From fractional solitons to Majorana fermions in a paradigmatic model of topological superconductivity

Majorana bound states are interesting candidates for applications in topological quantum computation. Low-energy models allowing one to grasp their properties are hence conceptually important. The usual scenario in these models is that two relevant gapped phases, separated by a gapless point, exist. In one of the phases, topological boundary states are absent, while the other one supports Majorana bound states. We show that a customary model violates this paradigm. The phase that should not host Majorana fermions supports a fractional soliton exponentially localized at only one end. By varying the parameters of the model, we describe analytically the transition between the fractional soliton and two Majorana fermions. Moreover, we provide a possible physical implementation of the model. We further characterize the symmetry of the superconducting pairing, showing that the odd-frequency component is intimately related to the spatial profile of the Majorana wave functions.

Figure 1

(a) The dispersion ${\epsilon }_{1/2}^{\left(P\right)}\left(k\right)$ in units of $v_F/L$, as a function of $k$, in units $1/L$, for $\mathrm{\Delta }=2v_F/L$ and $B=0.5v_F/L$. (b) The dispersion ${\epsilon }_{1/2}^{\left(P\right)}\left(k\right)$ in units of $v_F/L$ as a function of $k$ in units $1/L$, for $\mathrm{\Delta }=2v_F/L$ and $B=2v_F/L$.

Figure 2

Contour plot of ${\chi }_0\left(x\right)$, in units $L^{-1/2}$, as a function of $B$ in units $v_F/L$, and of $x$, in units $L$, for $\mathrm{\Delta }=7v_F/L$. The central white line corresponds to the gapless point.

Figure 3

The quantum spin Hall analogy of the model. The arrows in the central region indicate right- and left-moving particles, the arrows in the side blocks the magnetization of the barriers, and $B$ and $\mathrm{\Delta }$ the applied magnetic field and the superconducting pairing.