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Physical Review A

Interdimensional Scaling Laws from Spatially Anisotropic Interactions

Morgan K. Grover

Published Nov 1, 1973• 1 citation
Publisher

Abstract

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 $d=4$. From the known $d=2$ 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 $d=3$. Agreement with previous results for the $d=2,3$ 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 $d=2$.

DOI
10.1103/physreva.8.2754

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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 \ensuremathα=0\ensuremath{\alpha}=0\ensuremathα=0, \ensuremathβ=411\ensuremath{\beta}=\frac{4}{11}\ensuremathβ=114​, \ensuremathγ=1411\ensuremath{\gamma}=\frac{14}{11}\ensuremathγ=1114​, \ensuremathν=23\ensuremath{\nu}=\frac{2}{3}\ensuremathν=32​ 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−YX\ensuremath{-}YX\ensuremath−Y and Heisenberg models the results predict \ensuremathγ\ensuremath≈2.0and2.4\ensuremath{\gamma}\ensuremath{\approx}2.0 \mathrm{and} 2.4\ensuremathγ\ensuremath≈2.0and2.4, respectively, for d=2d=2d=2.
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, ($d\ensuremath{-}1$)-dimensional mean field, ($d\ensuremath{-}1$)-dimensional critical. By matching its behavior in the various regions we obtain equalities connecting the critical indices in ddd 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 $\ensuremath{\gamma}=\frac{5}{4}$.
This thesis discusses scaling and critical behavior of two different models. One model describes Ising dipoles, originates in condensed matter physics and depicts equilibrium critical phenomena. The other model, taken from the earth sciences, describes faulting instabilities and the resulting earthqnakes, and involves self-organized criticality — a nonequilibrium phenomenon. Both models are characterized by long range interactions, with a resulting sensitivity to boundary conditions. The ordering properties of Ising dipoles on lattices are studied in a mean field theory and by Monte Carlo simulations. The mean field theory is manifestly shape independent in zero external field. In the case of dipoles on a diluted lattice the mean field theory predicts a critical concentration above which the low temperature phase is ferroelectric (or anti-ferroelectric depending on the lattice structure). Extensive Monte Carlo simulation results are in agreement with those of mean field theory. We propose a finite size scaling form that includes logarithmic corrections for systems at the critical dimensionality. In the case of dipoles on a body centered tetragonal lattice we found that the finite scaling form significantly improved the data collapse over the scaling form with mean field exponents. With lattice parameters appropriate to the Ising ferromagnetic compound LiHoF4,we obtain a ferromagnetic transition temperature Tc= 1.51K in excellent agreement with experiment. This indicates that the material LiHoF4 is dominated by the dipole-dipole interaction; since in the simulations we only include dipole-dipole interactions. For dipoles on the simple cubic lattice, the ordered state is made up of anti-ferromagnetic rows. The critical exponents obtained by finite size scaling are ß~1/7,y ~ 8/7 and a ~4/7. These results are in good agreement with those of high temperature series expansions. A model of self—organized ruptures in an elastic medium is developed; and applied to earthquakes. In the model the local ruptures are represented by double couples to be consistent with elastic theory. The explicit form of this double couple source is derived. The system is driven by slowly increasing the shear stress. The model evolves towards a self-organized critical state in which the earthquake distribution follows the Gutenberg Richter law with an exponent in agreement with observational data. By modeling the local static fatigue for the rocks, we also obtained Omori’s law for the rate of aftershocks. The effects of annealing are investigated.
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 exponents for a ddd-dimensional system with an isotropic nnn-component order parameter and long-range attractive interactions decaying as $\frac{1}{{r}^{d+\ensuremath{\sigma}}} (\ensuremath{\sigma}>0)$ are derived, using the renormalization group approach, as power series in $\ensuremath{\epsilon}=2\ensuremath{\sigma}\ensuremath{-}d>0$ ($\ensuremath{\sigma}\ensuremath{\ne}2$, fixed) or $\ensuremath{\Delta}\ensuremath{\sigma}=\ensuremath{\sigma}\ensuremath{-}\frac{1}{2}d>0$ (ddd fixed) and, separately, to order 1n\frac{1}{n}n1​ for all ddd and $\ensuremath{\sigma}\ensuremath{\ne}2$. For $\ensuremath{\epsilon}<0$ the exponents have fixed ("classical") values; when $\ensuremath{\epsilon}=\ensuremath{\Delta}\ensuremath{\sigma}=0$ fractional powers of $\mathrm{ln}(\frac{\ensuremath{\Delta}T}{{T}_{c}})$ appear; when $\ensuremath{\sigma}>2$ the exponents assume their short-range values.
Extensive Monte Carlo simulations are employed in order to study the dynamic critical behavior of the one-dimensional Ising magnet, with algebraically decaying long-range interactions of the form 1rd+σ\frac{1}{r^{d+σ}}rd+σ1​, with σ=0.75σ=0.75σ=0.75. The critical temperature, as well as the critical exponents, is evaluated from the power-law behavior of suitable physical observables when the system is quenched from uncorrelated states, corresponding to infinite temperature, to the critical point. These results are compared with those obtained from the dynamic evolution of the system when it is suddenly annealed at the critical point from the ordered state. Also, the critical temperature in the infinite interaction limit is obtained by means of a finite-range scaling analysis of data measured with different cutoffs of the interaction range. All the estimated static critical exponents (γ/νγ/νγ/ν, β/νβ/νβ/ν, and 1/ν1/ν1/ν) are in good agreement with Renormalization Group (RG) predictions and previously reported numerical data obtained under equilibrium conditions. It is found that the dynamic exponent zzz is different for quenching and annealing experiments, most likely due to the influence of the Kosterlitz-Thouless transition occurring at relatively similar algebraic decay of the interactions with σ=1σ=1σ=1. However, for annealing experiments the measured exponent zzz is close to the RG predictions. On the other hand, the relevant exponents of the dynamic behavior (zzz and θθθ) are slightly different than the RG predictions, most likely due to the fact that they may depend on the especific dynamics used (Metropolis in the present paper).
The critical behavior of a system of nnn-component classical "spins"with anisotropic pair interactions is discussed using renormalization-group techniques for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. To first order in $\ensuremath{\epsilon}$ the correlation and susceptibility exponents in the isotropic limit are $2\ensuremath{\nu}=\ensuremath{\gamma}=1+[\frac{(n+2)}{2(n+8)}]\ensuremath{\epsilon}$, while the anisotropy or crossover exponent is $\ensuremath{\varphi}=1+[\frac{n}{2(n+8)}]\ensuremath{\epsilon}$. When $n\ensuremath{\rightarrow}\ensuremath{\infty}$ these expansions agree with exact spherical-model results. For n=3n=3n=3 and d=3d=3d=3 series expansions indicate $\ensuremath{\varphi}\ensuremath{\simeq}1.2$ (compared with $\ensuremath{\gamma}\ensuremath{\simeq}1.38$).