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arXiv:hep-lat/9507016 · Physical Review Letters

Evidence for Complex Subleading Exponents from the High-Temperature Expansion of Dyson's Hierarchical Ising Model

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

Published Dec 18, 1995• 26 citations
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Abstract

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.

DOI
10.1103/physrevlett.75.4555

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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.
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.
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.
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.
Using recently developed methods, we perform high-accuracy calculations of the susceptibility near ${\ensuremath{\beta}}_{c}$ for the D=3D=3D=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.
We study numerically the magnetic susceptibility of the hierarchical model with Ising spins (sigma=pm1\\sigma =\\pm 1sigma=pm1) above the critical temperature and for two values of the epsilon parameter. The integrations are performed exactly, using recursive methods which exploit the symmetries of the model. Lattices with up to 2182^18218 sites have been used. Surprisingly, the numerical data can be fitted very well with a simple power law of the form $(1- \\beta /\\beta _c )^{- \\gamma} $for the {\\it whole} temperature range. The numerical values for $\\gamma $ agree within a few percent with the values calculated with a high-temperature expansion but show significant discrepancies with the epsilon-expansion. We would appreciate comments about these results.
We construct series expansions for the scaling variables (which transform multiplicatively under a renormalization group (RG) transformation) in examples where the RG flows, going from an unstable (Wilson's) fixed point to a stable (high-temperature) fixed point, can be calculated numerically. The examples are Dyson's hierarchical model and a simplified version of it. We provide numerical evidence that the scaling variables about the two fixed points have overlapping domain of convergence. We show how quantities such as the magnetic susceptibility can be expressed in terms of these variables. This procedure provide accurate analytical expressions both in the critical and high-temperature region.