Topological Superconductivity and Superfluidity

Abstract

We construct time reversal invariant topological superconductors and superfluids in two and three dimensions which are analogous to the recently discovered quantum spin Hall and three-d $Z_2$ topological insulators respectively. These states have a full pairing gap in the bulk, gapless counter-propagating Majorana states at the boundary, and a pair of Majorana zero modes associated with each vortex. We show that the time reversal symmetry naturally emerges as a supersymmetry, which changes the parity of the fermion number associated with each time-reversal invariant vortex. In the presence of external T-breaking fields, non-local topological correlation is established among these fields, which is an experimentally observable manifestation of the emergent supersymmetry.

Figures (3)

• Figure 1: (Top row) Schematic comparison of $2d$ chiral superconductor and the QH state. In both systems, TR symmetry is broken and the edge states carry a definite chirality. (Bottom row) Schematic comparison of $2d$ TRI topological superconductor and the QSH insulator. Both systems preserve TR symmetry and have a helical pair of edge states, where opposite spin states counter-propagate. The dashed lines show that the edge states of the superconductors are Majorana fermions so that the $E<0$ part of the quasi-particle spectra are redundant. In terms of the edge state degrees of freedom, we have ${\rm (QSH)}={\rm (QH)}^2={\rm (Helical~SC)}^2={\rm (Chiral~SC)}^4$.
• Figure 2: Illustration of a 3d TRI topological superconductor with two TRI vortex rings which are (a) linked or (b) unlinked. The $E-k_\parallel$ dispersion relations show schematically the quasiparticle levels confined on the red vortex ring in both cases. "$\circ$" and "$\times$" stand for the quasiparticle levels that are Kramers' partners of each other. Only case (a) has a pair of Majorana zero modes located on each vortex ring.
• Figure 3: (a) Illustration of a 2d TRI topological superconductor with four TRI topological defects coupled to a TR-breaking field. The arrows show the sign of the TR-breaking field ${\rm sgn}(M({\bf r}_s))$ at each topological defect. In the two configurations shown, only the field around vortex 1 is flipped, leading to an opposite fermion number parity in the corresponding ground state $|G\rangle$ and $|G'\rangle$ (see text). (b) Illustration showing the flow of the energy levels when the upper configuration in figure (a) is deformed to the lower one. The flip of the TR-breaking field $M({\bf r}_1)$ leads to a level crossing at $M({\bf r}_1)=0$, where the fermion number parity in the ground state changes sign.