Topological invariant for generic one-dimensional time-reversal-symmetric superconductors in class DIII

A one-dimensional time-reversal-symmetric topological superconductor (symmetry class DIII) features a single Kramers pair of Majorana bound states at each of its ends. These holographic quasiparticles are non-Abelian anyons that obey Ising-type braiding statistics. In the special case where an additional $U\left(1\right)$ spin rotation symmetry is present, this state can be understood as two copies of a Majorana wire in symmetry class $D$, one copy for each spin block. We present a manifestly gauge invariant construction of the topological invariant for the generic case, i.e., in the absence of any additional symmetries like spin rotation symmetry. Furthermore, we show how the presence of inversion symmetry simplifies the calculation of the topological invariant. The proposed scheme is suitable for the classification of both interacting and disordered systems and allows for a straightforward numerical evaluation of the invariant since it does not rely on fixing a continuous phase relation between Bloch functions. Finally, we apply our method to compute the topological phase diagram of a Rashba wire with competing $s$-wave and $p$-wave superconducting pairing terms.

Figure 1

Schematic representation of a 1DTSC in class DIII, consisting of two Kramers partners (red and green) of Kitaev's Majorana chain in class $D$. The white dots denote the Majorana bound states at the ends. The ovals denote the lattice sites hosting two paired Majorans (connected black dots). The arrows denote the localization of the occupied quasiparticles illustrating the polarization of the chain with respect to the oval lattice sites.

Figure 2

Topological invariant $\nu$ as a function ${\alpha }_R$ and ${\Delta }_s$ at $\mu =0.5,{\Delta }_p=1$. Green denotes the nontrivial phase ($\nu =-1$), purple denotes the trivial phase ($\nu =1$). The orange line at ${\Delta }_s=0,{\alpha }_R>1$ indicates a metallic phase and the critical orange points at the phase boundary are also gapless.