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).
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. Rigorous 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 other 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 article is to present a configuration space counterpart, suitable for lattice formulations, of other RG equations formulated in momentum space (sometimes abbreviated as ERGE). § To whom correspondence should be addressed (yannick-meurice@uiowa.edu) Nonlinear Aspects of the RG Flows of Dyson’s Hierarchical Model 2
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.
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.
The random hierarchical model is defined by introducing random couplings between spin blocks in Dyson's original hierarchical model for continuous spins. The spin-distribution functions are now site dependent and functions of random parameters that we take to be a quadratic and quartic potential,
rl(m) and
Ul(m), respectively, for the mth spin at the
l level of the hierarchy. Renormalization-group recursion relations for the cumulant averages are derived in an expansion in ${\ensuremath{\epsilon}}^{\ensuremath{'}}=(2\ensuremath{\sigma}\ensuremath{-}1)\mathrm{ln}(2)$, where $\ensuremath{\sigma}$ is a parameter of the model and a measure of the interaction range. We find that only
rl=〈rl(m)〉, ${\ensuremath{\Delta}}_{l}=〈{[r_{l}^{}{}_{}{}^{(m)}\ensuremath{-}{r}_{l}]}^{2}〉$, and
Ul=〈Ul(m)〉 are relevant physical quantities within the ${\ensuremath{\epsilon}}^{\ensuremath{'}}$ expansion. By solving the equations for
Ul and ${\ensuremath{\Delta}}_{l}$ to $O({\ensuremath{\epsilon}}^{\ensuremath{'}})$ we obtain in general four fixed points of the recursion relations. For $\ensuremath{\sigma}>\frac{1}{2}$, there is no stable fixed point in the physical region $\ensuremath{\Delta}>~0$,
U> 0; thus the behavior of the system is still open to study. In the particular case of uniform block coupling but random spin distributions, we find a fixed point with ${U}^{*}\ensuremath{\approx}O({|{\ensuremath{\epsilon}}^{\ensuremath{'}}|}^{\frac{1}{2}})$, ${\ensuremath{\Delta}}^{*}\ensuremath{\approx}O({|{\ensuremath{\epsilon}}^{\ensuremath{'}}|}^{\frac{1}{2}})$ for ${\ensuremath{\epsilon}}^{\ensuremath{'}}<0$, but it is unstable.
Using recently developed methods, we perform high-accuracy calculations of the susceptibility near ${\ensuremath{\beta}}_{c}$ for the
D=3 version of Dyson's hierarchical model. Using linear fits, we estimate the leading (\ensuremath{\gamma}) and subleading (\ensuremath{\Delta}) exponents. Independent estimates are obtained by calculating the first two eigenvalues of the linearized renormalization group transformation. We found $\ensuremath{\gamma}=1.29914073\ifmmode\pm\else\textpm\fi{}{10}^{\ensuremath{-}8}$ and $\ensuremath{\Delta}=0.4259469\ifmmode\pm\else\textpm\fi{}{10}^{\ensuremath{-}7}$ independently of the choice of local integration measure (Ising or Landau-Ginzburg). After a suitable rescaling, the approximate fixed points for a large class of local measure coincide accurately with a fixed point constructed by Koch and Wittwer.