Flux-tunable Kitaev chain in a quantum dot array

1. In light of recent experimental developments with quantum-dot-based short Kitaev chains, the setup described in this work seems experimentally feasible and interesting for the Majorana community.
2. The manuscript is well written.

1. I do not see significant advancements in this work in terms of originality and methodology.

2. The authors do not articulate which of the 4 "expectations", required for SciPost Physics publications, are met by their work.

See further comments in "Report".

The authors study a model of a two-site Kitaev chain, where the two sites are single-orbital quantum dots with large Zeeman splitting and large on-site Coulomb repulsion, whereas tunnel coupling is mediated by a third dot embedded in a Josephson junction. A key finding, illustrated in Fig. 2a,c, is that by tuning the flux bias of the Josephson junction, the “topological dimerized limit” (aka “sweet spot”) of the two-site Kitaev chain can be reached. The authors also discuss “optimal sweet spots” in the parameter space, where the even-odd degeneracy of the sweet spot is robust against parameter fluctuations.

In light of recent experimental developments with quantum-dot-based short Kitaev chains, the setup described in this work seems experimentally feasible and interesting for the Majorana community. Also, the manuscript is well written. I have to add that I do not see significant advancements in this work in terms of originality and methodology.

My further comments are as follows. In my view, the authors reaction to the first 3 comments will be important to judge if this work should be published in SciPost Physics (cf. https://scipost.org/SciPostPhys/about#criteria).

1) The authors do not articulate which of the 4 "expectations", required for SciPost Physics publications, are met by their work. As a working hypothesis, I assume that they think that their work opens “a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work;”

2) I do understand the conceptual framework of the Kitaev chain, and how a realisation of it could lead to braiding-based robust control of Majorana qubits, and their protection against certain noise types. However, my understanding is that these features arise only in the non-interacting case. The authors study quantum dots with a large Hubbard-U interaction, i.e., away from the non-interacting case. Therefore I think the authors should discuss if their setup is still relevant for building protected Majorana qubits, and if it is, then how? If it turns out that they cannot justify the relevance of this setup for building protected Majorana qubits, then I doubt that the expectation I mentioned in 1) is fulfilled.

3) To my understanding, the robustness of a Majorana qubit against decoherence requires long chains. The authors present a two-site chain. Is their idea of using a flux-biased Josephson junction scalable to longer chains? If it is not, then this should be clearly discussed. If it is, then this should also be detailed, preferably with an image showing the geometry of a scaled-up device. Again, the response here affects how well the expectation I mentioned in 1) is fulfilled.

Further comments for the authors' consideration:

4) The term “sweet spot” is used in the text in different roles. (i) It refers to the “t = Delta” limit (fully dimerized limit) of the Kitaev chain, (ii) In Fig. 3 caption (and also in the text), it is also used as a parameter point (black cross) where the flux value needed to maintain the even-odd degeneracy is first-order insensitive to the common on-site energy of the outer dots, (iii) at the bottom of page 6, “sweet spot” is used to describe parameter points where the Majorana energy is insensitive to flux fluctuations. I find this confusing, and hence recommend to clarify the terminology, e.g., introduce different names for the different sweet spots (or at least use different qualifiers before “sweet spot”) .

5) In the abstract, the authors mention that they consider a “narrow” Josephson junction, but in the text they do not explain what that means and why narrowness is important. I recommend to either omit “narrow” or explain it.

6) In the introduction, the authors write that “The flux control method eliminates challenges associated with charge noise…”. Why is that true? I think charge noise is present in a gate-control quantum dot array, no matter if flux control or voltage control is used.

7) I recommend to the authors that they make an attempt to state the range of validity of the Hamiltonian defined in Eqs. (1)-(4), with the goal to allow experimentalists to judge whether their experimental setup is well described by this minimal Hamiltonian or not. E.g., I presume that Kondo-type correlations are not captured by this Hamiltonian, etc. Also, I recommend to comment on the feature that the middle dot has no Coulomb repulsion whereas the outer dots have strong Coulomb repulsion: what are the physical conditions in a device to achieve this setting, at least approximately?

