Majorana fermions in superconducting helical magnets

In a variety of rare-earth-based compounds, singlet superconductivity coexists with helical magnetism. Here we demonstrate that surfaces of these systems should generically host a finite density of zero-energy Majorana modes. In the limit of vanishing disorder, these modes lead to a divergent contribution to zero-energy density of states and to zero-temperature entropy proportional to the sample surface area. When confined to a wire geometry, a discrete number of Majorana modes can be isolated. The relatively large characteristic energy scales for superconductivity and magnetism, compared to other proposals, as well as the lack of need for fine tuning, make helical magnetic superconducting compounds favorable for the observation and experimental investigation of Majorana fermions.

Figure 1

2D cuts at $p_y=0$ through the Fermi surfaces of a helical magnet at different energies: (a) for the case $J<K/2$ and (b) for $J>K/2$. The outer/inner (blue/red) line corresponds to the lower/upper energy band. (c) The arrows indicate the spin polarizations of momentum eigenstates on the Fermi surfaces. The dashed lines parallel to the $p_z$ axis can have either two (upper line) or four (lower line) crossings with the Fermi surfaces. The former leads to undoubled Majorana surface modes, while the latter does not. Panel (d) illustrates the case of arbitrary orientation of the helical axis, and thus also of the $p$-wave superconductivity, relative to the surface normal ($\widehat{z}$). The undoubled Majorana modes exist in the shaded interval of momenta parallel to the surface: There, both the order parameter and the $\widehat{z}$ component of Fermi velocity change sign upon reflection from the interface.

Figure 2

Results of numerical simulations of model Eq. (1) on a square lattice with nearest neighbor hopping. The system is periodic in the $\widehat{x}$ direction and has open boundaries in the $\widehat{z}$ direction. The size of the system in the $\widehat{z}$ direction is 150 sites. The helical wavevector $K$ is either along $\widehat{z}$ (left column) or along $\widehat{x}$ (right column). The horizontal axes are the momentum along the edge (the lattice constant is the unit of length); the vertical axes are the energy in units of intersite hopping. Case (a) corresponds to the intermediate strength superconductivity, $\Delta /J=0.4$ [$J,K$ parameters from Fig. 1a, the third pair of Fermi surfaces from the top]. The range of momenta $p_x>0$ where unpaired Majorana modes are expected from the small-$\Delta$ consideration is shown by vertical yellow lines. For $K\left|\right|x$ there are no Majorana modes, as expected, since the sign of the order parameter does not change under reflection from the interface. The near-zero energy modes at $p_x=0$ are due to the gapless superconductivity in the bulk. In case (b) the amplitude of superconducting pairing has overcome the band splitting induced by magnetic ordering. A full superconducting gap opens and there are no zero-energy modes for either orientation of $K$ relative to the sample boundary. Case (c) corresponds to weak superconductivity, $\Delta /J=0.2$, for the parameters in Fig. 1b (the top pair of Fermi surfaces). The Majorana modes exist in the expected range of momenta $p_x$—marked by vertical (yellow) lines—as discussed in the main text.

Figure 3

Results of numerical simulations of model Eq. (1) of the main text on a square lattice with nearest neighbor hopping. The parameters are the same as in Fig. 2a: $J=0.25,K=1,\mu =-3.5,\Delta =0.1$. The system is periodic in the $\widehat{x}$ direction and has open boundaries in the $\widehat{z}$ direction. The horizontal axes are the momentum along the edge (the lattice constant is the unit of length); the vertical axes are the energy in units of intersite hopping. The size of the system in the $\widehat{z}$ direction is 150 sites. Different plots correspond to various angles $\theta$ between the helical wavevector $K$ and the $\widehat{z}$ axis ($K$ is assumed to be lying in the $\widehat{x}$-$\widehat{z}$ plane). The horizontal axes are the momentum along the edge (the lattice constant is the unit of length); the vertical axes are the energy in units of intersite hopping. Plots for $\theta =0$ and $\theta =\pi /2$ are the same as in Fig. 2a. For intermediate angles we see that the range of momenta where zero-energy Majorana modes are present changes gradually.