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Physical Review A

Interdimensional Scaling Laws

Y. Imry, G. Deutscher, David J. Bergman, Shlomo Alexander

Published Feb 1, 1973• 23 citations
Publisher

Abstract

We consider in detail the critical behavior of a $d$-dimensional system that is finite in one of its dimensions. Such a system can exhibit four different types of critical behavior: $d$-dimensional mean field, $d$-dimensional critical, ($d\ensuremath{-}1$)-dimensional mean field, ($d\ensuremath{-}1$)-dimensional critical. By matching its behavior in the various regions we obtain equalities connecting the critical indices in $d$ and $d\ensuremath{-}1$ dimensions. Assuming the usual two-dimensional critical indices of the Ising model, these relations lead to the following indices for the three-dimensional Ising model: $\ensuremath{\nu}=\frac{2}{3}$, $\ensuremath{\alpha}=0$, $\ensuremath{\beta}=\frac{3}{8}$, and $\ensuremath{\gamma}=\frac{5}{4}$.

DOI
10.1103/physreva.7.744

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We consider in detail the critical behavior of a ddd-dimensional system that is finite in one of its dimensions. Such a system can exhibit four different types of critical behavior: ddd-dimensional mean field, ddd-dimensional critical, (d\ensuremath−1d\ensuremath{-}1d\ensuremath−1)-dimensional mean field, (d\ensuremath−1d\ensuremath{-}1d\ensuremath−1)-dimensional critical. By matching its behavior in the various regions we obtain equalities connecting the critical indices in ddd and d\ensuremath−1d\ensuremath{-}1d\ensuremath−1 dimensions. Assuming the usual two-dimensional critical indices of the Ising model, these relations lead to the following indices for the three-dimensional Ising model: \ensuremathν=23\ensuremath{\nu}=\frac{2}{3}\ensuremathν=32​, \ensuremathα=0\ensuremath{\alpha}=0\ensuremathα=0, \ensuremathβ=38\ensuremath{\beta}=\frac{3}{8}\ensuremathβ=83​, and \ensuremathγ=54\ensuremath{\gamma}=\frac{5}{4}\ensuremathγ=45​.
The matching of leading singular contributions to the different mean-field or critical behaviors of a system with long-but-finite-ranged spatially anisotropic interactions leads to "scaling laws" connecting criticial exponents for different dimensions. The results are exact for mean-field theory (MFT) and spherical model exponents and predict MFT exponents for d=4d=4d=4. From the known d=2d=2d=2 Ising-model exponents, the scaling laws give $\ensuremath{\alpha}=0$, $\ensuremath{\beta}=\frac{4}{11}$, $\ensuremath{\gamma}=\frac{14}{11}$, $\ensuremath{\nu}=\frac{2}{3}$ for d=3d=3d=3. Agreement with previous results for the d=2,3d=2,3d=2,3 excluded volume problem is also quite good. For the $X\ensuremath{-}Y$ and Heisenberg models the results predict $\ensuremath{\gamma}\ensuremath{\approx}2.0 \mathrm{and} 2.4$, respectively, for d=2d=2d=2.
We have studied the critical phenomena of the Ising model with long-range interaction and the Ising model with spatially correlated random field, both of which are expected to change the effective dimension of the system continuously.The investigation was conducted from the numerical viewpoint by using the Markov chain Monte Carlo method.Second order phase transitions can be classified into the universality classes and the dimensionality of the Ising model changes the universality class: the one-dimensional Ising model has no long-range order at finite temperatures (lower critical dimension), four or higher dimensional model has the mean-field universality (upper critical dimension), two-dimensional Ising model belongs to different universality from either of them, and so on.Although both of the Ising models with long-range interaction and the Ising model with spatially correlated random field are believed to change their critical behavior from its real spatial dimension into different effective dimensions, their detailed behavior has not been known well because of the notorious difficulties in analysis.In this work, we have obtained the detailed behavior of these models by introducing and developing the various methods.To investigate the Ising model with long-range interaction, we introduced the Fukui-Todo method, which drastically reduces the computational cost from O(N 2 ) to O(N ), where N is the number of spins, and developed the generalized Ewald summation, the improved estimator, and the combined Binder ratio technique, which improves the convergence of the estimation.These techniques lead to the successful estimation of the boundary of the universality classes.We also investigated the Ising model with correlated random field by introducing the method for correlated random number generation and the various techniques for the precise analysis.Although the present numerical investigations suggested that the correlated random field causes a shift of effective dimensionality, they also revealed the difficulty for stronger random-field correlations because of the huge finite-size effects.i Prof. Synge
Critical exponents are calculated for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$ with $\ensuremath{\epsilon}$ small, using renormalization-group techniques. To order $\ensuremath{\epsilon}$ the exponent $\ensuremath{\gamma}$ is $1+\frac{1}{6}\ensuremath{\epsilon}$ for an Ising-like model and $1+\frac{1}{5}\ensuremath{\epsilon}$ for an XY\mathrm{XY}XY model.
"Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified."
Although the fully connected Ising model does not have a length scale, we show that the critical exponents for thermodynamic quantities such as the mean magnetization and the susceptibility can be obtained using finite size scaling with the scaling variable equal to N, the number of spins. Surprisingly, the mean value and the most probable value of the magnetization are found to scale differently with N at the critical temperature of the infinite system, and the magnetization probability distribution is not a Gaussian, even for large N. Similar results inconsistent with the usual understanding of mean-field theory are found at the spinodal. We relate these results to the breakdown of hyperscaling and show that hyperscaling can be restored by increasing N while holding the Ginzburg parameter rather than the temperature fixed, or by doing finite size scaling at the pseudocritical temperature where the susceptibility is a maximum for a given value of N. We conclude that finite size scaling for the fully connected Ising model yields different results depending on how the mean-field limit is approached.