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

Abstract

Abstract.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).

Lines of Fisher’s zeros as separatrices for complex renormalization group flows

“…The distribution of the Fisher's zeros is symmetric about the real axis and we only show the zeros in the upper half plane. For the D = 3 case, the system has a second order phase transition at β c ≃ 1.179 [14] and we can check the consistency of our results with finite size scaling [16,17]. In Fig.…”

“…For details, we refer to a recent review article [14]. The partition function is obtained by integrating e −βH with a local measure W 0 (φ).…”

Renormalization Group and Phase Transitions in Spin, Gauge, and QCD Like Theories

“…The partition function must be zero at the critical temperature. We know that the critical β for the three dimensional HM is around 1.179 [118]. In figure (2.9), we can see that the lowest Fisher zeros really accumulate around 1.179…”

“…reasons why we can do so, please refer to [118]. Usually we will take b D = 2 without further notice.…”

“…As mentioned above, the critical temperature β c has been known for a while [118]: β c = 1.17903017 · · · and in principle could be calculated to any significant digits. If the relation 2.46 holds, it is straight forward to get the critical exponent ν by performing a linear fit of the leading zeros(See Fig.…”