Nonlocality of local Andreev conductances as a probe for topological Majorana wires

We propose a protocol based only on local conductance measurements for distinguishing trivial from topological phases in realistic three-terminal superconducting nanowires coupled to normal leads, capable of hosting Majorana zero modes (MZMs). By using Green's functions and the scattering matrix approach, we calculate the conductance matrix and the local density of states (LDOS) as functions of the asymmetry in the couplings to the left $({\mathrm{\ensuremath{\Gamma}}}_L)$ and right $({\mathrm{\ensuremath{\Gamma}}}_R)$ leads. In the trivial phase, we find that the zero-bias local conductances are distinctively affected by variations in ${\mathrm{\ensuremath{\Gamma}}}_R$ (for fixed ${\mathrm{\ensuremath{\Gamma}}}_L$): while ${G}_LL$ is mostly constant, ${G}_RR$ decays exponentially as ${\mathrm{\ensuremath{\Gamma}}}_R$ is decreased. In the topological phase, surprisingly, ${G}_LL$ and ${G}_RR$ are both suppressed with ${G}_LL\ensuremath{\sim}{G}_RR$. This nonlocal suppression of ${G}_LL$ with ${\mathrm{\ensuremath{\Gamma}}}_R$ scales with the MZM hybridization energy ${\ensuremath{\varepsilon}}_m$ and arises from the emergence of a dip in the LDOS near zero energy at the left end of the wire, which affects the local Andreev reflection. We further exploit this nonlocality of the local Andreev processes and the gate-controlled suppression of the LDOS by proposing a Majorana-based transistor. Our results hold for zero and low electron temperatures $T<20\phantom{\rule{0.16em}{0ex}}\mathrm{mK}$. For $T=30,40\phantom{\rule{0.16em}{0ex}}\mathrm{mK}$, ${G}_LL$ and ${G}_RR$ become less correlated. As an additional nonlocal fingerprint of the topological phase at higher ${T}^{\ensuremath{'}}\mathrm{s}$, we predict modulations in our asymmetric conductance deviation $\ensuremath{\delta}{G}_LL^{\mathrm{asym}}={G}_LL^{{\mathrm{\ensuremath{\Gamma}}}_R={\mathrm{\ensuremath{\Gamma}}}_L}\ensuremath{-}{G}_LL^{{\mathrm{\ensuremath{\Gamma}}}_R\ensuremath{\ll}{\mathrm{\ensuremath{\Gamma}}}_L}$ that remains commensurate with the Majorana oscillations in ${\ensuremath{\varepsilon}}_m$ over the range $30<T<150\phantom{\rule{3.33333pt}{0ex}}\mathrm{mK}$.