Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior

@article{citeulike:2212838, abstract = {A generalization of the Ising model is solved; qualitatively; for its critical behavior. In the generalization the spin S n→ 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 S n→ =Σ m ψ m (n) S m ′ ; where the functions ψ m (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=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.}, author = {Wilson, Kenneth G. }, citeulike-article-id = {2212838}, doi = {http://dx.doi.org/10.1103/PhysRevB.4.3184}, journal = {Physical Review B}, keywords = {physics, statmech, theory}, month = {November}, number = {9}, pages = {3184+}, posted-at = {2008-01-10 02:10:20}, priority = {1}, publisher = {American Physical Society}, title = {Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior}, url = {http://dx.doi.org/10.1103/PhysRevB.4.3184}, volume = {4}, year = {1971} }

TY - JOUR ID - citeulike:2212838 L3 - citeulike-article-id:2212838 TI - Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior JF - Physical Review B VL - 4 IS - 9 SP - 3184 PB - American Physical Society N2 - A generalization of the Ising model is solved; qualitatively; for its critical behavior. In the generalization the spin S n→ 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 S n→ =Σ m ψ m (n) S m ′ ; where the functions ψ m (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=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. KW - physics KW - statmech KW - theory AU - Wilson, Kenneth PY - 1971/11/01/ UR - http://dx.doi.org/10.1103/PhysRevB.4.3184 ER -