In the last two decades, a vast variety of topological phases have been described, predicted, classified, proposed, and measured. While there is a certain unity in method and philosophy, the phenomenology differs wildly. This work deals with the simplest such case: fermions in one spatial dimension, in the presence of a symmetry group G, which contains anti-unitary symmetries. A complete classification of topological phases, in this case, is available. Nevertheless, these methods are to some extent lacking as they generally do not allow to determine the class of a given system easily. This paper will take up proposals for non-local order parameters defined through anti-unitary symmetries. They are shown to be homotopy invariants on a suitable set of ground states. For matrix product states, an interpretation of these invariants is provided: in particular, for a particle–hole symmetry, the invariant determines a real division super algebra such that the bond algebra is a matrix algebra over .
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.
For Hubbard models with strong interactions, particle–hole transformations are good symmetries when correlated hopping is negligible.55
The exponent is chosen as |ξ|2 instead of |ξ| for two reasons: (a) The degree of a vector is a 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, is a quadratic refinement of the bilinear braiding pairing , i.e., Q and B satisfy the relation 2 · B(x, y) = Q(x + y) − Q(x) − Q(y), where . This generalizes to other graduations than .
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.