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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?

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Yes large N is the same as naive mean field theory to the lowest order, which is just the saddle point approximation to the path integral after you get the action in the form where it is multiplied by an overall factor of N. What is different is it gives a systematic way to go beyond the lowest order and give 1/N corrections, and it also shows that as N increases the saddle point approximation gets better.

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.

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