Erratum: Heat-charge separation in a hybrid superconducting quantum Hall setup

After the publication of our paper, we became aware of a mistake in our numerical codes, which led to some changes in the ensemble-averaged Andreev transmission probability. Figure1in this Erratum, which replaces Fig. 2 in this paper, shows the correct behavior of the probability as a function of the energy in units of the superconducting gap, for three different values of the disorder strength. The difference with respect to Fig. 2 in this paper is that exactly vanishes at and reaches 1/2 over a finite energy range, whose extension depends on the value of. For the case, is reached at about, while for the value 1/2 is reached at a very small energy. (Figure presented). The fact that the Andreev transmission probability TA = 0 at E = 0, even in the presence of spin-mixing disorder, can be understood as follows. First of all, we notice that our system belongs to the symmetry class D, where time-reversal symmetry and spin-rotation symmetry are broken [1]. On the one hand, by using the particle-hole symmetry and unitarity of the scattering matrix, one can show [1] that TA is equal to either 0 or 1, in the case of a single open channel. This is indeed our case where only a single spin-polarized edge channel is available for transport. On the other hand, we expect our system to be in the topologically trivial phase, in which case TA must vanish [1] at E = 0. At energies E > 0, however, TA can deviate from 0, thus producing an ensemble-averaged Andreev transmission probability «TA» which eventually reaches a finite value, with a growth speed that depends on the disorder strength w. For very small values of w, i.e., w = 0.01t, «TA» remains vanishingly small in the whole range of energies explored in Fig. 1. For weak disorder, i.e., w = 0.2t, «TA» increases slowly, reaching a value close to 1/2 at E ∼ 0.2Δ. For strong disorder, w=0.7t, «TA» increases very rapidly to 1/2. The resulting charge currents, for the cases w = 0.2t and w = 0.7t, are plotted in Figs. 2 and 3, respectively, as functions of the voltage V and for a very low temperature, i.e., T = 1.0 × 10-2 Δ/κB. Such figures show that the charge current in lead 2 (circles) vanishes in the whole range of explored voltages only for the case of the strongest disorder (w = 0.7t), while for w = 0.2t, «I2 is strongly suppressed, but remains finite. Figure 3 replaces Fig. 3 in this paper, but it refers to a different value of disorder strength, i.e., w = 0.7t. In both cases, the heat current in the superconducting lead remains 0 since superconductors are poor heat conductors for temperatures below the gap. (Figure presented) In conclusion, the main result of the paper, i.e., the separation of charge and heat currents, is confirmed as long as a sufficiently strong spin-mixing disorder strength is taken into account. We kindly acknowledge Anton Akhmerov for drawing our attention to the fact that TA must vanish at E = 0. This observation led us to the discovery of a mistake in the numerical codes used for producing the figures in this paper. The software used in this article is openly available at Zenodo [2].