Wiedemann-Franz-type relation between shot noise and thermal conduction of Majorana surface states in a three-dimensional topological superconductor

We compare the thermal conductance $G_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}}$ (at temperature $T$) and the electrical shot noise power $P_{\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{t}}$ (at bias voltage $V\gg k_{\rm B}T/e$) of Majorana fermions on the two-dimensional surface of a three-dimensional topological superconductor. We present analytical and numerical calculations to demonstrate that, for a local coupling between the superconductor and metal contacts, $G_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}}\mathrm{/}{P}_{\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{t}}=\mathcal{L}T\mathrm{/}eV$ (with $\mathcal{L}$ the Lorenz number). This relation is ensured by the combination of electron-hole and time-reversal symmetries, irrespective of the microscopics of the surface Hamiltonian, and provides for a purely electrical way to detect the charge-neutral Majorana surface states. A surface of aspect ratio $W\mathrm{/}L\gg 1$ has the universal shot-noise power $P_{\rm shot}=(W/L)\times (e^2/h)\times (eV/2\pi)$