Soliton-induced Majorana fermions in a one-dimensional atomic topological superfluid

We theoretically investigate the behavior of dark solitons in a one-dimensional spin-orbit coupled atomic Fermi gas in harmonic traps by solving self-consistently the Bogoliubov–de Gennes equations. The dark soliton—to be created by phase imprinting in future experiments—is characterized by a real order parameter, which changes sign at a point node and hosts localized Andreev bound states near the node. By considering both cases of a single soliton and multiple solitons, we find that the energy of these bound states decreases to zero when the system is tuned to enter the topological superfluid phase by increasing an external Zeeman field. As a result, two Majorana fermions emerge in the vicinity of each soliton, in addition to the well-known Majorana fermions at the trap edges associated with the nontrivial topology of the superfluid. We propose that the soliton-induced Majorana fermions can be directly observed by using spatially resolved radio-frequency spectroscopy or indirectly probed by measuring the density profile at the point node. For the latter, the deep minimum in the density profile will disappear due to the occupation of the soliton-induced zero-energy Majorana fermion modes. Our prediction could be tested in a resonantly interacting spin-orbit coupled ${}^{40}\text{K}$ Fermi gas confined in a two-dimensional optical lattice.

Figure 1

Zero-temperature phase diagram without solitons at $\gamma =2.2$ and $\lambda =1.5E_F/k_F$, determined from the behavior of the two lowest-energy particle solutions (with $E_{\eta }\ge 0$) in the Bogoliubov quasiparticle spectrum (shown by the solid line and empty squares, respectively). The topological phase transition occurs at about $h\simeq 0.59E_F$. As a result, two zero-energy Majorana fermions emerge at the trap edges, as shown in the inset, plotting the local density of states for the case $h=0.8E_F$. They are highlighted by two gray circles.

Figure 2

(a) The solitonic pairing gap $\Delta \left(x\right)$ at zero Zeeman field $h=0$ and at different interaction strengths $\gamma$ and spin-orbit coupling strengths $\lambda$, when a $\pi$-phase jump is imprinted at $x_1=0$. (b) The solitonic $\Delta \left(x\right)$ at nonzero Zeeman fields [$h/E_F=0.4$ (thick solid line), $0.56$ (crosses), $0.58$ (daggers), or $0.8$ (dashed line)] and at $\gamma =2.2$ and $\lambda =1.5E_F/k_F$. For the case of $h=0.4E_F$, we also consider a soliton at $x_1=-0.5x_F$ and plot the result with a thin solid line.

Figure 3

(a) The energies of the three lowest-energy particle solutions with $E_{\eta }\ge 0$, as a function of the Zeeman field in the case of a single dark soliton at $x_1=0$. The solid and empty circles show Andreev bound states localized near the soliton, while the empty squares correspond to the lowest-energy particle state in the bulk. The vertical dashed line separates the BCS and topological superfluid phases. Inset: The minigap at $h=0$ and at different interaction parameters $\gamma$ and spin-orbit coupling strengths [$\lambda k_F/E_F=1.5$ (circles), $1.0$ (squares), and $0.0$ (daggers)], as a function of the square of the local pairing gap, ${\Delta }^2/{\epsilon }_F^2$. The dashed line is the linear fit $E_{\mathrm{minigap}}=0.26{\Delta }^2/{\epsilon }_F$. (b),(c) The spatial distribution of the Bogoliubov quasiparticle energy spectrum at $h=0.4E_F$ and $h=0.8E_F$, respectively. The Andreev bound states near the soliton are highlighted by empty and solid circles. Here, we approximately characterize the location of a quasiparticle by using its wave function, $\sqrt{\langle x^2\rangle }={\{\int dx{x}^2{\sum }_{\sigma }[u_{\sigma }^2\left(x\right)+{\nu }_{\sigma }^2\left(x\right)]\}}^{1/2}$, if the wave function is well localized at a certain point. Otherwise, $\sqrt{\langle x^2\rangle }$ characterizes the width of the wave function.

Figure 4

The wave functions of the lowest-energy Andreev bound states in the BCS superfluid phase at (a) $h=0.4E_F$ or in the topologically nontrivial superfluid phase at (b),(c) $h=0.8E_F$. In the latter case, the wave functions strongly interfere with that of Majorana fermions at the trap edges. The superposition persists when we place the dark soliton in a less symmetric position $x_1=-0.1x_F$.

Figure 5

Local density of states of a BCS superfluid at (a) $h=0.4E_F$ and of a topologically nontrivial superfluid at (b) $h=0.8E_F$, when a single dark soliton is created at $x_1=0$. In the topological phase, the two Majorana fermions near the point node of the dark soliton (at $x_1=0$) are not distinguishable because of the superposition of the wave functions. The color map indicates the magnitude of the local density of states in units of $n_F/E_F$.

Figure 6

The total density profile in the presence of a single dark soliton at $x_1=0$. The solid and dashed lines show the results at $h=0.4E_F$ and $h=0.8E_F$, respectively. Bottom inset: An enlarged view around the origin $x=0$, with additional results near the topological transition at $h=0.56E_F$ (crosses) and at $h=0.58E_F$ (daggers). Top inset: The oscillation amplitude $\Delta n$ of the density profile near the origin, $\Delta n=n_{\max }-n_{\min }$, where $n_{\max }$ and $n_{\min }$ are the maximum and minimum densities, respectively.

Figure 7

The (a) solitonic order parameter and (b) total density distribution in the presence of two dark solitons (placed at $x_1=-0.5x_F$ and $x_2=+0.5x_F$), at two Zeeman fields: $h=0.4E_F$ (solid line) and $h=0.8E_F$ (dashed line).

Figure 8

(a)–(c) The wave functions of three Majorana fermions of a topological superfluid at $h=0.8E_F$, in the presence of two dark solitons at $x_1=-0.5x_F$ and $x_2=+0.5x_F$. The energy of each Majorana fermion is indicated. (d) The corresponding local density of state within the bulk energy gap. The color indicates the magnitude of the local density of states; see, for example, Fig. 5.