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AIP conference proceedings

Series studies of critical exponents in continuous dimensions

John P. VAN DYKE, William J. Camp

Published Jan 1, 1976• 4 citations
Publisher

Abstract

Using results we have previously derived for the high‐temperature susceptibility expansion of classical models as a closed‐form function of lattice dimension, we study the dimensional dependence of the critical exponent γ and critical temperature, as well as the correction‐to‐scaling exponent Δ1, for Ising‐like (n=1) models. The numerical results are obtained by extrapolation of 10‐th order series on loose‐packed hypercubical lattices, and 8‐th order series on close‐packed hypertriangular lattices. The critical exponent γ increases monotonically with decreasing dimension, d, for d<4, and apparently tends to infinity at d=1; while the critical temperature decreases monotonically and smoothly to zero at d=1. Detailed contact is made with the ε‐expansion estimates for critical exponents obtained in the context of renormalization group theory.

DOI
10.1063/1.30414

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Using results we have previously derived for the high‐temperature susceptibility expansion of classical models as a closed‐form function of lattice dimension, we study the dimensional dependence of the critical exponent γ and critical temperature, as well as the correction‐to‐scaling exponent Δ1, for Ising‐like (n=1) models. The numerical results are obtained by extrapolation of 10‐th order series on loose‐packed hypercubical lattices, and 8‐th order series on close‐packed hypertriangular lattices. The critical exponent γ increases monotonically with decreasing dimension, d, for d<4, and apparently tends to infinity at d=1; while the critical temperature decreases monotonically and smoothly to zero at d=1. Detailed contact is made with the ε‐expansion estimates for critical exponents obtained in the context of renormalization group theory.
A high-temperature series expansion for the Edwards and Anderson spin-glass order-parameter susceptibility is computed for Ising spins on hypercubic lattices with nearest-neighbor interactions. The series is analyzed by Pad\'e approximants with Rudnick-Nelson-type corrections to scaling. The results agree with the first-order $\ensuremath{\epsilon}$ expansion of Harris, Lubensky, and Chen. The critical exponent ${\ensuremath{\gamma}}_{Q}$ increases monotonically with decreasing dimension, ddd, for d<6d<6d<6, and apparently tends to infinity at d=4d=4d=4; however, the critical temperature does not appear to go to zero at d=4d=4d=4.
We have studied the susceptibility of continuous‐spin Landau‐Wilson models on the FCC lattice using exact series expansions through 10‐th order in the inverse temperature. Both order‐disorder (double‐well) and displacive (single‐well) models have been considered. The exponent γ of the dominant singularity is found to be 1.25 for both types of model. The confluent singularity arising from corrections to scaling is found to have the universal value δ=0.5. We discuss in detail the crossover from Gaussian to Ising constant, and find a crossover exponent o/=1 in all dimensions.
High temperature series expansion for the critical exponents of the Ising model are reanalysed using a modified ratio method. The analysis shows that a minor modification of the ratio method yields for all lattices a value, for the exponent γ in three dimensions, close to 1.245, therefore lower than the quoted value 1.250, and much closer to the renormalization group (R.G.) value 1.241. The exponent is analysed in two ways : in one method γ is estimated directly while in the other one Tc, is calculated first. With these new values of Tc, the exponent α is recalculated and found to be very close to the R.G. value 0.110. The value of ν is not modified (0.638) and is therefore still a problem for hyperscaling, and is in disagreement with the R.G. value (0.630).
The Kadanoff lower-bound renormalization transformation is used to obtain approximate values of the critical exponents for the spin-1/2 Ising model on a simple hypercubic ddd-dimensional lattice for all integral values of z=2dz={2}^{d}z=2d between 3 and 16. The results for $\ensuremath{\gamma}, \ensuremath{\delta}$, and the thermal and magnetic correction to scaling exponents are compared with corresponding results obtained from the $\ensuremath{\epsilon}$ expansion and series calculations.
We have derived series in powers of K to 8‐th order for the two‐point function, susceptibility and free energy of the model Hamiltonian, ΣR⇒ {W[Q(R⇒)] + K/2 Σδ⇒ Q(R⇒)·Q(R⇒+δ⇒)}, where Q is a real tensor, and W is an arbitrary, even function of Q. Special cases include the spin‐S Ising model and the classical Heisenberg model, for both of which W = 0. Additional applications include models for structural phase transitions, tricritical points, He3‐He4 mixtures, liquid crystals, and the class of models employed in renormalization — group studies of critical phenomena. Detailed evaluation of susceptibility series have been carried out on the standard 2, 3, and 4‐dimensional lattices. The series for the scalar case on the triangular net is presented herein.
In this paper we reanalyze the series for the fluctuation of the order parameter, the fourth order fluctuation in the order parameter, and the specific heat for the spin [Formula: see text] model on three dimensional lattices. We find for the exponents, respectively, γ = 1.333 ± 0.001, γ2 = 4.725 ± 0.01, and α = 0.00 ± 0.04. We find no evidence for a correction to scaling term with exponent [Formula: see text] for the specific heat or for the susceptibility. Precise estimates for critical entropy and critical energy are obtained. From the critical amplitudes we obtain further support for the lattice–lattice scaling hypothesis and the one scale factor universality hypothesis.