I'm currently researching symmetry breaking and Goldstone's theorem for a project in my third year of my theoretical physics degree. So my knowledge isn't from a formal teaching but my own research.
I began with trying to understand Goldstone's theorem and from what I understand it's the idea that if a continuous symmetry is broken, you get massless scalar fields (Goldstone bosons). I've gone through the mathematics of this and it seems to make sense.
However, I'm looking into the Heisenberg model as a sort of real world example of Goldstone's theorem and I'm coming across issues. I guess I don't have a question per se but more looking to see if my understanding is correct. So the Heisenberg model says the Hamiltonian is made up from the spins of nearest neighbours in a lattice. Clearly this hamiltonian is symmetric under rotation (if you rotate all the spins by theta then the net energy will remain the same?). The ground state would be the state in which all the spins are pointing in the same direction and clearly there are an infinite number of these as they can point in any direction provided they're all pointing the same way. I have then heard that "choosing" a ground state spontaneously breaks this symmetry, is that because you've now collapsed from a infinite number of possible ground states which are invariant under rotation to a single state and hence if you rotate all the spins then it wouldn't be that specific state you selected? Furthermore, where does Goldstone's theorem come into this? I've heard something about spin waves, are these the goldstone bosons in this circumstance?
I hope somebody can help answer my questions or point me in the right direction. I've tried to explain myself clearly, whether or not that was achieved is a different story. Thank you all very much in advance.