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Physical Review Letters

Universality of the Exponent η to Order ε2 for a Class of Renormalization Groups

Prabodh Shukla, Melville S. Green

Published Feb 17, 1975• 20 citations
Publisher

Abstract

A formula for the exponent $\ensuremath{\eta}$ to order ${\ensuremath{\epsilon}}^{2}$ previously derived is generalized to a wider class of groups which includes the sharp-cutoff group and is shown to be universal to this order.

DOI
10.1103/physrevlett.34.436

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Physical Review Letters•Feb 17, 1975•15 citations
A formula for the exponent \ensuremathη\ensuremath{\eta}\ensuremathη to order \ensuremathϵ2{\ensuremath{\epsilon}}^{2}\ensuremathϵ2 previously derived is generalized to a wider class of groups which includes the sharp-cutoff group and is shown to be universal to this order.
A class of renormalization groups characterized as the general Gaussian renormalization group is defined. It includes the sharp cutoff renormalization group of Wilson, the linear renormalization group of Bell and Wilson, and the incomplete‐integration renormalization group of Wilson. The renormalization group approach to critical phenomena and the expansion around four dimensions is reviewed. The fixed point of the general Gaussian renormalization group is determined to order ε2, where ε=4−d (d is the dimensionality of the system), and the critical exponents are determined to the same order. The main conclusion is that the exponents do not depend on the cutoff functions of the renormalization group.
This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent $\ensuremath{\eta}$. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero $\ensuremath{\eta}$. They are intended to demonstrate, qualitatively, how nonzero values for $\ensuremath{\eta}$ are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$. The equations are solved both by $\ensuremath{\epsilon}$ expansion about four dimensions and by numerical integration in three dimensions. To order ${\ensuremath{\epsilon}}^{2}$ we obtain $\ensuremath{\eta}=0.05{\ensuremath{\epsilon}}^{2}$. Numerical results in three dimension are $\ensuremath{\eta}=0.058$, $\ensuremath{\nu}=0.588$, and $\ensuremath{\gamma}=1.14$. The relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$ is confirmed.
We compute the critical exponents ν, η and ω of O(N) models for various values of N by implementing the derivative expansion of the nonperturbative renormalization group up to next-to-next-to-leading order [usually denoted O(∂^{4})]. We analyze the behavior of this approximation scheme at successive orders and observe an apparent convergence with a small parameter, typically between 1/9 and 1/4, compatible with previous studies in the Ising case. This allows us to give well-grounded error bars. We obtain a determination of critical exponents with a precision which is similar or better than those obtained by most field-theoretical techniques. We also reach a better precision than Monte Carlo simulations in some physically relevant situations. In the O(2) case, where there is a long-standing controversy between Monte Carlo estimates and experiments for the specific heat exponent α, our results are compatible with those of Monte Carlo but clearly exclude experimental values.
We derive a renormalization group to calculate the nontrivial critical exponent of the divergent correlation length, thereby providing a universality classification of essential singularities in infinite-order phase transitions. This method thus resolves the vanishing scaling matrix problem. The exponent is obtained from the maximal eigenvalue of a scaling matrix in this renormalization group, as in the case of ordinary second-order phase transitions. We exhibit several nontrivial universality classes in infinite-order transitions different from the well known Berezinskiĭ-Kosterlitz-Thouless transition.
The determination of the critical exponents by means of the Exact Renormalizion Group approach is still a topic of debate. The general flow equation is by construction scheme independent, but the use of the truncated derivative expansion generates a model depencence in the determination of the universal quantities. We derive new nonperturbative flow equations for the one-component, Z_2 symmetric scalar field to the next-to-leading order of the derivative expansion by means of a class of proper time regulators. The critical exponents eta, nu and omega for the Wilson-Fisher fixed point are computed by numerical integration of the flow equations, without resorting to polynomial truncations. We show that by reducing the width of the cut-off employed, the critical exponents become rapidly insensitive to the cut-off width and converge to values that are in good agreement with the results of entirely different approaches.
The expression for the coefficient of ${\ensuremath{\epsilon}}^{2}$ in the $\ensuremath{\epsilon}$ expansion for the critical exponent $\ensuremath{\eta}$ of the Ising model has been obtained by Shukla and Green within the incomplete-integration method of Wilson. It was left by them in the form of an unevaluated integral, which appears to depend on parameters characterizing details of the renormalization-group procedure. The integral is evaluated here and found to equal 154\frac{1}{54}541​, regardless of the values of those parameters.