hetriamhageh6k20

2022-05-17

Consider one-dimensional ferromagnet namely N spin-1/2 objects placed around a circle with the Hamiltonian
$\mathcal{H}=-\mathcal{J}\sum _{n=1}^{N}{\stackrel{\to }{\mathcal{S}}}_{n}\cdot {\stackrel{\to }{\mathcal{S}}}_{n+1}$
where we assume the periodic boundary condition ${\stackrel{\to }{\mathcal{S}}}_{N+1}\equiv {\stackrel{\to }{\mathcal{S}}}_{1}$ and $\mathcal{J}>0$
I'm trying to show that total spin ket is a good quantum number that is they commute with $\mathcal{H}$ and finding out the energy corresponding to
$|{\psi }_{0}⟩=|↑{⟩}_{1}\otimes |↑{⟩}_{2}\otimes \cdots \otimes |↑{⟩}_{N}$
By definition:
${\mathcal{S}}^{2}={\left(\sum _{n}{\stackrel{\to }{\mathcal{S}}}_{n}\right)}^{2}=\sum _{n}{\mathcal{S}}_{n}^{2}+\sum _{i,j}{\stackrel{\mathcal{\to }}{\mathcal{S}}}_{i}\cdot {\stackrel{\mathcal{\to }}{\mathcal{S}}}_{j}$
The second term has our Hamiltonian but there are other terms also. I don't understand, How do I proceed from here?

allstylekvsvi

you should look at symmetries. The transformations that the total spin induces is a simultaneous rotations of all the spins. Such rotations leave the dot-product ${\stackrel{\to }{S}}_{i}\cdot {\stackrel{\to }{S}}_{j}$ unchanged. Therefore this is a symmetry of the system, which means that ${\stackrel{\to }{S}}_{\mathrm{t}\mathrm{o}\mathrm{t}}=\sum _{i}{\stackrel{\to }{S}}_{i}$ are conserved, and represent good quantum numbers (of course one can only pick ${S}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{2}$ and one of the vector component, say ${S}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{z}$)