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My idea is the following: We have a system with Hamiltonian $H$, and we know that there is spin wave in this system by some symmetry-breaking arguments. Now we start from the ground state $\left|\uparrow\uparrow\uparrow\cdots\right\rangle$, and I hope to express the Goldstone mode by an exponential of hermitian operator $\hat{O}\left(\mathbf{k}\right)$, where $\mathbf{k}$ is the wave vector:

$ \left|\mathbf{k}\right\rangle \sim e^{i\hat{O}\left(\mathbf{k}\right)}\left|\uparrow\uparrow\uparrow\cdots\right\rangle $

I wish to get the explicit form of $\hat{O}\left(\mathbf{k}\right)$ by making analogy to the quantum mechanical spin rotation $e^{-i\phi\hat{\mathbf{n}}\cdot\frac{\mathbf{\sigma}}{2}}\left|\uparrow\right\rangle$. Furthermore, is it possible to have some result like

$\left\langle \uparrow\uparrow\uparrow\cdots\right|e^{-i\hat{O}\left(\mathbf{k}\right)}\mathbf{S}\left(\mathbf{r}\right)e^{i\hat{O}\left(\mathbf{k}\right)}\left|\uparrow\uparrow\uparrow\cdots\right\rangle \sim\mathbf{S}_{0}e^{i\mathbf{k}\cdot\mathbf{r}}$?

Could someone offer some hint here, or reference containing the explicit expression? Thx :D

All the $\left|\uparrow\right\rangle$ kets are $\sigma_{z}$ eigen vectors, i.e. spin-z up.

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