Majorana bound states in open quasi-one-dimensional and two-dimensional systems with transverse Rashba coupling

We study the formation of Majorana states in quasi-one-dimensional (quasi-1D) and two-dimensional square lattices with open boundary conditions, with general anisotropic Rashba coupling, in the presence of an applied Zeeman field and in the proximity of a superconductor. For systems in which the length of the system is very large (quasi-1D) we calculate analytically the exact topological invariant, and we find a rich corresponding phase diagram which is strongly dependent on the width of the system. We compare our results with previous results based on a few-band approximation. We also investigate numerically open two-dimensional systems of finite length in both directions. We use the recently introduced generalized Majorana polarization, which can locally evaluate the Majorana character of a given state. We find that the formation of Majoranas depends strongly on the geometry of the system: for a very elongated wire the finite-size numerical phase diagram reproduces the analytical phase diagram for infinite systems, while if the length and the width are comparable, no Majorana states can form; however, one can show the formation of “quasi-Majorana” states that have a local Majorana character but no global Majorana symmetry.

Figure 1

Bulk topological phase diagram for (a) a strictly 1D single-channel wire and (b) a square lattice, where $\mathrm{\Delta }=0.4t$. White is the topologically trivial phase; light red is the bulk topologically nontrivial phase where a pair of Majorana edge states forms at the boundaries. The black lines show where the bulk gap closes.

Figure 2

Topological phase diagram as a function of $B$ and $\mu$ for a ladders with (a) $N_y=2$, (b) $N_y=3$, (c) $N_y=15$, and (d) $N_y=50$ with $\mathrm{\Delta }=0.4$. Light red is the topologically nontrivial phase, and white is topologically trivial. For (a) and (c) ${\alpha }_x={\alpha }_y=0.2$, and for (b) and (d) ${\alpha }_x={\alpha }_y=0.5$. For (a) and (b) gap closing lines belonging to the two different TRI momenta are shown as dashed and solid lines. In (c) and (d) the gap closing lines for the 2D bulk system are shown for comparison.

Figure 3

The phase diagram as a function of $B$ and $\mu$ for open systems of (a) $3\times 201$, (b) $101\times 101$, and (c) $51\times 201$. This has been calculated using the PH expectation value of the lowest-energy state summed over half of the system $C$ (see main text). In all the examples $\mathrm{\Delta }=0.4t$ and $\alpha =0.5t$. The analytical topological phase boundaries for quasi-1D (solid lines) and quasi-2D (dashed lines) are also plotted.

Figure 4

MP for the lowest-energy states for an open system of $51\times 201$ with $\mathrm{\Delta }=0.3t$ and $\alpha =0.5t$. We plot (a) a MBS $\mu =3.5t$ and $B=2.2t$ [corresponding to the red phase in Fig. 3], (b) a 1D edge state $\mu =3.5t$ and $B=2.3t$ [blue phase in Fig. 3], (c) a 2D edge state $\mu =3.5t$ and $B=t$ [yellow phase in Fig. 3], and (d) a bulk state $\mu =0.5t$ and $B=5t$ [blue phase in Fig. 3].

Figure 5

MP for lowest-energy state for an open system of $101\times 101$ with $\mathrm{\Delta }=0.3t$ and $\alpha =0.5t$. (a) $\mu =4t$ and $B=t$, and (b) $\mu =2t$ and $B=3t$, both in the 2D topologically nontrivial phase.

Figure 6

The local density of states at the edge of quasi-1D wires as a function of energy and $B$ with $\mathrm{\Delta }=0.4t,\mu =4t$, and $\alpha =0.5t$. (a) For $N_y=15$ and (b) for $N_y=51$; in all cases $N_x=201$.

Figure 7

A comparison of the exact and effective phase diagrams as a function of $B$ and $\mu$. We take $\mathrm{\Delta }=0.25{\ell }_{n^{*}}$ and $\alpha =0.1{\ell }_{n^{*}}$. (a) and (b) A comparison of the full and multiband effective topological phase diagrams for $N_y=50$ and $n^{*}=6,8$ respectively. The white (topologically trivial) and dark purple (topologically nontrivial) phases denote the regions where both calculations agree. The pink regions are predicted to be topological by the exact calculation and nontopological by the multiband approximation, while the light purple regions are predicted to be nontopological by the exact calculation and topological by the multiband approximation. (c) A comparison between the phase diagrams obtained by the multiband approximation with $n^{*}=10$ for systems with $N_y=100$ and $N_y=101$. The small pink regions are the extratopological regions corresponding to $N_y=101$. (d) The exact topological phase diagram for $N_y=100$ with the region where the multiband theory is working [corresponding to the results plotted in (c)] boxed by the small rectangle .

Figure 8

Topological phase diagram as a function of $B$ and $\mu$ for (a) and (c) $N_y=3$ and (b) and (d) $N_y=15$, with transverse ${\alpha }_y=0$ and $\mathrm{\Delta }=0.4$. For (a) and (b) light red is topologically nontrivial, and white is topologically trivial, where we have plotted the parity of the $\mathbb{Z}$ invariant which is a ${\mathbb{Z}}_2$ invariant. (c) and (d) show the number of MBS pairs $M$.

Figure 9

Number of edge-state pairs $N$ as a function of $B$ and $\mu$ for a system with $N_y=4,\mathrm{\Delta }=0.4$ and (a) ${\alpha }_y=0$ and (b) ${\alpha }_y=0.3$. For ${\alpha }_y=0$ many of these states are the protected MBS, which become gapped for ${\alpha }_y\ne 0$.

Figure 10

Topological phase diagram as a function of $B$ and $\mu$ for systems with $t_x=t,t_y=0.2t,{\alpha }_{x,y}=0.5t_{x,y}$, and $\mathrm{\Delta }=0.4t$. Light red is topologically nontrivial, and white is topologically trivial. Plotted are phase diagrams for (a) 15 wires and (b) 16 wires.

Figure 11

Topological phase diagram as a function of $B$ and $\mu$ for systems with $t_x=t,t_y=20t,{\alpha }_{x,y}=0.5t_{x,y}$, and $\mathrm{\Delta }=0.4t$. Light red is topologically nontrivial, and white is topologically trivial. Plotted are phase diagrams for (a) 15 wires and (b) 16 wires.

Figure 12

Topological phase diagram as a function of $B$ and $\mu$ for a system with three wires with ${\alpha }_x={\alpha }_y=0.5$ and $\mathrm{\Delta }=0.4$. Light red is topologically nontrivial, and white is topologically trivial. Plotted are phase diagrams for (a) $t^{\prime }=0.3$, (b) $t^{\prime }=0.7$, and (c) $t^{\prime }=1$ (the fully periodic nanotube). Note these are all 1D topological phase diagrams.

Figure 13

Topological phase diagram as a function of $B$ and $\mu$ for a system with six wires with ${\alpha }_x={\alpha }_y=0.5$ and $\mathrm{\Delta }=0.4$. Light red is topologically nontrivial, and white is topologically trivial. The coupling $t^{\prime }$ has the same values as in Fig. 12.

Figure 14

MP for a system with six wires with ${\alpha }_x={\alpha }_y=0.5$ and $\mathrm{\Delta }=0.4$. We show the transition from the MBS in an open system to a non-MBS in the periodic system. The state is destroyed by leaking along the weakly coupled edges. The coupling is $t^{\prime }=0,0.4t_y,t_y$ for (a), (b), and (c) respectively. The system size is $N_x=60$ and $N_y=6$ (see Fig. 13).