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arXiv:hep-lat/9402020

Algebraic Computation of the Hierarchical Renormalization Group Fixed Points and their $ε$-Expansions

K. Pinn, Andreas Pordt, C. Wieczerkowski

Published Feb 28, 1994• 0 citations
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Abstract

Nontrivial fixed points of the hierarchical renormalization group are computed by numerically solving a system of quadratic equations for the coupling constants. This approach avoids a fine tuning of relevant parameters. We study the eigenvalues of the renormalization group transformation, linearized around the non-trivial fixed points. The numerical results are compared with $ε$-expansion.

DOI
10.48550/arxiv.hep-lat/9402020

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Algebraic Computation of the Hierarchical Renormalization Group Fixed Points and their εεε-Expansions
Nontrivial fixed points of the hierarchical renormalization group are computed by numerically solving a system of quadratic equations for the coupling constants. This approach avoids a fine tuning of relevant parameters. We study the eigenvalues of the renormalization group transformation, linearized around the non-trivial fixed points. The numerical results are compared with εεε-expansion.
We consider the problem of calculating the nonuniversal parameters entering into the scaling laws of renormalized quantities. We examine the renormalization group transformation of Dyson's hierarchical model and a simplified version of the model. We introduce the hierarchical model and explain the use of approximations which aid in extracting numerical information about the susceptibility. We then use these approximations to relate the hierarchical model to a simplified, one- parameter model. In the simplified model, we can express the magnetic susceptibility in terms of dual expansions, one about the high-temperature fixed point, and the other about the critical point. With the introduction of two dual scaling variables which transform covariantly under the renormalization group transformation, we are able to calculate all of the nonuniversal quantities. Extensions of these techniques to multi-parameter models, such as the hierarchical model, are discussed.
We construct series expansions for the scaling variables (which transform multiplicatively under a renormalization group (RG) transformation) in examples where the RG flows, going from an unstable (Wilson's) fixed point to a stable (high-temperature) fixed point, can be calculated numerically. The examples are Dyson's hierarchical model and a simplified version of it. We provide numerical evidence that the scaling variables about the two fixed points have overlapping domain of convergence. We show how quantities such as the magnetic susceptibility can be expressed in terms of these variables. This procedure provide accurate analytical expressions both in the critical and high-temperature region.
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.
We review recent results concerning the renormalization group (RG) transformation of Dyson's hierarchical model (HM). This model can be seen as an approximation of a scalar field theory on a lattice. We introduce the HM and show that its large group of symmetry simplifies drastically the blockspinning procedure. Several equivalent forms of the recursion formula are presented with unified notations. Rigourous and numerical results concerning the recursion formula are summarized. It is pointed out that the recursion formula of the HM is inequivalent to both Wilson's approximate recursion formula and Polchinski's equation in the local potential approximation (despite the very small difference with the exponents of the latter). We draw a comparison between the RG of the HM and functional RG equations in the local potential approximation. The construction of the linear and nonlinear scaling variables is discussed in an operational way. We describe the calculation of non-universal critical amplitudes in terms of the scaling variables of two fixed points. This question appears as a problem of interpolation between these fixed points. Universal amplitude ratios are calculated. We discuss the large- N limit and the complex singularities of the critical potential calculable in this limit. The interpolation between the HM and more conventional lattice models is presented as a symmetry breaking problem. We briefly introduce models with an approximate supersymmetry. One important goal of this review is to present a configuration space counterpart, suitable for lattice formulations, of functional RG equations formulated in momentum space (often called exact RG equations and abbreviated ERGE).
A modification of the Wilson-Kogut smooth-cutoff formulation of the renormalization group is developed. The resulting hierarchy of equations is truncated at four levels, with the renormalization-group parameters being approximated by the leading terms of a mometum-space Taylor series. The equations are then solved for d=3d=3d=3, N=1N=1N=1 systems. A line of fixed points with associated $\ensuremath{\eta}\ensuremath{\cong}0.045$ is obtained. The largest two eigenvalues of the near-fixed-point hierarchy are ${y}_{1}\ensuremath{\cong}1.6$ and ${y}_{2}\ensuremath{\cong}\ensuremath{-}1.6$.