Pairing Obstructions in Topological Superconductors

The modern understanding of topological insulators is based on Wannier obstructions in position space. Motivated by this insight, we study topological superconductors from a position-space perspective. For a one-dimensional superconductor, we show that the wave function of an individual Cooper pair decays exponentially with separation in the trivial phase and polynomially in the topological phase. For the position-space Majorana representation, we show that the topological phase is characterized by a nonzero Majorana polarization, which captures an irremovable and quantized separation of Majorana Wannier centers from the atomic positions. We apply our results to diagnose second-order topological superconducting phases in two dimensions. Our work establishes a vantage point for the generalization of topological quantum chemistry to superconductivity.

Figure 1

Position-space picture of insulators and superconductors in 1D, with atomic sites indicated by vertical lines. (a) The ground state of an insulator as a product over Wannier states. All 1D electronic band structures (absent symmetries) can be realized by exponentially localized Wannier functions. In contrast to superconductors, there are therefore no topological insulators in 1D. (b) The fixed-particle number component of the superconducting ground state as a product of Cooper pair states with wave functions decaying exponentially in the trivial phase and polynomially in the topological phase. (c) The ground state as a product over Majorana Wannier functions with quantized Majorana charge. In the topological phase, Majoranas from different atomic sites are paired up, leaving behind unpaired Majorana zero modes at the boundaries.

Figure 2

Comparison of 1D topological insulators and superconductors. (a) In the Su-Schrieffer-Heeger (SSH) model [58], spatial inversion symmetry is required to pin electrons to the unit cell edges, leading to midgap states in an open geometry. However, it is possible to choose an inversion-symmetric enlarged unit cell that has no end states. (b) In the Kitaev chain [59], Majorana modes are paired across unit cells, leaving behind unpaired Majorana zero modes in an open geometry. Since unit cells may not cut through atomic sites, any enlarged unit cell shows the same topological behavior.

Figure 3

Real-space structure of a $C_4$-symmetric second-order topological superconductor. (a) Majorana corner modes in a square-shaped sample geometry. (b) Naively, the two ground state Cooper pair wave functions [generalizations of Eq. (4)] $g_{\mathbf{r}{\mathbf{r}}^{\prime }}$ and ${\tilde{g}}_{\mathbf{r}{\mathbf{r}}^{\prime }}$ decay exponentially with electron separation. (c) When chosen as eigenfunctions of $C_4$ symmetry, the Cooper pair wave functions of each $C_4$ subsector, $g_{\mathbf{r}{\mathbf{r}}^{\prime }}^{C_4,1}$ and $g_{\mathbf{r}{\mathbf{r}}^{\prime }}^{C_4,2}$, necessarily decay polynomially as a consequence of the nontrivial second-order topology.