How to generalize the non-linear sigma model for general magnetic structure? Non-linear sigma model(NLSM) is an effective model to describe magnetic behavior, which only has some phenomenology parameter $g$, $c$, and $\lambda$:$$L=\sum_i \frac{1}{g}[(\partial_\tau n_i)^2+(\partial_r n_i)^2+i\lambda(n_i^2-1)]$$
However, most textbook only gives this model from the Néel anti-ferromagnetism, i.e. via the Haldane mapping:
$$\mathbf{\Omega}_{\mathbf{r}}=(-1)^{\mathbf{r}} \mathbf{n}_{\mathbf{r}} \sqrt{1-\mathbf{L}_{\mathbf{r}}^{2}}+\mathbf{L}_{\mathbf{r}}$$
and gives the final result:
$$S=\frac{1}{2 g} \sum_{q} n_{-q}\left(v^{2}+c^{2} q^{2}+\eta^{2}\right) n_{q} $$
with the dispersion $\epsilon_q=\sqrt{c^2q^2+\eta^{2}}$, where $\eta$ can be seem as the inverse of correlation length.

Here is my question: can this derivation and conclusion be generalized to all the other magnetic configuration?  Or, are there any reference or books about this problem?
For example, if I need a low-energy model to describe the ferromagnetic fluctuation $(0,0)$, or stripe anti-ferromagnetism $(\pi,0)$ fluctuation, can I just naively copy the conclusion above, i.e. the same dispersion? Since I note if I take the limit $\eta\rightarrow 0$, which means long-range ordered established, the dispersion $\epsilon_q \sim c|q|$ is linear in term of $q$, but we know the magnon dispersion for ferromagnetism is quadratic。
 A: There is some confusion in your post. 
First $\lambda$ is not a parameter but a field that should be integrated over. In fact the NLSM looks like a free theory for a 3D vector $n(x)$ with the exception that the vector is constrained to have norm 1. This is the origin of the non-linearity. Strictly speaking what you write in the first equation is a linear theory. 
A common approach to solve the NLSM is given by various large-N expansion. In the limit one obtains indeed a free massive theory, where the mass (your $\eta$) depends on the cutoff. This leads to a relativistic dispersion as you write. 
As for the effective model, it is essentially recognized that the NLSM represents the low energy theory of various antiferromagnetic models (with the inclusion possibly of a topological term). 
The one that you write is in 1+1 dimension (one spatial, one temporal), so it describes the low energy sector of the antiferromagnetic Heisenberg model in 1D for integer spin $S$. The prediction is that it has a unique ground state and a mass gap which implies correlation decay exponentially with the distance. 
For half-integer spins you get an additional topological term and the physics changes completely. In particular there are vanishing excitation with linear dispersion and the correlation decay algebraically. These results though are obtained via the Bethe Ansatz, not through the field theory that in this case is too complicated. 
In any case the simple idea behind this mapping is that after factoring the antiferromagnetic behavior (the $(-1)^x$) the field $n(x)$ is sufficiently smooth. 
Note that the limit $\eta\to 0$ does not makes sense. The mass term $\eta$ in the NLSM emerges out of the non-linearity and it only depends on the spin $S$. 
If you are interested in the effective theory for ferromagnetism I would look somewhere else. In this case the landau theory with order parameter given again by a $3D$ vector is the standard choice. 
Now, as per your question

Can this derivation and conclusion be generalized to all the other magnetic configuration? 

Clearly not as you also say. For example the dispersion of the antiferromagnetic Heisenberg model in 1D (that can be solved by Bethe Ansatz) is linear in $k$, while for the ferromagnetic model the dispersion is quadratic (and the ground state there is degenerate at finite size). 

Or, are there any reference or books about this problem?

A good book that treats a lot of quantum magnetism is Interacting Electrons and Quantum Magnetism by Assa Auerbach. 
