Gate-defined Kondo lattices with valley-helical quantum dot arrays

I thank the referee for appreciating the originality of the work. I address the minor revision suggestions listed by the referee below:

1. The discussion of the effective model (eqn. (1-4)) should be expanded. Also, indicating the individual terms of the Hamiltonian (eq (1)), i.e., decoupled helical states and dot levels, in Figures 2 or 3, e.g., via dashed lines, would ease the reading.

I expanded the discussion of the effective model. I also added the dashed lines in Fig. 3. It is impossible to do it in Figure 2 since this comes from a microscopic simulation where I cannot completely decouple the degrees of freedom.

1. The perturbative expansions leading to the Kondo lattice (eq. (6)) and Heisenberg exchange term (eq. (7)) should be included in the main text or the appendix. It seems that the author has the channel-symmetric two-channel Kondo model in mind for the effective model (eq. (6)), but details of the derivation, which could help to verify this, are unfortunately absent.

I added references now where this derivation is done. The SU(4) Kondo model was fully derived for carbon nanotubes. Although there the dispersive states are not valley helical, the left and right movers are decoupled in a similar way to the valleys in the present manuscript. The Heisenberg model was recently derived in the context of gate-defined quantum dots. Also, a similar derivation on valley-helical channels coupled to localized states was previously done in twisted bilayer graphene and strained graphene superlattices. We also added these references.

1. The discussion of the effective Kondo-Heisenberg model should be expanded: In which parameter regime is the obtained effective low-energy model justified? Which constraints does this impose on the exploitable parameter regime of the proposed Kondo-Heisenberg model for realistic values of the tuning parameters (gate voltage, layout and size of patterned middle layers etc.)?

We have now added a discussion on how the effective model parameters depend on the microscopic parameters. This should provide a route for tuning these parameters in experiments. While I believe these considerations are general, an actual estimation of parameter windows can only be correctly made with realistic electrostatic simulations and possibly additional experimental input. Therefore, I limited the discussion to the expected scaling. The Kondo-Heiseberg model is justified as long as $\Delta, \Gamma \ll V$, $\chi \gg a$ (or $1/K\chi \ll 1$), and intervalley scattering caused by disorder is negligible. I went over the text to ensure that these requirements are clear.

1. Typos and inconsistent notation, in particular in the equations, have to be removed, see eqns. (2,3,5).

I went over the equations and fixed the typos.

I thank the referee for highlighting how the proposal in the manuscript addresses issues faced by the experimental community in other Kondo lattice platforms.

The referee says that I do not discuss the experimental feasibility of the device. This is in fact true for a reason: while there are, in principle, no practical limitations on implementing a setup with multiple back gates, no experimental works have tried it yet. Discussing with experimentalists in the field, I could not spot a fundamental issue with the proposal, but there are no previous works to back it up. I hope the manuscript will encourage experimentalists to try similar setups and address the feasibility of multiple back-gate devices. Finally, following the suggestion from referee 1, I also added an Appendix on transport across a split-gate device that addresses my own comment on the possible challenges.

I address the minor revision suggestions listed by the referee below:

There should be a discussion on how variations in electric field strength, the distance between dots (related to the structure of the gates), and the interaction strength U affect the parameters of the Kondo lattice model. Given the claim that the Kondo lattice parameters are electrically tunable, it is important to demonstrate how these parameters can be modified.

We now provide an estimation of how the effective model parameters scale with the microscopic parameters based on the numerics. I believe this discussion provides a direct recipe for tuning the effective parameters.

There should be a discussion on how disorder will affect the arising effective Kondo lattice model.

As the referee pointed out, (short-range) disorder is detrimental to the valley-helical transport. However, a series of recent experiments have observed valley-dependent phenomena in gate-defined nanostructures. These existing works suggest that short-range scattering sources, which would cause large momentum transfer and break valley conservation, are negligible. It is therefore reasonable to assume that the only sources of scattering are due to the confining electrostatic potential. Due to screening effects, this potential varies on a scale much larger than the lattice constant, and therefore does not cause intervalley scattering. I mention the absence of intervalley scattering in experiments at the end of the second paragraph of Sec. 2.

I would appreciate it if the author could provide further details on the methodologies and calculations.

I indeed missed details on the tight-binding calculations performed in Kwant. The code used in this manuscript was heavily based on the simulations done in a previous work, which we now refer to in Appendix A. In this previous work, we also developed the scaling model for the electronic structure that we use in the manuscript. Hopefully, this additional reference and the added comments in the Appendix provide enough information. I will be happy to provide additional information if the referee still misses something specific.

I thank the referee for recommending publication. I went over the text to fix the typos. Moreover, I added simulations of a split-gate geometry to the appendix. While this geometry probably cannot reproduce the Kondo physics because the larger size of the quantum dots likely suppresses the charging energy, we show that the setup still works as a valley filter. We also related our simulations to recent experiments.

Furthermore, I fixed the typos pointed out by the referee and added the missing references.