Majorana Corner Modes with Solitons in an Attractive Hubbard-Hofstadter Model of Cold Atom Optical Lattices

Higher-order topological superconductors hosting Majorana-Kramers pairs (MKPs) as corner modes have recently been proposed in a two-dimensional quantum spin Hall insulator proximity-coupled to unconventional cuprate or iron-based superconductors. Here, we show that such MKPs can be realized using a conventional $s$-wave superfluid with a soliton in cold atom systems governed by the Hubbard-Hofstadter model. The MKPs emerge in the presence of interaction at the “corners” defined by the intersections of line solitons and the one-dimensional edges of the system. Our scheme is based on the recently realized cold atom Hubbard-Hofstadter lattice and will pave the way for observing possible higher-order topological superfluidity with conventional $s$-wave superfluids or superconductors.

Figure 1

Band structure of the noninteracting two-component Hofstadter model $H_0$, with $\alpha =1/3$ and periodic boundary conditions in the $y$ direction. (a) System with $N_x=92$ and hard confinement in the $x$ direction. (b) System with $N_x=92$ and soft (Gaussian) confinement (see Supplemental Material [75]). The bulk states are shown in blue, and the red curves represent the gapless edge modes.

Figure 2

Left: The mean-field phase diagram obtained by plotting the self-consistent value of the pairing order parameter $\mathrm{\Delta }$ for $k_{B}T={10}^{-4}t$. The dashed white line indicates the phase boundary. Center: Chemical potential as a function of the filling factor for $\mathrm{\Delta }=0$. Right: Mean-field $\mathrm{\Delta }$ as a function of the interaction strength for two different filling factors. The $n_0=1$ line shows $\mathrm{\Delta }\ne 0$ (i.e., superfluid phase) all the way to $U\sim 0$, while for $n_0=2/3$ one needs $U\sim 3t$ to enter the superfluid phase. The green dotted lines mark the band edges of the bulk spectrum in Fig. 1.

Figure 3

Position-dependent pairing potential $\mathrm{\Delta }(x,y)$ for a strongly interacting system with $U=3.5t$, i.e., in the SF phase. The pairing potential is obtained as the self-consistent solution of the mean-field equations (Eqs. S7–S10) for a finite system with $N_x\times N_y=50\times 34$ and soft confinement (see Supplemental Material [75]) at finite temperature $k_{B}T=0.01t$. The total number of particles is fixed: $N=800$. Top: Self-consistent solution with a constant phase. The (self-consistent) chemical potential is $\mu =-1.250t$. Bottom: Self-consistent solution with a line soliton. The chemical potential is $\mu =-1.248t$. Note that $\mathrm{\Delta }(x,y)$ is nonzero in the bulk—consistent with the phase diagram in Fig. 2—as well as on the boundary of the system, except along the line soliton.

Figure 4

Top: Low-energy spectrum of the Hofstadter-Hubbard model with a strong interaction ($U=3.5t$) within the mean-field approximation for a system with constant phase pairing potential and parameters corresponding to Fig. 3 (top). Bottom: The same but for parameters corresponding to the bottom panel in Fig. 3. Note that in the presence of a line soliton the system hosts two pairs of zero-energy Majorana bound states (red dots). The insets plot the wave functions of the states marked by arrows.