8) The text says “As depicted in Fig. 3(d), the wavefunctions of the two Majoranas are completely localized on the left and right dots, respectively”, but in Fig. 3(d) the majority of the wavefunction is on the middle dot. I recommend to resolve this contradiction.

In this work the authors present an interesting idea of using a Josephson junctions to tune the coupling between two quantum dots to create a Kitaev chain. The advantage of the setup proposed is that the coupling between the quantum dots can be controlled by tuning the flux threading the Josephson junction allowing to tune via the flux the topological character of the chain.

The work is well written and is innovative and I find that it could be published in SciPost Physics.

There are few items that I think the authors should address in a revised version of the manuscript.

1. It would be helpful if in Fig. 1 the authors would also include a schematic of the real system giving a sense of its geometry.

2. A more substantial question relates to the type of Josephson junction needed to realize the setup. In the text the authors mention that they "use a planar Josephson junction". Is this necessary? Must the Josephson junction be very wide and therefore planar? Or the requirement is simply that the Junction must be large enough to be able to tune the flux without having to use very large magnetic fields? What about the length of the junction? Does it have to be short?

3. On page 4 Delta_CAR and Delta_ECT are expressed in terms of the coherence factors u and v, but these are not defined in the text.

4. Units should be added for all the axes of all the panels of Fig.3.

5. Given the order of the discussion in the main text the columns of Fig. 3 should be switched.

6. A couple of typos (for instance has->is 3rd line on page 7).

The paper presents an investigation into a three-quantum-dot setup resembling a Josephson junction. The authors explore various aspects such as the impact of phase difference ($\phi$) and coupling strengths ($\Gamma_{1,2}$) on the system's behavior, with a particular focus on the equality of the CAR and ECT amplitudes at a specific value of $\phi$. The presentation is generally clear, and the paper adequately references and cites relevant literature. However, several concerns and areas for improvement have been identified:

1. Inconsistency between Fig. 1-b and the Hamiltonian defined in Eq. 3, Eq. 5, or Eq. 8. The schematic in Fig. 1-b, particularly the orange representation, appears to deviate from the described Hamiltonian formulations.

2. Ambiguities in Fig. 2. It would enhance clarity if the authors compared the energy spectrum and electron-hole components (u, v) of Eq. 5 with the effective Hamiltonian (Eq. 8) and included these comparisons in panels a and b, respectively. Currently, panels a and b seem to derive from Eq. 5, while panels c and d derive from the effective Hamiltonian, which could be confusing for readers.

3. The effective Hamiltonian (Eq. 8) should be elaborated with details upon to ensure self-consistency, especially considering its role in benchmarking with the numerical Exact Diagonalization (ED) results.

4. The theory developed by the authors regarding the limit of large U in the outer dots appears to exhibit characteristics that may persist even in the single-particle picture, particularly under the extreme condition of U=0, accompanied by a substantial Zeeman splitting. It would be valuable for the authors to provide a more detailed explanation and elaboration on this aspect.

5. Revisiting the Local conductance presented in Fig. 3 using standard transport methods, such as Non-Equilibrium Green's Function (NEGF), holds the potential to offer valuable insights, particularly in light of the absence of interactions in the outer dots, as mentioned in the previous comment regarding U=0.

6. The formulation related to Local conductance in Fig. 3 needs detailed explanation, which was unfortunately omitted in the current version.

7. The introduction of a coupling strength $\Gamma_{LD}=0.01$ (Fig. 3) requires clarification in comparison with the tunneling Hamiltonian defined in Eq. 4.

8. Discrepancies between the description of Majorna wave functions in Fig. 3-d and the text. It appears that the wavefunctions for cases with $\mu_M=0.2$ and $\mu_M=0.5$ are differently interpreted in the figure and the accompanying explanation, necessitating clarification.

Taking into account the aforementioned comments, I suggest that the manuscript could be suitable for publication in SciPost Phys. Core rather than SciPost Phys, provided that the authors address the outlined concerns and make necessary revisions to enhance clarity and coherence.