# Difference between mean-field theory and large $N$ of $CP^{N-1}$

I am reading the Ch.14 of Auerbach, Interaction electrons and quantum magnetism about $$CP^{N-1}$$ which describes non-linear $$\sigma$$ model.

The complex field is $$\mathbf{z}=\left(z_{1}, z_{2}, \ldots, z_{N}\right)$$ with the $$constraint:|\mathbf{z}|^{2}=\sum_{m=1}^{N}\left|z_{m}\right|^{2}=\frac{N}{2}$$. Then we can generalize from $$CP^{1}$$ model :\begin{aligned} Z_{C P 1}=& \int \mathcal{D}^{2} \mathbf{z} \mathcal{D} \mathbf{A} \mathcal{D} \lambda \exp \left\{-\int d^{d}_{x}\left[\frac{2 \Lambda^{d-2}}{f} \sum_{\mu}\left|\left(\partial_{\mu}-i A_{\mu}\right) \mathbf{z}\right|^{2}-i \lambda\left(|\mathbf{z}|^{2}-1\right)\right]\right\} \end{aligned}

to $$CP^{N-1}$$ model:

\begin{aligned} Z_{C P^{N-1}} &=\int \mathcal{D} \mathbf{A} \mathcal{D} \lambda \exp (-N \mathcal{S}[\mathbf{A}, \lambda]) \\ \mathcal{S}[\mathbf{A}, \lambda] &=\operatorname{Tr} \ln \left[-\left(\partial_{\mu}-i A_{\mu}\right)^{2}+\lambda\right]-\frac{i \Lambda^{d-2}}{f} \int d^{d} x \lambda \end{aligned}

Since there exists the coefficient $$N$$ in front of $$S$$, we can just computing the saddle point and ignore the correction(e.g. Gaussian fluctuation):$$\left.\frac{\delta \mathcal{S}}{\delta \overline{\mathbf{A}}}\right|_{\overline{\mathbf{A}}_{\mathbf{0}, \lambda_{0}}}=\left.\frac{\delta \mathcal{S}}{\delta \lambda}\right|_{\overline{\mathbf{A}}_{\mathbf{0}, \lambda_{0}}}=0$$

Finally, we can obtain the "large-N mean-field Hamiltonian":$$H_{MFT}^N=-\frac{2 \Lambda^{d-2}}{f} \sum_{\mathbf{k}}\left(|\mathbf{k}|^{2}+i\lambda_0\right) \mathbf{z}_{\mathbf{k}}^{*} \mathbf{z}_{\mathbf{k}}-\frac{N \mathcal{N} \Lambda^{d-2}}{f} i\lambda_0$$

Then I have two question:

(1) It seems like the large-N (i.e. $$CP^1\rightarrow CP^{N-1}$$) just add a coefficient $$N$$ in front of origin action. So I could obtain the same Hamiltonian if I make naive MFT for $$CP^1$$model directly. But we know the "large-U MFT" is different from "naive MFT" (e.g. Mean field theory = large-$N$ approximation?) So I am confused what is my fault?

(2)I find the constraint of large-N sometimes is arbitrary, for example, $$|\mathbf{z}|^{2}=\sum_{m=1}^{N}\left|z_{m}\right|^{2}=\frac{N}{2}$$ or $$1$$ in different reference. Does it lead to some distinction?

For your second question, the constraint appears in the Lagrange multiplier term like $$\lambda(|z|^2-1)$$. If $$|z|^2=N/2$$, this term needs to be changed to $$\lambda(|z|^2-N/2)$$ (you have some mistakes in your formulation of the question). We can change the constraint to be whatever we like by rescaling the $$z$$ fields, but this will introduce a factor in the kinetic term, which effectively rescales the parameter $$f$$.
If you do it correctly, you will find that if you use a constraint like $$|z|^2=1$$ that does not scale like N, you will need to assume $$f$$ scales with N in order to get the total action to be proportional to N, which again is the beginning of the saddle point approximation.