Josephson junctions of topological nodal superconductors

We thank the referee for their comments and suggestions which helped us to improve the quality of our work. Our detailed response as well as the changes we intend to incorporate are detailed below.

1. The form of the barrier used in this work need not be non-superconducting. Only a weak link is required such that the phase change of the superconducting pairing across the junction is abrupt and not continuous. Josephson junctions in mono- or few-layer films can be realized by naturally occurring cracks in TMD flakes, see e.g. in https://arxiv.org/pdf/2106.11662.pdf, or by dynamically generated lines of reduced superconductivity, which arise due to the presence of screening currents generated by flux piercing see e.g. https://arxiv.org/pdf/2204.02888.pdf.

2. We address the effects of a finite junction width in our detailed response to referee A point 1. Regarding the possibility to control $k_y$: we note that one does not need to experimentally control $k_y$. Rather, for a wide and smooth junction, the Josephson current would be a sum of contributions from all transverse channels. In the nodal topological superconducting phase, some of these contributions would be $4\pi$ periodic which should be reflected in a fourier analysis of the Josephson signal. In addition, we expect that one way to control the relative contributions is by considering junctions at different orientations with respect to the x-axis. We intend to explore the detailed effect of junction orientation as well as other symmetry breaking perturbations which can be controlled experimentally in future work.

3. We have now added in the manuscript the following short summary of the method used to calculate the Josephson Energies. “To summarize, the Josephson energies are calculated using the following steps. Given an energy E and transverse momentum $k_y$, we invert the dispersion relation which results in four values of $k_x (E, k_y)$. Using these we construct the general wave function on either side of the barrier (Eqs. 10(a) and 10(b)) and match the boundary conditions at the junction, x = 0 (Eqs. 9(a) and 9(b)). This results in a matrix equation given in Eq. 11(a), which has solutions only when the determinant of M given by Eq. 11(b) vanishes. This condition gives us $E_J(\phi)$ for each $k_y$.” There is no simple analytical expression for $E_J(\phi)$

4. We have now edited the plots to be more readable and clearer than earlier.

5. This has been corrected

6. We have changed Fig 3 to explicitly indicate the different topological regions as a function of ky.

We thank the referee for their comments and suggestions which helped us to improve the quality of our work. Our detailed response as well as the changes we intend to incorporate are detailed below.

1. In this work we have assumed that the electron momentum parallel to the junction is a conserved quantity. This allowed us to use $k_y$ as a parameter. As the referee has correctly pointed out, this treatment of a Josephson Junction is justified only when there is perfect translational invariance along the y-direction. However, in real systems the sample has a finite width. This would have two effects: 1) the confinement in the y-direction gives rise to quantization of $k_y$. If the width of the sample is $d_y$, ky is quantized as $\pi n/d_y$ where $n$ is an integer. 2) the open boundary condition will mix states with $\pm k_y $. The former would have the effect that only a discrete set of $k_y$ values are being sampled. As the topological regions II and IV occur in narrow strips in $k_y$, the effect would be noticeable if the junctions are wide enough such that the discretization is smaller than the splitting between the nodal points controlled by the in plane field. Regarding the latter effect, we note that the Hamiltonian, and consequently the winding numbers are symmetric under $k_y \rightarrow -k_y $. We therefore do not expect mixing between $\pm k_y $ to affect the results. We agree with the referee that the study of more realistic Josephson junction with finite width, orientation mismatch and disorder is an interesting topic which we intend to explore in a future work.

2. We thank the referee for this comment. We intend to include an animation that shows the full evolution with $k_y$ as supplemental material.

3. As we explain in the manuscript, the $4\pi$ periodicity can be understood as resulting from the appearance of Majorana zero mode at the two ends of the junction when the junction is oriented in the x-direction. We expect that, as is the situation with any nodal material (e.g. graphene), the edge states will not be present when the boundary orientation projects nodes of opposite charge onto each other, which in our case would be along the $y-$direction. Therefore we expect that the properties and stability of the topological regions II and VI will indeed depend on the orientation of the boundary with respect to the x-axis. In fact, to the extent that orientations can be experimentally controlled, we expect this to be a useful control knob that can be used to detect these effects. The effect of a general junction orientation with respect to the crystal on the Josephson response is an interesting topic which we intend to explore in a future work.

4. The system we consider here is a 1H-NbSe2 monolayer.

5. We have now modified Fig 1 such that the nodal points are visible more clearly.