Impurity probe of topological superfluids in one-dimensional spin-orbit-coupled atomic Fermi gases

We investigate theoretically nonmagnetic impurity scattering in a one-dimensional atomic topological superfluid in harmonic traps by solving self-consistently the microscopic Bogoliubov-de Gennes equation. In sharp contrast to topologically trivial Bardeen-Cooper-Schrieffer $s$-wave superfluid, topological superfluid can host a midgap state that is bound to localized nonmagnetic impurity. For strong impurity scattering, the bound state becomes universal, with nearly zero energy and a wave function that closely follows the symmetry of that of Majorana fermions. We propose that the observation of such a universal bound state could be useful evidence for characterizing the topological nature of topological superfluids. Our prediction is applicable to an ultracold, resonantly interacting Fermi gas of ${}^{40}$K atoms with spin-orbit coupling confined in a two-dimensional optical lattice.

Figure 1

Phase diagram at $T=0$ (solid line) and $T=0.3T_F$ (gray dashed line) determined from the behavior of the lowest energy in the quasiparticle spectrum. With increasing the effective Zeeman field, the Fermi cloud changes from a standard BCS superfluid to a topologically nontrivial superfluid. The phase-transition point is slightly affected by finite temperature. The inset shows the energy spectrum at $h=1.2E_F$ as a function of the position of quasiparticles. A zero-energy quasiparticle (i.e., Majorana fermion) at the trap edge has been highlighted by a big dark circle. Here, we characterize approximately the position of a quasiparticle by using its wave function: $\langle x^2\rangle =\int dx{x}^2{\sum }_{\sigma }[u_{\sigma }^2\left(x\right)+{\nu }_{\sigma }^2\left(x\right)]$.

Figure 2

(a) The pairing gap distribution function $\Delta \left(x\right)$ (dot-dashed line) and the criterion for a local topological phase $h>h_c\left(x\right)$ (solid line) at $h=1.2E_F$. The shaded crosshatching highlights the topologically nontrivial area. (b) The linear contour plot of the local density of state at $h=1.2E_F$. At each trap edge, a series of edge states, including the zero-energy Majorana fermion mode, is clearly visible.

Figure 3

Density profile and pairing gap distribution in the presence of a strong attractive nonmagnetic impurity with scattering potential strength $V_{\mathrm{imp}}=-0.30x_F{E}_F$. The inset shows the spatial distribution of the Bogoliubov quasiparticle energy spectrum. The midgap bound state near the impurity site $x=0$ is highlighted by big blue circles.

Figure 4

(a) Density of state for a topological superfluid ($h=1.2E_F$) at $x=0$, $0.05x_F$, and $0.1x_F$ (from bottom to top). For better illustration, the curves have been offset. The magnitude of the local density of state at $x=0$ and $0.1x_F$ has been enlarged by a factor of 10. (b) Linear contour plot of the local density of state. The impurity-induced bound state is clearly visible near $x=0$ and $\omega =0$.

Figure 5

The dependence of the bound-state energy on the impurity strength for a topological superfluid at $h=1.2E_F$. The solid and empty circles show the results for attractive and repulsive potential scattering, respectively. The dashed lines gives the bound-state energy at the infinitely large impurity strength, $E\simeq \pm 0.113{\Delta }_0$, obtained by an extrapolation. Here, ${\Delta }_0\simeq 0.464E_F$ is the pairing gap at the trap center without impurity.

Figure 6

Wave function of the universal bound state with energy $E\simeq \pm 0.113{\Delta }_0$ for a topological superfluid at $h=1.2E_F$. The wave function at $\pm E$ may be regarded as the bond and antibond superposition of two Majorana wave functions, which satisfy, respectively, the symmetry $u_{\sigma }\left(x\right)={\nu }_{\sigma }^{*}\left(x\right)$ (on the left side with $x<0$) and $u_{\sigma }\left(x\right)=-{\nu }_{\sigma }^{*}\left(x\right)$ (on the right side with $x>0$).

Figure 7

Linear contour plot of the density of state at $h=1.2E_F$ for an attractive or a repulsive Gaussian-shaped impurity scattering potential. Here, we take $d=0.1x_F$ and $V_{\mathrm{imp}}=\pm 0.30x_F{E}_F$.

Figure 8

The dependence of the pairing gap function on the high-energy cutoff $E_c$. Here, we consider a spin-orbit-coupled Fermi gas of $N=100$ atoms in harmonic traps at zero temperature. The dimensionless interaction parameter is $\gamma =\pi$. The spin-orbit coupling is taken to be $\lambda k_F/E_F=1$ and $h=1.2E_F$. At these typical parameters, the results become independent of the cutoff energy $E_c$ once it is larger than $6E_F$.