Phase-tunable second-order topological superconductor

Two-dimensional second-order topological superconductors (SOTSCs) have gapped bulk and edge states, with zero-energy Majorana bound states localized at corners. Motivated by recent advances in Majorana nanowire experiments, we propose to realize a tunable SOTSC as a two-dimensional nanowire array. We show that the coupling between the Majorana modes of adjacent wires can be controlled by phase-biasing the device, allowing one to access a variety of topological phases. We characterize the system using scattering theory, which provides access to its transport properties and its topological invariants. The setup is robust against disorder, both in the nanowires themselves and in the Josephson junctions formed between adjacent wires. Further, we identify a parameter regime in which an initially trivial system is rendered topological upon adding disorder, providing an example of a second-order topological Anderson phase.

Figure 1

System composed of two coupled wires with opposite superconducting phases. Each nanowire (yellow) is covered by a superconductor (blue), and hosts a pair of MBSs (red). The thick black lines indicate the coupling between the wires.

Figure 2

Spectrum of the two-wire system as a function of the phase difference between wires. We color the bulk modes in green and edge modes in blue (and red, if the mirror symmetry is broken). In (a)–(c), the Zeeman field points in the $x$ direction; (a) $t_y\ne 0$ and $\beta =0$, (b) $t_y=0$ and $\beta \ne 0$, and (c) $t_y\ne 0$ and $\beta \ne 0$. (d)–(f) The Zeeman field points in the $z$ direction; (d) $t_y\ne 0$ and $\beta =0$, (e) $t_y=0$ and $\beta \ne 0$, and (f) $t_y\ne 0$ and $\beta \ne 0$. In all cases, we consider two wires that are 80 sites long. For panels (a)–(e), the mirror-symmetric boundary modes are represented with blue, whereas for panel (f), this symmetry is broken and the Majorana modes localized on the right (left) boundary of the system are shown in red (blue), respectively.

Figure 3

Array of nanowires realizing a SOTSC. Panels (a) and (b) show the model given by the Hamiltonian equations (3) and (4), whereas panels (c) and (d) show the setup corresponding to Eqs. (3) and (5). The Zeeman field points in the $x$ direction for panels (a) and (c), whereas it points in the $z$ direction for panels (b) and (d). In all cases, superconducting bridges (blue) connect pairs of wires, leading to a vanishing phase difference between them. Phase-biasing the device such that unconnected superconductors have a small, positive phase difference (for instance, by means of a supercurrent $I_{\mathrm{in}/\mathrm{out}}$) will then only alter the MBS coupling between disconnected pairs of wires. The resulting dimerization pattern (solid and dashed lines) drives the system into topologically different phases. These include a trivial phase (a), a nontrivial SOTSC with four Majorana corner modes [red, panel (c)], as well as SOTSC phases with two corner modes, on the left (b) or right (d) edge. Reversing the direction of the supercurrent (blue arrows) causes topological phase transitions, changing the corner mode distribution from that of panel (a) to that of panel (c), and from panel (b) to panel (d), respectively. Similarly, rotating the direction of the Zeeman field amounts to interchanging (a) with (b) and (c) with (d).

Figure 4

(a), (b)/(c), (d) The spectrum and the spatial distribution of four/two gapless modes for a system consisting of $40\times 40$ sites, with the Zeeman field in the $x$ /$z$ direction. In both cases, the phase difference is $\varphi =4.5$.

Figure 5

(a) The topological invariant (blue) and the thermal conductance (red) as functions of the superconducting phase difference, for a system size of $80\times 80$ sites. (b) The Andreev conductance in a SOTSC phase, once $\varphi =4.5$. (c) The Andreev conductance in a trivial phase, once $\varphi =1.5$. All calculations are done for the Zeeman field in the $x$ direction.

Figure 6

The topological invariant (blue) and the thermal conductance (red) as functions of the superconducting phase, for the system size $80\times 80$ sites and $V_z\ne 0$. Panels (a)/(b) are obtained using left/right leads.

Figure 7

(a)–(d) Phase diagrams, obtained as a function of phase difference (horizontal axis) and disorder strength (vertical axis), either in the chemical potential $\mu$ [(a), (b)] or in the phases $\varphi$ [(c), (d)]. In the top panels, the color scale indicates the topological invariant, whereas in the bottom panels it denotes the thermal conductance. Panels (a) and (c) correspond to the system described by Eqs. (3) and (4), while (b) and (d) correspond to Eqs. (3) and (5). These phase diagrams are obtained by averaging topological invariants obtained from 301 disorder realizations. (e)–(g) The averaged value of $x$-edge/$y$-edge/bulk conductance, respectively, for the disorder in the chemical potential $\mu$. (h) The averaged $y$-edge conductance, corresponding to panels (c) and (d). In all calculations, the system size is $80\times 80$ sites.

Figure 8

(a) To compare analytics and numerics, we plot the absolute value of the first component of a Majorana wave function localized around $x=0$. The analytical solution [Eq. (A11)] is represented by a red line while the numerical wave function, calculated using Kwant, is given in blue. For the numerical solution, we consider an 800-site-long wire with the lattice constant $a=1$ and the hopping strength $t_x=10$. Other parameters are $m=-1/20,\mu =0,\alpha =9,\mathrm{\Delta }=1,\varphi =0$, and $V_z=2$. (b) For $t_y=0.08$ and $\beta =0$, the analytical solution $t_{y}sin(\varphi /2)$ given with a blue line matches a numerical calculation, represented with black points. (c) For $\beta =0.08$ and $t_y=0$, the analytical solution $\beta \mathrm{\Delta }cos(\varphi /2)/V_z$ plotted in blue matches a numerical simulation, represented with black points. Intrawire parameters used in (b) and (c) are $\mu =0,t_x=1.7,\alpha =2.5,\mathrm{\Delta }=2.5,$ and $V_z=4$.

Figure 9

(a)–(d) Phase diagrams, once there is disorder in the chemical potential $\mu$ [(a), (b)] or in the phases $\varphi$ [(c), (d)]. Panels (a) and (c) are obtained from reflection matrices of the left leads, while (b) and (d) are recovered from reflection matrices of the right leads. These phase diagrams are obtained by averaging topological invariants obtained from 301 disorder realizations. (e), (f) The averaged value of $x$-/$y$-edge conductance, respectively, for disorder in the chemical potential $\mu$ and calculated from the left leads. (g) The bulk conductance calculated using periodic boundary conditions in the $y$ direction. (h) The averaged $y$-edge conductance, corresponding to panel (d). In all calculations, the system size is $80\times 80$ sites for the parameters defined in the main text.