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Physical review. B, Solid state

Approximate renormalization-group calculations of critical exponents in continuous dimensions

Sheldon L. Katz, Michel Droz, J. D. Gunton

Published Feb 1, 1977• 19 citations
Publisher

Abstract

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 $d$-dimensional lattice for all integral values of $z={2}^{d}$ 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.

DOI
10.1103/physrevb.15.1597

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Physical review. B, Solid state•Apr 1, 1977•46 citations
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 \ensuremathγ,\ensuremathδ\ensuremath{\gamma}, \ensuremath{\delta}\ensuremathγ,\ensuremathδ, and the thermal and magnetic correction to scaling exponents are compared with corresponding results obtained from the \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion and series calculations.
The "subcritical" exponent ${\ensuremath{\Delta}}_{2s}$ for corrections to asymptotic scaling in the renormalization-group theory of critical phenomena, has been calculated for the continuous-spin versions of the classical, d=3d=3d=3 Ising, $X\ensuremath{-}Y$, and Heisenberg models by numerical integrations of Wilson's approximate recursion formula. The results are ${\ensuremath{\Delta}}_{2s}(\mathrm{Ising})=\ensuremath{-}0.640$, ${\ensuremath{\Delta}}_{2s}(X\ensuremath{-}Y)=\ensuremath{-}0.644$, and ${\ensuremath{\Delta}}_{2s}(\mathrm{Heisenberg})=\ensuremath{-}0.647$. Comparison is made with results from perturbation expansions in $\ensuremath{\epsilon}=4\ensuremath{-}d$ (ddd being the dimensionality), high-temperature series expansions, and experimental measurements.
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 scaling-field representation of Wilson's exact renormalization-group equation is derived and used for the approximate calculation of critical exponents for the continuous-spin Ising model in three dimensions. The truncation of the hierarchy of scaling-field equations that retains only the four most relevant scaling fields yields the critical exponents $\ensuremath{\nu}\ensuremath{\approx}0.60$, $\ensuremath{\eta}\ensuremath{\approx}0.06$, and ${\ensuremath{\Delta}}_{1}\ensuremath{\approx}\ensuremath{-}0.47$ (where ${\ensuremath{\Delta}}_{1}$ denotes Wortis's correction-to-scaling exponent).
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.
Accurate calculations of the thermodynamic properties of a system near its critical point are now possible using finite-lattice approximations to renormalization groups. In order to investigate some features of such approximations, we introduce a linear-renormalization-group transformation appropriate to a system of continuous spins like the Gaussian model of Berlin and Kac. We solve the renormalization-group equations for the Gaussian model exactly, and then study in detail a finite-lattice approximation to the renormalization group for the three-dimensional case. The renormalization-group transformation contains a parameter bbb. The exact transformation has fixed points only for $b={2}^{\ensuremath{-}\frac{5}{2}}$. The approximate transformation has fixed points for a range of values of bbb around ${2}^{\ensuremath{-}\frac{5}{2}}$. Eigenvalues of the transformation, which determine critical exponents, depend on the value of bbb used in the calculation. We study the problem of identifying the correct value of bbb.
The critical behavior of the sss-state Ashkin-Teller-Potts model on a square lattice is studied for s=3,4,and5s=3, 4, \mathrm{and} 5s=3,4,and5, using a variational renormalization-group transformation of the type introduced by Kadanoff. For the s=3s=3s=3 and s=4s=4s=4 models, the transformation correctly predicts a continuous phase transition. The calculated values of the critical exponents and the transition temperature are consistent with the results obtained from other approaches. For s=5s=5s=5, however, the transformation fails to detect the first-order transition that is known to occur.