Topological phase diagram of a three-terminal Josephson junction: From the conventional to the Majorana regime

We study the evolution of averaged transconductances in three-terminal Josephson junctions when the superconducting leads are led throughout a topological phase transition from an $s$-wave to a $p$-wave (Majorana) phase by an in-plane magnetic field $B_x$. We provide a complete description of this transition as a function of $B_x$ and a magnetic flux $\mathrm{\Phi }$ threading the junction. For that we use a spinful model within a formalism that allows us to treat on an equal footing the contribution to the transconductance from both the Andreev subgap levels and the continuum spectrum. We unveil a fractionalization in the quantization of the transconductance due to the presence of Majorana quasiparticles, reflecting the effective pumping of half a Cooper pair charge in the $p$-wave regime.

Figure 1

Scheme of a Josephson trijunction threaded by an external magnetic flux $\mathrm{\Phi }$. The superconducting leads (with phases ${\phi }_{\nu }$) have Rashba spin-orbit coupling and can be driven into a topological (Majorana) regime by an in-plane magnetic field $B_x$.

Figure 2

(a) Zero-temperature average transconductance $G_{12}$ as a function of the external flux $\mathrm{\Phi }$ and the in-plane Zeeman field $B_x$ (in units of the critical field $B_c=\sqrt{{\mathrm{\Delta }}^2+{\mu }^2}$). (b) $G_{12}$ as a function of the external flux $\mathrm{\Phi }$ for different values of $B_x$. Notice that the effective pumped charge $q^{★}$ changes from $2e$ to $e$ when $B_x$ becomes larger than $B_c$ ($p$-wave or Majorana phase).

Figure 3

Andreev subgap spectrum along the path ${\phi }_2=-{\phi }_1$ as a function of the Zeeman field $B_x$ for $\mu =3.2\mathrm{\Delta }$ and spin-orbit coupling $\alpha =50\mathrm{\Delta }$. The central magnetic flux was taken to be ${\mathrm{\Phi }}_c^{\left(1\right)}=0.53$.

Figure 4

Andreev subgap spectrum along ${\phi }_2$ for ${\phi }_1=2\pi$ as a function of the Zeeman field $B_x$ for $\mu =3.2\mathrm{\Delta }$ and spin-orbit coupling $\alpha =50\mathrm{\Delta }$. The central magnetic flux was taken to be $\mathrm{\Phi }=0$.

Figure 5

Upper panels show the spectral density of the three-terminal Josephson junction in the spin-degenerate case ($B_x=0$) along the path ${\phi }_2=-{\phi }_1$ for different fluxes threading the central scattering region between the leads (a) $\mathrm{\Phi }=0$, (b) $\mathrm{\Phi }=0.3$, and (c) $\mathrm{\Phi }=0.53$. Lower panels (d), (e), and (f) show the eigenvalues of $H_{\text{top}}^J\left(\mathit{\phi }\right)=-{\mathcal{G}}_J^{-1}(\omega =0)$ for this same path with the same corresponding fluxes. Solid lines indicate the bands topologically connected with the Andreev bound states, and dashed lines indicate the eigenvalues bearing the topological information of the continuum states.

Figure 6

Chern numbers $C_{12}$ of the Andreev bound states (ABS) and the continuum states. The lower panel shows the sum of both contributions.