Josephson junction arrays as a platform for topological phases of matter

Two-dimensional arrays of superconductors separated by normal metallic regions exhibit rich phenomenology and a high degree of controllability. We establish such systems as platforms for topological phases of matter, and in particular chiral topological superconductivity. We propose and theoretically analyze several minimal models for this chiral phase based on commonly available superconductor-semiconductor heterostructures. The topological transitions can be adjusted using a time-reversal-symmetry breaking knob, which can be activated by controlling the phases in the islands, introducing flux through the system, or applying an in-plane exchange field. We demonstrate transport signatures of the chiral topological phase that are unlikely to be mimicked by local nontopological effects. The flexibility and tunability of our platforms, along with the clear-cut experimental fingerprints, make for a viable playground for exploring chiral superconductivity in two dimensions.

Figure 1

Chiral topological superconductivity in a phase-controlled Josephson junction array. (a) The proposed configuration, consisting of a spin-orbit coupled 2DEG (blue) covered by superconducting islands (gray). The islands form a superlattice with the repeating superconducting phases $\{0,0,{\varphi }_1,{\varphi }_2\}$; the unit cell is shown in a black frame. (b) Topological phase diagram of the model as a function of the phases ${\varphi }_1$ and ${\varphi }_2$. Colors indicate the Chern number $\mathcal{C}$, whereas the intensity corresponds to the bulk gap relative to the pairing gap $\mathrm{\Delta }$. The topologically nontrivial states with $\mathcal{C}=\pm 1$ appear only in regions where phase winding occurs, which are marked by solid lines for clarity. Notice that the chirality (the sign of the Chern number) flips when changing the phases from forming a vortex to an antivortex in each unit cell. In the tight-binding model, each island is represented by a single site, and the parameters are $t=1$, $\mathrm{\Delta }=0.5$, $\mu =1.15$, $\eta =0.5$, and $\lambda =0.2\pi$.

Figure 2

Four phase-biased islands. (a) Energy spectrum (lowest 40 levels) as a function of the phase bias $\varphi$, showing many Andreev bound states. When the phase winds ($\varphi >\pi /2$), the lowest Andreev bound states are close to zero energy. (b) Wavefunction of the lowest-energy state at $\varphi =3\pi /4$, showing anisotropic localization at the junction between the four islands. The tight-binding parameters are $t=1$, $\lambda =\pi /5$, $\mu =-3$, $\mathrm{\Delta }=0.3$.

Figure 3

Gapless superconductivity in the phase-controlled Josephson junction array of Fig. 1. Setting ${\varphi }_1=-{\varphi }_2=\varphi$, we calculated the single-particle energy gap (blue) and the pair correlation function (green). For both quantities, we looked for the lowest values across the two-dimensional Brillouin zone. At a certain region (shaded), the single-particle gap vanishes while the pair correlation function remains finite at all momenta. This is the characteristic behavior of a gapless superconductor. (Inset) $E=0$ Fermi surface at $\varphi =3\pi /4$ (marked by a dash line) inside the gapless phase.

Figure 4

Chiral topological superconductivity in a Josephson junction array with applied magnetic flux. (a) Proposed configuration, where each island (square) is described by four sites (circles). The islands are connected to their neighbors by Josephson coupling $t_{\mathrm{J}}$, and the intraisland tunneling is either $t$ (solid lines) or $t^{\prime }$ (dashed lines). The islands' phases (depicted by green, gray, blue, or purple shading) are chosen in accordance with the checkerboard vortex lattice appropriate for $h/4e$ flux per plaquette (vortices are shown as winding arrows). (b) Topological phase diagram as a function of $t/t^{\prime }$, featuring gapped topological (blue), gapped nontopological (gray), and gapless (white) phases. The diagram shows the parity of the Chern number $\mathcal{C}$ (which is $-1$ for the topologically nontrivial phases), multiplied by the bulk energy gap in units of the pairing gap $\mathrm{\Delta }$. The tight-binding parameters $t=1$, $\mathrm{\Delta }=0.1$, $\mu =0.42$, $\lambda =0.4\pi$, and $t_{\mathrm{J}}=t$.

