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Renormalization group and critical phenomena. II: Phase-space cell analysis of critical behavior. (English) Zbl 1236.82016

Summary: A generalization of the Ising model is solved, qualitatively, for its critical behavior. In the generalization the spin \(s_{\vec n}\) at a lattice site \(\vec n\) can take on any value from \(-\infty\) to \(\infty\). The interaction contains a quartic term in order not to be pure Gaussian. The interaction is investigated by making a change of variable \(s_{\vec n}=\sum_m\psi_m(\vec n)s_m'\), where the functions \(\psi_m(\vec n)\) are localized wavepacket functions. There are a set of orthogonal wave-packet functions for each order-of-magnitude range of the momentum \(\vec k\). An effective interaction is defined by integrating out the wave-packet variables with momentum of order 1, leaving unintegrated the variables with momentum \(<0.5\). Then the variables with momentum between 0.25 and 0.5 are integrated, etc. The integrals are computed qualitatively. The result is to give a recursion formula for a sequence of effective Landau-Ginsberg-type interactions. Solution of the recursion formula gives the following exponents: \(\eta=0, \gamma=1.22, \nu=0.61\) for three dimensions. In five dimensions or higher one gets \(\eta=0, \gamma=1\), and \(\nu=1/2\), as in the Gaussian model (at least for a small quartic term). Small corrections neglected in the analysis may make changes (probably small) in the exponents for three dimensions.
For part I, cf. Phys. Rev. B (3) 4, No. 9, 3174–3183 (1971; Zbl 1236.82017).

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B28 Renormalization group methods in equilibrium statistical mechanics

Citations:

Zbl 1236.82017
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References:

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