Multiterminal Josephson junctions with tunable topological properties

Abstract

Since the discovery of the Andreev reflection process at normal-metal/superconductor junctions and the corresponding Andreev bound states in superconductor/normal-metal/superconductor junctions, various multiterminal Josephson junctions have been studied to explore many exotic phases of quantum matter, where the formation of Andreev bound states in the normal region account for dissipationless supercurrent and play a central role in determining exotic properties. Recently, an intriguing aspect of the multiterminal Josephson junctions has been proposed to study the topological properties, wherein the Andreev bound states acquire topological characteristics upon tuning the phase differences of superconducting terminals. In this work, we investigate topologically non-trivial phases in four-terminal Josephson junctions based on square and graphene lattices. Additionally, we apply a gating potential that smoothly drives the Andreev bound states from a topologically non-trivial state to a trivial state. Furthermore, we observe that the gating potential in our setup produces the similar physics of the topological Andreev bound states of the double (single) quantum-dot multiterminal Josephson junctions when the gating potential is small (large) compared to the superconducting gap.

Figures (5)

• Figure 1: (a) Schematic illustration of the multiterminal Josephson junctions setup. We consider a system with four superconducting terminals attached with a scattering region, forming Josephson junctions. The scattering region is indicated in light-green with length $L$ and width $W$ whereas the four superconducting (SC) leads are depicted in the light-blue with phases $\phi_\alpha$ (for $\alpha=0$ to $3$). These semi-infinite horizontal and vertical leads have the corresponding width $W_s$ and length $L_s$ respectively. Additionally, a schematic of externally applied gating potential, acting only in the scattering region, is shown in orange color, increasing in strength along the diagonals. (b) We setup the system based on square and graphene lattices and employ the KWANT Groth2014 to numerically simulate the tight-binding model Hamiltonians in Eqs. \ref{['eq:hamilt-sq']} and \ref{['eq:hamilt-gr']}. (c) A typical color plot and contour plot for the applied gating potential of the form $V(x,y) = V_g x^2 y^2$. (d) A discretized mesh-grid in ($\phi_1, \phi_2$) space is used to numerically calculate the Chern number, in Eq. \ref{['eq:chern-discrete']}, by using the Fukui, Hatsugai, and Suzuki method Fukui2005.
• Figure 2: (a) Topological phase diagram for the Chern number $C_{12}^{\mathrm{GS}}$ in (i) and the corresponding phase boundaries from minimum gap $\delta_{\mathrm{min}}$ (of the lowest ABS energy) in (iii) are plotted in the parameters space $(\phi_3, \eta)$ when the gating potential is set to zero, i.e., $V_g=0$; whereas, they are also shown for a few selected values of $\eta$ in (ii) and (iv). (b)-(c) At a fixed $\eta=0.22$, the Andreev-energy dispersions $\varepsilon_{\pm1}$ (in unit of $\Delta$) and the Berry curvature $B_{12}$ (for the lower band) with respect to the phase differences $\phi_1$ and $\phi_2$ for a set of different $\phi_3$ values. The chosen $\phi_3$ values are shown in the corresponding plots of (b) and (c); while particularly for the $\varepsilon_{\pm1}$, they are also shown in (a)-(ii) as inverted black triangles.
• Figure 3: Topological phases for Chern number $C_{12}^{\mathrm{GS}}$ and the corresponding phase boundaries from $\delta_{\mathrm{min}}$ with respect to $\phi_3$ and $V_g$ for fixed values of $\eta=0.3$ and $0.6$.
• Figure 4: (a)-(b)Topological non-trivial phases for the graphene lattice, obtained from the Chern number $C_{12}^{\mathrm{GS}}$ and corresponding phase boundaries from $\delta_{\mathrm{min}}$, in parameters space of $\phi_3$ and $\eta$ when the gating potential is absent $V_g=0$. For fixed $\eta=0.55$ in (c)-(d) and $\eta=0.75$ in (e)-(f), the topologically non-trivial phases are robust and evolve upon tuning $V_g$.
• Figure 5: Chern number $C_{12}^{\mathrm{GS}}$ vs. $\phi_3$ obtained from the scattering matrix (SM) method (solid line) and the BdG method (markers) for both the lattices in the absence and presence of gating potential $V_g$.