Non-local order parameters for fermion chains via the partial transpose

This work is in one spatial dimension, where long-range-entangled (LRE) or symmetry protected (SPT) and short-range entangled (SRE) or topologically ordered topological phases can be treated simultaneously.^9,17 These distinctions become more relevant in higher dimensions.

Another approach is to impose regularity conditions on the path directly.¹ Another interesting restriction, which is not pursued here, is that the microscopic degrees of freedom are kept constant along the path.²⁷ 

This approximation is much better for exponentially correlated states.^11,50

For Hubbard models with strong interactions, particle–hole transformations are good symmetries when correlated hopping is negligible.⁵⁵ 

This rests on the hereditariness of cones in C*-algebras [see, e.g., Ref. 12 (Sec. 1.5)].

The exponent is chosen as |ξ|² instead of |ξ| for two reasons: (a) The degree of a vector is a $Z2$ variable. Hence, i^|ξ| is ill-defined. Consider, e.g., i^|L(ξ)| = (−1)^|L‖ξ|i^|L|i^|ξ| ≠ i^|L|i|ξ|. This ambiguity is removed by using |ξ|². (b) More formally, $Q:Z2→Z4,x↦x2$ is a quadratic refinement of the bilinear braiding pairing $B:Z2×Z2→Z2,(x,y)↦xy$⁠, i.e., Q and B satisfy the relation 2 · B(x, y) = Q(x + y) − Q(x) − Q(y), where $2⋅:Z2→Z4,x↦2x$⁠. This generalizes to other graduations than $Z2$⁠.

There are two additional assumptions necessary: (i) unitarity, which comes for free for the physicist, and (ii) invertibility, which models the absence of topological order.