Figure 5

Chiral topological superconductivity in a Josephson junction array with an in-plane exchange field. (a) The proposed configuration, with superconducting islands (gray) deposited on top of a Rashba 2DEG (blue). Several unit cells are indicated in frames, and they repeat themselves in a brick-wall lattice. An in-plane exchange field $B$ is applied at an angle $\theta$ with respect to the horizontal axis $\widehat{x}$. (b) Topological phase diagram as a function of $B$ and $\theta$. Colors indicate the Chern number $\mathcal{C}$, whereas the intensity corresponds to the bulk gap relative to the pairing gap $\mathrm{\Delta }$. We find that $\theta =\frac{\pi }{2},\frac{3\pi }{2}$, i.e., along the elongated islands ($\widehat{y}$), are most favorable for a chiral topological state ($\mathcal{C}=\pm 1$) with a large gap. In the tight-binding simulation, we modeled islands that are twice as long as they are wide, and used the parameters $t=1$, ${\mu }_{\mathrm{S}}=-1.2$ (chemical potential in the superconducting regions), ${\mu }_{\mathrm{N}}=3.3$ (chemical potential in the normal regions), $\mathrm{\Delta }=0.2$, and $\lambda =0.2\pi$.

Figure 6

Transport signatures of chiral topological superconductivity. (a) Proposed measurement setup: a variant of the phase-controlled island array of Sec. 2 is contacted by three tunnel probes (L, B, T) and one strongly coupled lead (R). We envision the L lead as the source and the rest as drains. The chiral edge mode appearing in the topological phase is illustrated in dashed lines. (b) Transport simulation results at zero bias voltage as a function of $\varphi$. The nonlocal conductances $G_{\mathrm{LB}}$ and $G_{\mathrm{LT}}$ onset only in the topological phases, and they appear for Chern numbers $\mathcal{C}=+1,-1$ respectively, due to the chiral nature of the edge mode. The inset shows the local conductance $G_{\mathrm{LL}}$, which also onsets only in the topological phase (the $\mathcal{C}=0$ phase is fully gapped), but does not differentiate between the chiralities $\mathcal{C}=\pm 1$. The tight-binding parameters are the same as in Fig. 8, and we use a lattice of $300\times 300$ unit cells. The left, top, and bottom leads are four sites wide, whereas the right lead is ten sites wide.

Figure 7

Two-dimensional topological superconductivity from an array of parallel one-dimensional topological superconductors. When each 1D system is tuned to the critical point, it hosts two gapless Majorana modes. Coupling these left- and right-movers (blue and green arrows) appropriately, as shown by the dashed lines, results in a gapless chiral edge mode.

Figure 8

2D topological superconductivity in a phase-controlled Josephson junction array where each unit cell contains three islands. (a) The proposed geometry: 2DEG (blue) with superconducting islands (gray, green, and purple, with the color coding representing the superconducting phases) on top. The unit cell is depicted in a black frame. (b) Topological phase diagram of the model as a function of the phases ${\varphi }_1$ and ${\varphi }_2$. Colors indicate the Chern number $\mathcal{C}$, whereas the intensity corresponds to the bulk gap relative to the pairing gap $\mathrm{\Delta }$. As in the model of Fig. 1, the topologically nontrivial states with $\mathcal{C}=\pm 1$ appear only in regions where phase winding occurs, which are marked by solid lines for clarity. The tight-binding parameters in the simulation are $t=1$, $t^{\prime }=0.5$, $\mu =-1$, $\mathrm{\Delta }=0.5$, $\lambda =0.3\pi$, and the two left islands in each unit cell form an angle $\theta =0.3\pi$ with respect to the right island.