Quantized phase-coherent heat transport of counterpropagating Majorana modes

We demonstrate that phase-coherent heat transport constitutes a powerful tool to probe Majorana physics in topological Josephson junctions. We predict that the thermal conductance transverse to the direction of the superconducting phase bias is universally quantized by half the thermal conductance quantum at phase difference $\varphi =\pi$. This is a direct consequence of the parity-protected counterpropagating Majorana modes which are hosted at the superconducting interfaces. Away from $\varphi =\pi$, we find a strong suppression of the thermal conductance due to the opening of a gap in the Andreev spectrum. This behavior is very robust with respect to the presence of magnetic fields. It is in direct contrast to the thermal conductance of a trivial Josephson junction which is suppressed at any phase difference $\varphi$. Thus, thermal transport can provide strong evidence for the existence of Majorana modes in topological Josephson junctions.

Figure 1

(a) Short S-TI-S Josephson junction of width $W$ embedded in a point contact geometry to measure heat transport perpendicular to the direction of phase bias $\varphi$. Thermal transport is driven by a temperature difference $\mathrm{\Delta }T$ between the two TI leads and occurs via two counterpropagating modes (red and blue arrow) in the leads and in the superconducting region. (b) Quasiparticle transmission $\mathcal{T}$ given by Eq. (9) as a function of energy $\varepsilon$ and phase difference $\varphi$ for $\mu =10{\mathrm{\Delta }}_0$ and $Z_0=\pi /4$. For $\varphi =\pi$, we find perfect transmission while for $\varphi \ne \pi$, the transmission is exponentially suppressed at low energies.

Figure 2

Heat conductance $\kappa \left(\varphi \right)$ in units of the thermal conductance quantum $G_Q={\pi }^2{k}_B^{2}T/\left(3h\right)$ in the absence of any Zeeman field. (a) Width dependence at fixed base temperature $k_{B}T=0.1{\mathrm{\Delta }}_0$. (b) Temperature dependence at fixed width $W=\xi /2$. The thermal conductance $\kappa (\varphi =\pi )=G_Q/2$ is quantized independently of the chosen parameters. Other parameters as in Fig. 1.

Figure 3

(a) Heat conductance $\kappa \left(\varphi \right)$ in units of the thermal conductance quantum $G_Q$ for an in-plane Zeeman field $Z_{xy}=\pi /5$. While a finite $Z_x$ broadens the conductance peak, a finite $Z_y$ leads to a phase shift. The thermal conductance is quantized even in the presence of a finite Zeeman field. The remaining parameters are chosen as in Fig. 2 with $W=\xi /2$ and $k_{B}T=0.1{\mathrm{\Delta }}_0$.