Editing N-vector model (section)

==Continuum limit==

The [[continuum limit]] can be understood to be the [[sigma model]].  This can be easily obtained by writing the Hamiltonian in terms of the product 
:<math>-\tfrac{1}{2}(\mathbf{s}_i - \mathbf{s}_j) \cdot (\mathbf{s}_i - \mathbf{s}_j) = \mathbf{s}_i \cdot \mathbf{s}_j - 1</math>
where <math>\mathbf{s}_i \cdot \mathbf{s}_i=1</math> is the "bulk magnetization" term. Dropping this term as an overall constant factor added to the energy, the limit is obtained by defining the Newton [[finite difference]] as 
 
:<math>\delta_h[\mathbf{s}](i,j)=\frac{\mathbf{s}_i - \mathbf{s}_j}{h}</math>
on neighboring lattice locations <math>i,j.</math> Then <math>\delta_h[\mathbf{s}]\to\nabla_\mu\mathbf{s}</math> in the limit <math>h\to 0</math>, where <math>\nabla_\mu</math> is the [[gradient]] in the <math>(i,j)\to\mu</math> direction. Thus, in the limit,

:<math>-\mathbf{s}_i\cdot \mathbf{s}_j\to \tfrac{1}{2}\nabla_\mu\mathbf{s} \cdot \nabla_\mu\mathbf{s}</math>

which can be recognized as the kinetic energy of the field <math>\mathbf{s}</math> in the [[sigma model]].  One still has two possibilities for the spin <math>\mathbf{s}</math>: it is either taken from a discrete set of spins (the [[Potts model]]) or it is taken as a point on the [[sphere]] <math>S^{n-1}</math>; that is, <math>\mathbf{s}</math> is a continuously-valued vector of unit length. In the later case, this is referred to as the <math>O(n)</math> non-linear sigma model, as the [[rotation group]] <math>O(n)</math> is group of [[isometries]] of <math>S^{n-1}</math>, and obviously, <math>S^{n-1}</math> isn't "flat", ''i.e.'' isn't a [[field (physics)|linear field]].