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A one dimensional time reversal symmetric topological superconductor(symmetry class DIII) features a single Kramers pair of Majorana bound statesat each of its ends. These holographic quasiparticles are non-Abelian anyonsthat obey Ising-type braiding statistics. In the special case where anadditional U(1) spin rotation symmetry is present, this state can be understoodas two copies of a Majorana wire in symmetry class D, one copy for each spinblock. We present a manifestly gauge invariant construction of the topologicalinvariant for the generic case, i.e., in the absence of any additionalsymmetries like spin rotation symmetry. Furthermore, we show how the presenceof inversion symmetry simplifies the calcuation of the topological invariant.The proposed scheme is suitable for the classification of both interacting anddisordered systems and allows for a straightforward numerical evaluation of theinvariant since it does not rely on fixing a continuous phase relation betweenBloch functions. Finally, we apply our method to compute the topological phasediagram of a Rashba wire with competing s-wave and p-wave superconductingpairing terms.

Topological invariant for generic 1D time reversal symmetric superconductors in class DIII