Coordinate Bethe Ansatz

Question

I had a couple of questions regarding the coordinate Bethe ansatz after trying to follow along in some lecture notes.

1. The first one regards showing that the $H_{XXX}$ Hamiltonian preserves down spins. This was claimed without proof. The way I figured to go about this was to show that $S^{z} = \sum_{i}S^{z}_{i}$ commutes with the Hamiltonian. We also note that $\left[S^{a}_{i}, S^{b}_{j} \right] = \delta_{ij} \epsilon^{abc}S^{c}_{i}$. My work is as follows:

$\left[S^{z}, H \right] = \left[S^{z},\sum_{i}J_{x}S_{i}^{x}S^{x}_{i+1} + J_{y}S_{i}^{y}S^{y}_{i+1} + J_{z}S_{i}^{z}S^{z}_{i+1} \right] = \sum_{i}J_{x}(\left[S^{z}_{i},S^{x}_{i} \right]S_{i+1}^{x} + S_{i}^{x}\left[S_{i}^{z},S_{i+1}^{x} \right]) + J_{y}(\left[S^{z}_{i},S^{y}_{i} \right]S_{i+1}^{y} + S_{i}^{y}\left[S_{i}^{z},S_{i+1}^{y} \right]) + J_{z}(\left[S^{z}_{i},S^{z}_{i} \right]S_{i+1}^{z} + S_{i}^{z}\left[S_{i}^{z},S_{i+1}^{z} \right]) = J_{x}\left[S_{i}^{z}, S_{i}^{x} \right]S^{x}_{i+1} + J_{y}\left[S_{i}^{z}, S^{y}_{i} \right]S^{z}_{i+1} = J(S^{y}_{i}S^{x}_{i+1} - S^{x}_{i}S^{y}_{i+1}).$

where I applied two commutator identities to go to the second equality, I applied the spin commutator to get to the third equality, and then applied it again to get to the fourth equality (also setting $J_{x}=J_{y}$ for XXX and XXZ). Am I allowed to re-define the indicies in the first term, i.e. $x \rightarrow y$, $y \rightarrow x$ so that

$J(S^{x}_{i}S^{y}_{i+1} - S^{x}_{i}S^{y}_{i+1}) = 0$? If so, what are the conditions by which I would know how one could rename indicies in this way?

2. The kets for spin are given by: $S^{+}\vert \uparrow \rangle = 0, S^{-}\vert\uparrow\rangle = \vert\downarrow \rangle, S^{+}\vert\downarrow \rangle = \vert \uparrow \rangle, S^{-}\vert \downarrow \rangle = 0, S^{z}\vert\uparrow\rangle = \frac{1}{2}\vert\uparrow\rangle, S^{z}\vert\downarrow\rangle = -\frac{1}{2}\vert\downarrow\rangle $

I also note that $\vert n_{1}, n_{2}, ..., n_{N} \rangle = S^{-}_{n_{1}}S^{-}_{n_{2}}...S^{-}_{n_{N}}\vert\Omega \rangle$ and $\vert\Omega\rangle = \vert \uparrow \uparrow ...\rangle$ is the vacuum eigenvector of all spins up.

I was wondering if someone would be able to explain in a bit more detail how to compute

$\sum_{l=1}^{L}S^{+}_{l}S^{-}_{l+1}\vert n \rangle$,

$\sum_{l=1}^{L}S^{-}_{l}S^{+}_{l+1}\vert n \rangle$,

$\sum_{l=1}^{L}S^{z}_{l}S^{z}_{l+1}\vert n \rangle$.

I tried computed the first one and obtained $\vert n+1 \rangle$ by the first $S^{-}_{l+1}$ acting on the $\vert n \rangle$ turning a spin from up to down at the (n+1)-th site, and then the $S^{+}_{l}$ operator acting on the ket to flip the spin at the nth site from down to up. Hence we are left with an (n+1)-ket. When I tried doing the second one, it appeared that the whole ket was killed when I applied the first operator. I also didn't obtain the third value of $\frac{L-4}{4}$, so I am definitely missing something and would appreciate some help.

Answer 1

1. You're missing some terms in the third equality: the sum disappeared and you've dropped the terms where you take the commutators with the second spin operator in each product.

2. a) That's correct. b) The result is nonzero if $l+1=n$. Hint: first work out the commutators between $S^+_i, S^-_j, S^z_k$ and then do the computation more carefully. c) Hint: what is the eigenvalue of $S^z_i$ on $\vert \Omega\rangle$?