Why we call ''nonlinear'' sigma model?

[ndung200790]

Please teach me this:
The linear sigma model L(Lagrangian)=$\frac{1}{2}$($\delta$$_{\mu}$$\Phi$$^{i}$)$^{2}$ + $\frac{1}{2}$$\mu$$^{2}$($\Phi$$^{i}$)$^{2}$ -
$\frac{\lambda}{4}$(($\Phi$$^{i}$)$^{2}$)$^{2}$.
The nonlinear sigma model:
L=f$_{ij}$({$\Phi$$^{i}$})$\delta$$_{\mu}$
$\Phi$$^{i}$$\delta$$^{\mu}$$\Phi$${j}$.
After put condition O(N) symmetry,we have Lagrangian(because after the putting f=constant):L=$\frac{1}{2g^{2}}$/$\delta$$_{\mu}$n/$^{2}$.
.Then the nonlinear model is a special case of the linear sigma model.So I do not understand why we call it the ''nonlinear'' model?

 

[ndung200790]

At the moment I think that the term nonlinear sigma model has origination from history.It was first considered as an alternative description of spontaneous symmetry breaking.In the nonlinear model the sigma field is constant(the fluctuation of the field is zero,then they call ''nonlinear'').But in the linear sigma model, the fluctuation sigma field is nonzero(then they call ''linear'') plus the expectation at ground state of field(constant).Is that correct?