Renormalization group and critical phenomena. 2. Phase space cell analysis of critical behavior
Jun, 197122 pages
Published in:
- Phys.Rev.B 4 (1971) 3184-3205
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Abstract: (APS)
A generalization of the Ising model is solved, qualitatively, for its critical behavior. In the generalization the spin Sn→ at a lattice site n→ can take on any value from −∞ to ∞. The interaction contains a quartic term in order not to be pure Gaussian. The interaction is investigated by making a change of variable Sn→=Σmψm(n)Sm′, where the functions ψm(n→) are localized wavepacket functions. There are a set of orthogonal wave-packet functions for each order-of-magnitude range of the momentum 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: η=0, γ=1.22, ν=0.61 for three dimensions. In five dimensions or higher one gets η=0, γ=1, and ν=12, 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.References(11)
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