Odd-Parity Topological Superconductors: Theory and Application to Cux_xBi2_2Se3_3

Topological superconductors have been theoretically predicted as a new class of time-reversal-invariant superconductors which are fully gapped in the bulk but have protected gapless surface Andreev bound states. In this work, we provide a simple criterion that directly identifies this topological phase in \textit{odd-parity} superconductors. We next propose a two-orbital $U-V$ pairing model for the newly discovered superconductor Cu$_x$Bi$_2$Se$_3$%. Due to its peculiar three-dimensional Dirac band structure, we find that an inter-orbital triplet pairing with odd-parity is favored in a significant part of the phase diagram, and therefore gives rise to a topological superconductor phase. Finally we propose sharp experimental tests of such a pairing symmetry.Comment: 4.1 pages, 2 figure

From one-dimensional charge conserving superconductors to the gapless Haldane phase

We develop a framework to analyze one-dimensional topological superconductors with charge conservation. In particular, we consider models with $N$ flavors of fermions and $(\mathbb{Z}_2)^N$ symmetry, associated with the conservation of the fermionic parity of each flavor. For a single flavor, we recover the result that a distinct topological phase with exponentially localized zero modes does not exist due to absence of a gap to single particles in the bulk. For $N>1$, however, we show that the ends of the system can host low-energy, exponentially-localized modes. The analysis can readily be generalized to systems in other symmetry classes. To illustrate these ideas, we focus on lattice models with $SO\left(N\right)$ symmetric interactions, and study the phase transition between the trivial and the topological gapless phases using bosonization and a weak-coupling renormalization group analysis. As a concrete example, we study in detail the case of $N=3$. We show that in this case, the topologically non-trivial superconducting phase corresponds to a gapless analogue of the Haldane phase in spin-1 chains. In this phase, although the bulk is gapless to single particle excitations, the ends host spin-$1/2$ degrees of freedom which are exponentially localized and protected by the spin gap in the bulk. We obtain the full phase diagram of the model numerically, using density matrix renormalization group calculations. Within this model, we identify the self-dual line studied by Andrei and Destri [Nucl. Phys. B, 231(3), 445-480 (1984)], as a first-order transition line between the gapless Haldane phase and a trivial gapless phase. This allows us to identify the propagating spin-$1/2$ kinks in the Andrei-Destri model as the topological end-modes present at the domain walls between the two phases