Detecting the Majorana Fermion Surface State of ${}^3\mathrm{He}\mathrm{\text{−}}B$ through Spin Relaxation

The Majorana fermion, which can be useful for topological quantum computation, has eluded detection. ${}^3\mathrm{He}\mathrm{\text{−}}B$, recently shown to be a time-reversal invariant topological superfluid, has a gapless Majorana fermion surface state. We show here that an electron spin relaxation experiment can detect this surface state—its Majorana nature through the Zeeman field direction dependence of the relaxation time $1/T_1\propto {sin}^2\theta$, where $\theta$ is the angle between the field and the surface normal. We propose an experimental setup where an electron inside a nanobubble is injected below the ${}^3\mathrm{He}$ liquid surface.

Figure 1

Illustration of the surface state of the ${}^3\mathrm{He}\mathrm{\text{−}}B$ phase consisting of a single Majorana cone, where the $E<0$ part of the quasiparticle spectrum (with the dashed boundary) is redundant. Also shown are the dimensions of the bubble electron when we apply a perpendicular electric field of $150\mathrm{V}/\mathrm{cm}$. Note how small the size and depth of the bubble are compared to the depth $\xi$ of the surface state, for which we take the weak coupling approximation $\hslash v_F/\Delta$ as in Eq. (3).

Figure 2

The electron spin relaxation rate (in ${10}^{-3}\mathrm{Hz}$) due to the surface state through dipole-dipole interaction for the magnetic field applied parallel to the surface. From the top to the bottom curve, the applied perpendicular electric field is $150\mathrm{V}/\mathrm{cm}$, $10\mathrm{V}/\mathrm{cm}$, and $1.5\mathrm{V}/\mathrm{cm}$ respectively, which gives us the bubble depth $|b|=22.5\mathrm{nm}$, 87.4 nm, and 225.2 nm, respectively (for consistency with Eq. (3) the depth of the surface state is set $\xi =\hslash v_F/\Delta 237\mathrm{nm}$).