About spin chain string order

Question

We know that the string order of a spin chain is defined as $$\mathcal{O}^\alpha=\lim_{i-j\to\infty}\left\langle S_i^\alpha\prod_{k=i+1}^{j-1}\exp(i\pi S_k^\alpha)\ S_j^\alpha \right\rangle$$ now consider a spin-1 chain and the state to be $|+1,+1,\cdots,+1\rangle$. Then the string order should be $\langle1\cdot(-1)^{j-i-1}\cdot1\rangle$, then this number is not defined.

I think the way to avoid this is by consider that any state cannot be so pure so that as $i-j\to\infty$ this number could have exponential decay or something like that, making the string order turns into 0. I'm highly likely to be wrong, but what the correct way to explain this?

Edit: as suggested, the string order might only be able to study the ground state topological properties, and for systems (Hamiltonian) with different class of symmetry, the string order parameter is different. However, what if one want to study an eigenstate's topological properties in a non-equilibrium scenario?

Answer 1

Your mistake is in claiming that that is 'the' string order. There is no such unique string order; rather, it depends on the phase of matter one wants to probe.

Let me illustrate the point with an example that is more familiar: suppose someone tells you that 'the' symmetry-breaking order parameter is $$\lim_{|i-j| \to \infty}\langle S^x_i S^x_j \rangle. $$ And then suppose that same person complains that this is not well-defined for an anti-ferromagnet, where the above quantity would have an alternating sign on every other site.

Your answer to that person would be simple: firstly, for different symmetry-breaking orders you could need different order parameters (in general one tries to find some operator $\mathcal O$ which does not commute with the symmetry yet which would obtain a finite expectation value in a symmetry-broken ground state, and then one is interested in the long-range two-point correlator of that operator). Secondly, it is understood that one tries to peel off short-distance phenomena, as they are irrelevant and often easy to avoid given the physics at hand. In particular, if one expects an anti-ferromagnetic phase, one would instead propose $\lim_{|i-j| \to \infty} (-1)^{i-j} \langle S^x_i S^x_j \rangle$.

Having cleared that issue up, the natural question you might then wonder is "but how does one in general figure out the correct string order parameter for a given symmetry-protected topological (SPT) phase?" That is a good question, and its answer has been figured out (at least for a huge class of SPT phases). I can for example recommend https://arxiv.org/abs/1204.0704 for further details on this.