michael_nielsen

A few tidbits: the model is spin s=1. The s=1 spins in the chain are modelled as the symmetric subspace of 2 spin 1/2 particles, i.e., the space spanned by ^^, vv, and ^v+v^, in an obvious notation.

Does the definition of the projector (oval shaped symbol) on this page help answer your question about what * should be? https://en.m.wikipedia.org/wiki/AKLT_model  (I did that write up for Wikipedia because it’s such an interesting model and I was surprised it wasn’t already on there!)

I'm afraid I found that unclear. Is it saying that you construct the product of the spin singlet states, and then project onto the symmetric subspace of the paired spin 1/2's?

So in the case of n = 3 spin 1's, represented as 6 spin 1/2s, the ground state would be in P^{otimes 3} * otimes s otimes s otimes *, where P projects on the symmetric subpace of two spin 1/2s, s is the usual spin singlet ^v-v^, and the * states are arbitrary spin 1/2 states.

It seems unclear why the ground state is two-dimensional, since the end spins both have 2 unspecified degrees of freedom.

No, your intuition is right the ground state space is approximately four dimensional for a long, open-boundary AKLT chain. For a periodic ring there is a unique ground state.