Axion topological field theory of topological superconductors

Topological superconductors are gapped superconductors with gapless and topologically robust quasiparticles propagating on the boundary. In this paper, we present a topological field theory description of three-dimensional time-reversal invariant topological superconductors. In our theory the topological superconductor is characterized by a topological coupling between the electromagnetic field and the superconducting phase fluctuation, which has the same form as the coupling of “axions” with an Abelian gauge field. As a physical consequence of our theory, we predict the level crossing induced by the crossing of special “chiral” vortex lines, which can be realized by considering $s$-wave superconductors in proximity with the topological superconductor. Our theory can also be generalized to the coupling with a gravitational field.

Figure 1

Representation of the (3+1)-d topological superconductor on the spatial manifold $M_3$ as the boundary state of a (4+1)-d topological insulator defined on $M_3\times I$.

Figure 2

(a) A system with two vertical vortex lines (lines with double arrow) and four horizontal chiral vortex lines (lines with single arrow). The two blue (red) horizontal lines are chiral vortices of ${\theta }_L$ (${\theta }_R$), respectively (see text). (b) When one vertical vortex line moves across the two horizontal chiral vortex lines, two Majorana fermion zero modes (marked by “$\otimes$” at the vortex lines) are created along the horizontal vortex lines. (c) When both vertical vortex lines cross the chiral vortex lines, the Majorana fermions disappear. (d) The same process happens to the right-handed chiral vortex lines when the two vertical vortex lines are moved across them. Due to periodic boundary condition, the configurations in (a) and (d) are considered to be the same. In terms of energy spectrum, this process corresponds to the level crossing illustrated in (e), with a,b,c,d marking the times corresponding to the configurations shown in (a)–(d). The solid and dashed curves are particle-hole partners of each other, and the blue (red) curves correspond to Majorana modes on the blue (red) chiral vortex lines, respectively.

Figure 3

(a) Illustration of the trijunction configuration with two $s$-wave superconductors on top of a topological superconductor (TSC). The phase of the $s$-wave superconductors are $-\pi /2$ in the blue region and $\pi /2$ in the yellow region. The vectors $(0,\pi )$, $(\pi /2,\pi /2)$, etc. stand for the value of $({\theta }_L,{\theta }_R)$ in the corresponding regions. (b) The evolution of phases ${\theta }_L$ (blue solid line) and ${\theta }_R$ (red dashed line) along the red circle path marked in panel (a). In such a configuration ${\theta }_R$ has winding number $-1$ while ${\theta }_L$ has winding number 0. Depending on the sign of the phase gradient between the two $s$-wave superconductors (the position of the vertical path connecting $\pi /2$ and $-\pi /2$ relative to the origin), one may also have winding number 0 for ${\theta }_R$ and 1 for ${\theta }_L$. Independent from this choice, there is always a Majorana-Weyl fermion propagating along the junction, as is illustrated by the dashed line in panel (a).