The high-temperature expansion of the hierarchical Ising model: From Poincaré symmetry to an algebraic algorithm

Meurice, Yannick

doi:10.1063/1.531087

Cited by 9 publications

(13 citation statements)

References 14 publications

(18 reference statements)

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“…We see that this quantity grows linearly with the dimension. The explicit calculation of the high-temperature expansion [1] suggests that when D → ∞, i.e. when c → 2, we have…”

Section: The Phase Structure Of the Approximated Modelsmentioning

confidence: 99%

Guide to precision calculations in Dyson’s hierarchical scalar field theory

et al. 1998

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The goal of this article is to provide a practical method to calculate, in a scalar 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 phase to perform high-precision calculations of the unsubtracted Green's functions at zero momentum in dimension 3, 4, and 5. We use the well-known correspondence between statistical mechanics and field theory in which the large cutoff limit is obtained by letting ␤ reach a critical value ␤ c ͑with up to 16 significant digits in our actual calculations͒. We show that the round-off errors on the magnetic susceptibility grow like (␤ c Ϫ␤)Ϫ1 near criticality. We show that the systematic errors ͑finite truncations and volume͒ can be controlled with an exponential precision and reduced to a level lower than the numerical errors. We justify the use of the truncation for calculations of the high-temperature expansion. We calculate the dimensionless renormalized coupling constant corresponding to the 4-point function and show that when ␤→␤ c , this quantity tends to a fixed value which can be determined accurately when Dϭ3 ͑hyperscaling holds͒, and goes to zero like ͓Ln(␤ c Ϫ␤)͔ Ϫ1 when Dϭ4. ͓S0556-2821͑98͒05510-6͔

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“…We see that this quantity grows linearly with the dimension. The explicit calculation of the high-temperature expansion [1] suggests that when D → ∞, i.e. when c → 2, we have…”

Section: The Phase Structure Of the Approximated Modelsmentioning

confidence: 99%

Guide to precision calculations in Dyson’s hierarchical scalar field theory

et al. 1998

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“…We give here a presentation [66] that does not require a detailed knowledge of the p-adic numbers. We will rewrite the hamiltonian of the HM given in equation (3.1) using a function v(x, y) which specifies the level l at which x and y start to differ.…”

Section: Scalar Models On Ultrametric Spacesmentioning

confidence: 99%

“…In particular models of random walks over the p-adic numbers were considered [62,63,64] and it was recognized that it was possible to reformulate the HM as a scalar model on the 2-adic fractions [65]. This reformulation was used in high temperature (HT) expansions [66], helps understanding the absence of certain Feynman graphs [3] and suggests ways to improve the hierarchical approximation [21]. This is discussed in more detail in section 14.…”

Section: Motivationsmentioning

confidence: 99%

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

Meurice¹

2007

J. Phys. A: Math. Theor.

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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).

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“…This observation can be substantiated by using exact results at finite volume [10] for low order coefficients, or by displaying the values of higher order coefficients at successive iterations as in Figure 1 of Ref. [9].…”

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confidence: 99%

“…This method has been presented in Ref. [9] and checked using results obtained with conventional graphical methods [10]. For the sake of briefness, we shall follow exactly the set-up and the notations of Refs.…”

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confidence: 99%