = -\partial{\log\chi(\tau)}/\partial{\log(\tau)}
\end{equation}
(which has quite different properties from $\gamma_{t}$ at high temperatures).


At first sight the sets of infinite series of corrections in \ref{chiwegner} appear rather forbidding.
However from inspection of $S=1/2$ high temperature series expansions (HTSE) there are exact closure rules at infinite temperature :
$C_{\chi}[1 + a_{\chi} + \cdots] \equiv 1$
and $\gamma_{eff}(1) \equiv z\beta_c$ \cite{fahnle:84}
where $z$ is the coordination number and the $[\cdots]$ are evaluated here as the sum of all the higher order terms setting $\tau=1$.
It turns out that for the 3d Ising models which we will discuss explicitly, over a wide temperature region above $T_c$ the leading confluent correction term dominates. All the remaining terms can be collected together into a single effective correction term $k_{\chi}\tau^{\lambda_{\chi}}$,
giving a compact approximate expression
\begin{equation}
\chi(\tau)\tau^{\gamma} = C_{\chi}[1 + a_{\chi}\tau^{\theta} + k_{\chi}\tau^{\lambda_{\chi}}]
\label{chitau}
\end{equation}
so


%\begin{equation}
$\gamma_{eff}(\tau) = \gamma -[ a_{\chi}\theta\tau^{\theta} + k_{\chi}\lambda_{\chi}\tau^{\lambda_{\chi}}]$
%\label{gammatau}
%\end{equation}

The closure rules then become
$C_{\chi}[1+ a_{\chi}+ k_{\chi}]= 1$
 and
$\gamma -[a_{\chi}\theta + k_{\chi}\lambda_{\chi}]C_{\chi} = z\beta_c$.
Once $C_{\chi}$ and $a_{\chi}$ are estimated from data close to criticality, $k_{\chi}$ and $\lambda_{\chi}$ are fixed also, so the entire temperature dependencies of $\chi(\tau)$ and of $\gamma_{eff}(\tau)$ are determined.

While not rigorous this {\it ansatz} gives a representation of the exact behavior which turns out to be accurate at the $10^{-3}$ level over the whole temperature range above $T_c$ for all the models we have studied.

The temperature dependent susceptibility $\chi(\beta,L)$ has been evaluated on diamond, sc, bcc and fcc lattices with near neighbor interactions, see \cite{haggkvist:07,lundow:09,campbell:10} where the numerical techniques are described, and on sc lattices with equivalent interactions up to second, third, fourth or fifth neighbor following \cite{domb:66,luijten:97,orkoulas:00}. The coordination numbers are $z = 4, 6, 8, 12, 18, 26, 32$ and $56$ respectively. All these models are in the 3d short range interaction Ising universality class so the numerical values for the universal 3d Ising exponents are fixed at $\gamma=1.2371$, $\theta=0.50$ and $\nu=0.630$ \cite{pelissetto:02,deng:03}. The critical inverse temperatures $\beta_c$ for the various models were evaluated from the present data using the Binder cumulant $g(\beta,L)$ and the parameter $W(\beta,L)$ introduced in \cite{lundow:10}. The $\beta_c$ values obtained from the finite size scaling analyses are in full agreement with previous estimates \cite{luijten:99}.

The susceptibility results for various lattice sizes $L$ are exhibited in Figure 1 in the form of plots of $\chi(\tau)\tau^{\gamma}$ against $\tau^{\theta}$.

\begin{figure}
  \includegraphics[width=3.5in]{highT_fig1}
\caption{(Color online) The normalized reduced susceptibility $\chi(\tau)\tau^{\gamma}$ as a function of $\tau^{\theta}$ for the diamond, sc, bcc, fcc, $z18, z26, z32$ and $z56$ sc Ising models, top to bottom (black, red, green, blue, cyan, pink, yellow, olive). The exponents are taken to be $\gamma = 1.2371$ and $\theta =0.50$.   }
  \protect\label{fig:1}
\end{figure}
\begin{table}[htbp]
  \caption{\label{Table:I} Values of the fitting parameters for $\chi(\tau)$ in each of the models, as defined in Eqn. \ref{chitau}.}
  \begin{ruledtabular}
\begin{tabular}{cccccc}

$z$&$\beta_c$&$C_{\chi}$&$a_{\chi}$&$k_{\chi}$&$\lambda_{\chi}$ \\
$4$&$0.3697$&$1.245$&$-0.1382$&$-0.059$&$2$\\
$6$&$0.221655$&$1.116$&$-0.0914$&$-0.0125$&$3$\\
$8$&$0.15737$&$1.032$&$-0.039$&$0.008$&$1.15$\\
$12$&$0.102067$&$1.022$&$-0.062$&$0.04$&$1.5$ \\
$18$&$0.06442$&$0.871$&$0.111$&$0.038$&$0.9$\\
$26$&$0.0430385$&$0.765$&$0.302$&$0.009$&$1$\\
$32$&$0.0343267$&$0.703$&$0.428$&$-0.006$&$3$\\
$56$&$0.0189291$&$0.575$&$0.800$&$-0.06$&$1.5$\\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}
  \includegraphics[width=3.5in]{highT_fig2}
\caption{(Color online) The effective exponent $\gamma_{eff}(\tau)$ as a function of $\tau^{\theta}$ for the diamond, sc, bcc, fcc, $z18, z26, z32$ and $z56$ sc Ising models, top to bottom (black, red, green, blue, cyan, pink, yellow, olive). }
  \protect\label{fig:2}
\end{figure}
The critical parameter estimates $\beta_c(z), C_{\chi}(z)$ and $a_{\chi}(z)$ and the approximate effective parameters $k_{\chi}(z)$ and $\lambda_{\chi}(z)$ are given in Table I. (As the second correction term is always weak, the values of $\lambda$ are not very precise). All the models, including those with longer range interactions, follow the critical scaling rules up to infinite temperature, with a gradual evolution of the main critical parameters as $z$ increases
but without a trace of a crossover to mean-field behavior at high $T$. There seems no obvious reason to expect a breakdown in these rules however large the range of interactions, as long as there is a cut-off so that the range remains finite. (However if interactions fall off algebraically and sufficiently slowly, the models will leave the finite-range universality class \cite{fisher:72,suzuki:72}).

