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

Abstract

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.

21 pages, 10 Postcript figures, latex file, uses revtex

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• Evidence for Complex Subleading Exponents from the High-Temperature Expansion of the Hierarchical Ising Model
• Specific Heat Exponent for the 3-d Ising Model from a 24-th Order High Temperature Series
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• Non-Universal Quantities from Dual RG Transformations
• Hyperscaling in the Broken Symmetry Phase of Dyson's Hierarchical Model
• A Check of a D=4 Field-Theoretical Calculation Using the High-Temperature Expansion for Dyson's Hierarchical Model
• Non-Gaussian numerical errors versus mass hierarchy
• Universality in nontrivial continuum limits: a model calculation
• Small numerators canceling small denominators of the high-temperature scaling variables: a systematic explanation in arbitrary dimensions
• Complex singularities of the critical potential in the large-N limit
• Universality, Scaling and Triviality in a Hierarchical Field Theory