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A Millenium Problem in Mathematical Physics, -Analysis of 2 Dimensional O(N) Spin Model by Renormalization Group Methed-

Hold Date
2013-01-24 16:00〜2013-01-24 17:30
Seminar Room 6, Faculty of Mathematics building, Ito Campus
Object person
ITO, Keiichi (Setsunan University)

There are unsolved problems in mathematical physics which have been repelling our efforts so long time. Among them, existence or non-existence of phase transitions in some statistical models is related to the basic ansatz (confinements and mass generations) in modern physics and still open. Here we discuss spin models in two-dimensions.

This system is well known for N = 1 (Ising model) and for N = 2 (XY model), but very few are known for N ≥ 3 we discuss. In this model, we have neither adequate definition of domain walls which is essential to study the Ising model nor duality transformation which is crucial to study Abelian models (XY model).

For N ≥ 3, we need some new tricks and calculations. We first introduce partly a Fourier transformation to the O(N) spin variables and obtain a system consisting of the O(N) spins ϕ ∈ RN and an auxiliary field ψ ∈ R. This trick yields a Gaussian system perturbed by an extracted functional determinant. But this new system has long range interactions, which makes our analysis hard. Therefore we apply multi-scale analysis (block spin transformation) to this system. We integrate the system by degrees of freedom of short ranges recursively. Thus we obtain a flow of effective actions. The flow moves toward the high-temperature region for any initial inverse temperature β > 0. This means that there exist no phase transitions if N is sufficiently large, no matter how large β is.