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The elusive asymptotic behavior of the high-temperature expansion of the hierarchical Ising model

Yannick Meurice, G. Ordaz

arXiv:hep-lat/9409021Journal of Statistical PhysicsJan 1, 19968 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.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

Nonuniversal quantities from dual renormalization group transformations

Yannick Meurice, S. Niermann

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.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.

arXiv:hep-lat/9904002Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsSep 1, 19999 citations

The hierarchical model and the renormalization group

Giovanni Gallavotti, H. J. F. Knops

Rivista Del Nuovo CimentoJul 1, 197515 citations

A numerical study of the hierarchical Ising model: High-temperature versus epsilon expansion

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

arXiv:hep-lat/9401016Journal of Statistical PhysicsNov 1, 19948 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

The oscillatory behavior of the high-temperature expansion of Dyson's hierarchical model: A renormalization group analysis

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