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"Total spin conservation" means global $SU(2)$ spin-rotation symmetry (a continuous symmetry) of the Heisenberg model, and "spin wave" indicates an ordered ground state that spontaneously breaks the spin-rotation symmetry. Thus, according to Goldstone theorem, there must be a gapless mode for spin wave.

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There are two ways to deal with a linear term in $\phi$: Complete the square, as was suggested in the comments. This is very often possible, but sometimes you do not want to do that. Interpret it as an interaction term with a $\phi$ particle popping out of the vacuum or vanishing. This will lead to non-zero tadpoles in your Feynman diagrams, so additional ...

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The first assumption is that whatever vev the higgs picks up is constant in space, because this has less energy than one that increases the kinetic term in the Lagrangian. So we can do one global transformation to make the vev be in the second component only. You can imagine doing this prior to symmetry breaking, if you know what it is going to be ahead of ...

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Well, after symmetry breaking, all that remains is electromagnetic $U(1)$, so the only generator that is truly a symmetry generator is $Q$. The fermions couple to the "Higgs" via the Yukawa coupling: $\mathcal{L}_y = -y_e^{ij} \bar L_{L,i} \Phi e_{R,j} - y_u^{ij} \bar Q_{L,i} \tilde{\Phi} u_{R,j} - y_d^{ij} \bar Q_{L,i} \Phi d_{R,j} + h.c.\,$ which mixes ...

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