arXiv:hep-lat/9904002 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Nonuniversal quantities from dual renormalization group transformations

Yannick Meurice, S. Niermann

Published Sep 1, 1999• 9 citations

Abstract

Using a simplified version of the renormalization group (RG) transformation of Dyson's hierarchical model, we show that one can calculate all the nonuniversal quantities entering into the scaling laws by combining an expansion about the high-temperature fixed point with a dual expansion about the critical point. The magnetic susceptibility is expressed in terms of two dual quantities transforming covariantly under an RG transformation and has a smooth behavior in the high-temperature limit. Using the analogy with Hamiltonian mechanics, the simplified example discussed here is similar to the anharmonic oscillator, while more realistic examples can be thought of as coupled oscillators, allowing resonance phenomena.

DOI: 10.1103/physreve.60.2612

Related papers

Nonlinear aspects of the renormalization group

S. Niermann

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.

Dec 1, 20000 citations

Examples of overlapping convergent expansions of scaling variables

Yannick Meurice, S. Niermann

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.

arXiv:hep-lat/0007037Jul 25, 20000 citations

The Oscillatory Behavior of the High-Temperature Expansion of Dyson's Hierarchical Model: A Renormalization Group Analysis

Meurice, Y, Niermann, S M, Ordaz, G

We calculate 800 coefficients of the high-temperature expansion of the magnetic susceptibility of Dyson's hierarchical model with a Landau-Ginzburg measure. Log-periodic corrections to the scaling laws appear as in the case of a Ising measure. The period of oscillation appears to be a universal quantity given in good approximation by the logarithm of the largest eigenvalue of the linearized RG transformation, in agreement with a possibility suggested by K. Wilson and developed by Niemeijer and van Leeuwen. We estimate $\\gamma $ to be 1.300 (with a systematic error of the order of 0.002) in good agreement with the results obtained with other methods such as the $\\epsilon $-expansion. We briefly discuss the relationship between the oscillations and the zeros of the partition function near the critical point in the complex temperature plane.

Aug 6, 19968 citations

Evidence for Complex Subleading Exponents from the High-Temperature Expansion of Dyson's Hierarchical Ising Model

Yannick Meurice, G. Ordaz, V.G.J. Rodgers

Using a renormalization group method, we calculate 800 high-temperature coefficients of the magnetic susceptibility of the hierarchical Ising model. The conventional quantities obtained from differences of ratios of coefficients show unexpected smooth oscillations with a period growing logarithmically and can be fitted assuming corrections to the scaling laws with complex exponents.

arXiv:hep-lat/9507016Physical Review LettersDec 18, 199526 citations

Nonlinear aspects of the renormalization group flows of Dyson's hierarchical model

Y Meurice

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).

arXiv:hep-th/0701191Journal of Physics A Mathematical and TheoreticalMay 22, 200727 citations

Non-perturbative methods for hierarchical models

M. B. Oktay

The goal of this thesis is to provide a practical method to calculate, in scalar field theory, accurate numerical values of the renormalized quantities which could be used to test any kind of approximate calculation. We use finite truncations of the Fourier transform of the recursion formula for Dyson's hierarchical model in the symmetric and broken phases to perform high precision calculations of the Green's functions at zero momentum. We use the well-known correspondence between statistical mechanics and field theory in which the large cut-off limit is obtained by letting β reach a critical value βc. We show that the round-off errors on the magnetic susceptibility grow like (βc − β) −1 near criticality. We show that the systematic errors (finite truncation and volume) can be controlled with an exponential precision and reduced to a level lower than numerical errors. We probe the numerical errors made in Renormalization Group (RG) calculations by varying slightly the resealing factor of the fields and resealing back in order to get the same (if there were no round-off errors) magnetic susceptibility. We find a significant departure from Gaussian behavior. In addition, the average value differs from the exact answer by a quantity which is of the same order as the standard deviation. We calculate the critical exponents γ and the subleading exponent Δ of Dyson's model by direct fits of the zero-momentum two-point function calculated with an Ising and a Landau-Ginzburg measure and by linearization about the non-trivial fixed point for O(N) model up to N = 5. We extend the method to the low temperature phase and show that accurate values of the connected one-, two- and three- point functions can be obtained at large volume and in a limited range of constant external field coupled linearly to the field variable. We show that the leading exponents obey the scaling and hyperscaling relations. We discuss the concept of universality in terms of critical exponents, correction amplitudes and the universal coupling constant for different Landau-Ginzburg measures. We compare our accurate numerical results with those obtained from perturbative methods.

Jan 1, 20011 citation