\begin{figure}
  \includegraphics[width=3.5in]{highT_fig3}
  \caption{(Color online) The normalized reduced susceptibility $T\chi_{T}\tau^{\gamma}$ of Ni as a function of $\tau^{\theta}$ with the Heisenberg exponents $\gamma = 1.349$ and $\theta=0.55$. The experimental data are taken from \cite{weiss:26} (red circles) and \cite{fallot:44} (black squares). Following \cite{fallot:44} and \cite{souletie:83}, a small temperature independent term has been subtracted from the raw $\chi_{T}$ data.}
  \protect\label{fig:3}
\end{figure}
Experiments can now be examined using the same approach. The venerable experimental data for the susceptibility of Ni tabulated by Weiss and Forrer \cite{weiss:26} and by Fallot \cite{fallot:44} are exhibited in Figure 3 in the same form as that used for the numerical data in Figure 1. Here we consider Ni as a Heisenberg local moment system and so use the Heisenberg exponent values $\gamma= 1.396$ and $\theta = 0.55$ \cite{compostrini:02}. Following Fallot himself \cite{fallot:44} and \cite{souletie:83}, we have subtracted out a small temperature independent susceptibility term, which could well come from an orbital contribution (see \cite{dupree:79} for the case of Co). As in the Ising models with higher coordination numbers shown in Figure 1, the normalized reduced susceptibility increases almost linearly with $\tau^\theta$ over the wide range of temperatures covered which extends to $3T_c$, i.e. $\tau^{\theta} \sim 0.8$.  The ratio between the asymptotic critical value of $\chi(\tau)\tau^{\gamma}$ and the estimated extrapolated infinite temperature value (equal to $1$ for spin $1/2$ in the appropriate units) can be taken as a measure of the effective $C_{\chi}$. The observed ratio is about $0.60$ for Ni.
\begin{figure}
  \includegraphics[width=3.5in]{highT_fig4}
  \caption{(Color online) The normalized reduced compressibility $TK_{T}\tau^{\gamma}$ of Xe as a function of $\tau^{\theta}$. Experimental data taken from \cite{guttinger:81}.}
  \protect\label{fig:4}
\end{figure}
As an example of a gas-liquid transition we consider the isothermal compressibility of Xe on the critical isochore, for which results from careful experiments based on light scattering techniques are tabulated in \cite{guttinger:81}. In Figure 4 these data are plotted in the form $TK_{T}\tau^{\gamma}$ against $\tau^{\theta}$ with the 3d Ising exponents. Again the figure shows an essentially linear increase of $TK_{T}\tau^{\gamma}$ with  $\tau^{\theta}$ just as in the numerical plots for the Ising models with larger coordination numbers. The ratio of the critical limit to the extrapolated high temperature limit is about $C_{\chi} \sim 0.55$.
\begin{figure}
  \includegraphics[width=3.5in]{highT_fig5}
  \caption{(Color online) The effective critical amplitudes $C_{\chi}(z)$ for the Ising models as a function of $z$, plus points for Ni and Xe obtained from the plots in Figures 3 and 4.}
 \protect\label{fig:5}
\end{figure}
In Figure 5 the values of $C_{\chi}$ for the numerical models are plotted against $z$ , and the effective values for Ni and for Xe are indicated by arrows.
From this figure we can estimate the effective coordination number $z$ for Ni and for Xe. The Ni experimental data should in principle be compared to numerical results for Heisenberg spins on an fcc lattice for different $z$ rather than for Ising spins on an sc lattice, but assuming that the numerical $C_{\chi}(z)$ plot will be similar, one can estimate $z \sim 50$, or in other words the effective interactions extend to about two lattice spacings.

For Xe the effective $C_{\chi}$  corresponds to an effective coordination number $z \sim 70$. Inspection of data expressed graphically in terms of $\gamma_{t}$ \cite{sengers:09} suggests that some other liquids can show much smaller effective $z$ values.

The effective interaction range is a fundamental parameter for understanding the magnetism of metallic ferromagnets which can often be considered either from a band or from a local moment perspective. For the liquids it should be possible to characterize effective interaction ranges and to link them to the interatomic potentials used for calculating structure functions.


In conclusion, the analysis given above leads to an overall physical scenario in which the RGT formalism can be applied from criticality up to infinite temperature without the need to invoke any form of crossover at high temperature. Differences in behavior from model to model and from system to system within a universality class reflect essentially variations of the non-universal confluent correction term amplitude, which increases systematically with increasing coordination number.

We acknowledge gratefully a very interesting discussion of the experimental high temperature effective exponents with Ralph Chamberlin.

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\end{document}