Anderson localization of Cooper pairs and Majorana fermions in an ultracold atomic Fermi gas with synthetic spin-orbit coupling

We theoretically investigate two-particle and ground-state many-particle Anderson localizations of a spin-orbit coupled ultracold atomic Fermi gas trapped in a quasiperiodic potential and subjected to an out-of-plane Zeeman field. We solve exactly the two-particle problem in a finite length system by exact diagonalization and solve approximately the ground-state many-particle problem within the mean-field Bogoliubov-de Gennes approach. At a small Zeeman field, the localization properties of the system are similar to that of a Fermi gas with conventional $s$-wave interactions. As the disorder strength increases, the two-particle binding energy increases and the fermionic superfluidity of many particles disappears above a threshold. At a large Zeeman field, where the interatomic interaction behaves effectively like a $p$-wave interaction, the binding energy decreases with increasing disorder strength, and the resulting topological superfluidity shows a much more robust stability against disorder than the conventional $s$-wave superfluidity. We also analyze the localization properties of the emergent Majorana fermions in the topological phase. Our results could be experimentally examined in future cold-atom experiments, where the spin-orbit coupling can be induced artificially by using two Raman lasers, and the quasiperiodic potential can be created by using bichromatic optical lattices.

Figure 1

Two possible phase diagrams for a spin-orbit coupled Fermi gas in a disordered potential. In the first-type diagram (a), as the disorder strength increases, a conventional $s$-wave superfluid turns into a topological superfluid before losing its superfluidity. The transition from a topological superfluid to a gapped insulator is direct. In contrast, in the second-type diagram (b), with increasing disorder strength a topological superfluid first becomes a conventional superfluid and then translates into a gapped insulator.

Figure 2

(a) Anderson localization of Cooper pairs, indicated by the inverse participation ratio (IPR) of the ground-state two-particle wave function at $L=F_{l=10}=89$. The black circles with line shows the phase boundary in the rational limit $L\to \infty$. (b) Binding energy of the ground-state Cooper pair as a function of the disorder strength at several Zeeman fields. Here we set $U=4.5t$ and $\lambda =2t$.

Figure 3

Left panel: Evolution of the superfluid stiffness as a function of the disorder strength (a) or Zeeman field strength (c). Right panel: The lowest or the second lowest quasiparticle excitation energy as a function of the disorder strength (b) or Zeeman field strength (d), under an open boundary condition. Here we take $L=89$ and set $U=4.5t$ and $\lambda =2t$. Moreover, we use an average (quarter) filling factor $n=N/L=44/89$.

Figure 4

Phase diagram of a 1D spin-orbit coupled Fermi gas in bichromatic optical lattices at $U=4.5t$ and $\lambda =2t$. The color bar indicates the superfluid stiffness. A vanishingly small superfluid stiffness (i.e., the red dotted line) determines the phase transition to a normal state ($V_c$), due to the Anderson localization of the ground state of many particles. The blue squares with dashed line ($V_c^{*}$) enclose the phase space, where the superfluid is topologically nontrivial and Majorana fermions exist at the two open edges. We use an average filling factor $n=N/L=44/89$, as in Fig. 3, except for the four points (green empty squares), which show the critical disorder strength $V_c$ at a smaller filling factor $n=22/89$.

Figure 5

Upper panel: Evolution of the three lowest quasiparticle excitation energies as a function of the disorder strength, under an open boundary condition and in the presence of a soliton at the center of the system. Lower panel: The wave functions of the two Majorana fermions ($E_1$ and $E_2$) at the disorder strength $V=0.6t$ (b) and $V=1.2t$ (c). Here we fix the Zeeman field strength to $h_z=1.8t$. The other parameters are the same as in Fig. 3.

Figure 6

The superfluid stiffness (left $y$ axis) and the mean IPR of Bogoliubov quasiparticle wave functions (right $y$ axis) as a function of the disorder strength at $h_z=0.9t$ and $h_z=1.8t$, as indicated. Here we take $L=89$ and set $U=4.5t$ and $\lambda =2t$. The average filling factor is $n=N/L=44/89$.