Confined vs. extended Dirac surface states in topological crystalline insulator nanowires

We have considered all issues and criticisms raised by the reviewers, and we have modified our manuscript to address them. Please see the referees reports for detailed answers.

1. In the abstract, we replaced the words “pronounced and unique” in the word “novel”.

2. In the first paragraph of the introduction, references were added, as well as an addition to the last sentence: “…and have currently been achieved mainly in Graphene-based hetero-structures and devices.״

3. In the second paragraph of the introduction, rephrased the sentence: “Until now such a configuration of surface states has required complex hetero-structures, coupling strong topological insulator surfaces to magnetism or a dimensional reduction, and hence not been realized in other topological phases.” to “Until now such a configuration of surface states has required coupling to magnetism, complex hetero-structures, or a dimensional reduction.”

4. In Section II, a clarification regarding the definition of the flux through the wire was added, along with a small changes in notations: "$\phi=\frac{\Phi}{\Phi_{0}}$ is the number of flux quanta through the wire's cross section, $\Phi=B\cdot A$ is the total magnetic flux through the wire for a magnetic field $B$ and a wire cross section $A$. ${\Phi_{0}=\frac{h}{e}}$ is the magnetic flux quantum.״ Accordingly, the notions of $\Phi$, $\phi$ , and $\phi_{0}$ in the older version were replaced with $\phi$, $\Phi$ , and $\Phi_{0}$ respectively in the current version, in this paragraph and in equations 1 and 2, to be consistent with the notation of $\phi$ in the rest of the manuscript.

5. A comment regarding the Zeeman effect was added to Section II: “The addition of a Zeeman term to this model may change the values of these parameters but not the essence of the effect, as discussed in the supplementary material.”

6. In the last paragraph of Section III.A, rephrased the sentence “​​Our analysis of the (100) wire is compatible with predictions for AB as a sum of those arising from projections of several Dirac points with a finite size gap tunable by flux, as described in Section I” to “Our analysis of the (100) wire is compatible with the predictions presented in Section I, for AB oscillations as a sum of those arising from projections of several Dirac points with a finite size gap tunable by flux”.

7. In the last paragraph of Section III.A, rephrased the sentence: “The physics is richer due to an interplay of the surface and hinge modes at the corners of the wire, clearly demonstrates that the surface is 2D in character, and sensitive to flux via modification of the boundary conditions.” to “The interplay of the surface and hinge modes at the corners of the wire clearly demonstrates that the surface is 2D in character, and sensitive to flux via modification of the boundary conditions.”

8. Two paragraphs of discussions regarding the optional measurement techniques to probing the effects were added to the Summary and Conclusion section:
“The predictions of AB responses in the different wire setups can be ideally and directly probed in scanning tunneling microscopy of SnTe nanowires. Using spectroscopic mapping, both the evolution of induced surface gaps and the formation of hinge states, as well as the interplay among them, can be visualized as a function of threaded magnetic field. Our theoretical results thus mark a clear and unique path for the exploration and characterization of topological boundary modes in experiment.
In addition, presence or absence of AB oscillations in the spectrum may be also probed in transport measurements, similar to experiments performed on strong TI nanowires. Ref.[51] reported measurements of AB oscillations in single crystalline SnTe nanowires of a typical width of 59nm and at temperatures as high as 30K. The surfaces of the wires, however, are not specified and therefore the surface theory is unknown. The absence of AB oscillations will be measured in a geometry that introduces masses in the wire's corners, like the (110) wire presented, with the chemical inside the energy range that is affected by the gap induced by symmetry breaking. Since the gap at the corners is caused by a local perturbation due to the breaking of symmetries near the hinges, it is not expected to depend on the overall wire’s dimensions, but on the details of the hinge and the particular bulk material. From the spectra of the (110) wire in fig.4, the energy scale in which there are no AB oscillations, namely the energy cutoff above which bands start to respond to flux in the usual way, is approximately 150meV. In transport experiments on strong TI nanowires, gate voltage control over the chemical potential with a resolution of the order of 10meV was achieved. Although the level of control over the chemical potential is determined by the specific details of the experiment, such as the density of states of the bands near the Fermi level, the dimensions of the device and its fabrication process, a similar resolution in SnTe nanowires may be accessible.”

9. In the last paragraph of the Summary and Conclusions sections, a few references were added.

10. A subsection regarding the effects of a Zeeman coupling on the results was added to the Supplementary materials:
“In the low energy model in section II and in the numerical calculations, we neglected the Zeeman effect, coupling the magnetic field to the electron spin.
First, we stress that the magnitude of the magnetic field required to see AB oscillations in nanowires is small: considering a typical cross-section of $60\times 60 nm^2$, the field required for one flux quantum is B=1.1 T. This statement is independent of the type of wire, and depends only on the wire dimensions. For these field values, and for a g factor g=57 measured for SnTe, the Zeeman splitting is $\frac{g\mu_{B}B}{2}=1.9 meV$, which is small compared to the sub-band spacing in our case.
Next, we point out that Zeeman coupling is also relevant for strong TI wires, where it breaks TR symmetry. For example, in TI wires made from BiSbSe the g factor is estimated to be approximately 23. In experiments done on such wires, AB oscillations were observable but the details are slightly modified compared to predictions, as was also discussed in theoretical works. We believe this should also be the case for our system as we now explain.
The changes induced by the Zeeman coupling will modify the details of the observed band structure, but not the main conclusions of our work. This can be seen as follows: A Zeeman term can be added to the surface Hamiltonian which may change the surface theory and as a result, the effects discussed in the paper. We consider, for example, adding a Zeeman term to the low energy model of the surfaces of SnTe. Since the magnetic field is parallel to the wire, it is parallel to each one of its facets. Such a term does not gap the surface Dirac cone, but shifts their location, as detailed in ref.[41] and specifically its correction [54]. The low energy model of SnTe surfaces as presented in this reference is of the from $\textbf{k}\times\boldsymbol{\sigma}=k_{x}\sigma_{z}-k_{z}\sigma_{x}$. If the wire is along the z axis, and a Zeeman term $g\mu_BB\sigma_z$ is added, the location of the Dirac cone in the surface theory will shift along the x coordinate. Performing a coordinate transformation and taking x around the wire as in section II, this shift will modify the flux values needed for gap closing. In terms of the general low energy model presented in section II, a Zeeman term will shift the location of the Dirac cones along kz.”