Title:A Millennium Problem in Mathematical Physics  
- Analysis of  Two-dimensional $O(N)$ Spin Models 
by the Renormalization Group Method -- 

Abstract:


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 $N\geq 3$ 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 $N\geq 3$, 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 $\phi\in R^{N}$ and an auxiliary field 
$\psi \in R$.  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 $\beta>0$. This means  that 
there exist no phase transitions if $N$ is sufficiently 
large,  no matter how large $\beta$ is.