Majorana Modes in Driven-Dissipative Atomic Superfluids with a Zero Chern Number

We investigate dissipation-induced $p$-wave paired states of fermions in two dimensions and show the existence of spatially separated Majorana zero modes in a phase with vanishing Chern number. We construct an explicit and natural model of a dissipative vortex that traps a single of these modes, and establish its topological origin by mapping the problem to a chiral one-dimensional wire where we observe a nonequilibrium topological phase transition characterized by an abrupt change of a topological invariant (winding number). We show that the existence of a single Majorana zero mode in the vortex core is intimately tied to the dissipative nature of our model. Engineered dissipation opens up possibilities for experimentally realizing such states with no Hamiltonian counterpart.

Figure 1

Mechanism ensuring the existence of a single MZM in the core of an odd dissipative vortex. (a), Left: $\ell =1$ vortex on the plane, characterized by a core (blue gradient) and a phase factor premultiplying the $p$-wave operator ${\partial }_x+i{\partial }_y$. Right: Mapping from the (grey) annular region around the vortex core to the cylinder—where the vortex phase disappears from the relevant $p$-wave operator—and reduction to a (chiral) 1D wire problem (thick red). (b) Topological phase diagram for the planar model of Eq. (2) with no vortex (dotted blue, characterized by the Chern number ${\nu }_{2\mathrm{D}}$) and with a single $\ell =1$ vortex (red, characterized by the winding number ${\nu }_{1\mathrm{D}}$). (c) Dissipative boundary conditions of the 1D wire inherited from the vortex core profile (see text).

Figure 2

Numerical results for a single dissipative vortex with $\ell =1$ on a square lattice of $35\times 35$ sites with unit spacing and $\kappa =1$. Left: Typical form of the MZMs localized in the vortex core and on the edge, respectively (here for $\beta =3$), and localization length scale $\xi$ associated with the MZM trapped in the core. Right: Low-lying part of the damping and purity spectra for $\beta =2$, both featuring a gap with two zero eigenvalues. All results were obtained for a vortex profile $f(r)=(r/d)e^{-(r/\xi {)}^2}$ with $d=10$ and $\xi =20$.