Rigorous characterization of correlation-length trends across topological sectors

Characterize rigorously why the topological-index triples (+1,+1,+1) and (-1,-1,-1) appear easier to realize physically, as suggested by their comparatively shorter correlation lengths in the sampled matrix product states.

Background

The paper constructs symmetry-preserving matrix product state representatives for all eight triples of hidden-order, time-reversal, and reflection indices $(\sigma_H,\sigma_T,\sigma_R)\in\mathbb{Z}_2^3$. Numerical sampling for spins 1, 2, and 3 reports correlation lengths for representatives in each sector. The author observes a trend in which the sectors $(+1,+1,+1)$ and $(-1,-1,-1)$ tend to have shorter correlation lengths than the sectors with mixed or “flipped” indices, suggesting that these two sectors may be easier to realize physically.

The unresolved problem is to determine whether this observed trend has a rigorous explanation or characterization. The paper does not establish a theorem relating the topological-index triple to correlation length or to physical realizability in the nearest-neighbor $SU(2)$-invariant phase diagram.