Emergence of quasiparticle Bloch states in artificial crystals crafted atom-by-atom

1. High quality experimental data
2. Detailed investigation of 1D and 2D lattices

1. Some detailed information missing
2. Missing discussion of past literature
3. Similar type of experiment has been recently reported

In the manuscript by J. Girovsky et al, the authors investigate the modifications and emergence of band structure which results from coupling Cl vacancies in ordered arrays on the Cu(100). The experiments are based on low-temperature scanning tunneling microscopy and spectroscopy, and they corroborate the electronic structure of the artificial lattices with a tight-binding model. Indeed, an artificial lattice has been constructed on this system and published, as cited, but I find the results here innovative enough, especially the level of experimental detail, to merit publication. I find the experimental data of high quality, and the visualization of the electronic structure of the engineered structures very clear. The conclusions are sound and worthy of publication. I have some minor comments and questions which I detail below. The authors should address these points in a revised manuscript, before publication.

1. The authors fail to compare/contrast and cite the original work of N. Nilius et al, Science, 297, 1853 (2002), which was the first work of this kind. This paper originally looked at the development of 1D band structure in a nearly identical experiment. Before publication, this paper should be adequately cited and discussed in context of the new findings here.
2. Why do the authors not see the development of standing waves in the 1D chains? I find it peculiar that the length dependence of the electronic structure saturates already at six atoms? Is there some explanation for this?
3. I am missing a value of k_F, or some reference to a wavelength here? How does this compare to the length of the 1D and 2D structures?
4. On page 2, the authors write “bulk limit.” As this is not a 3D structure, I find the use of the word bulk a bit misleading. I would suggest something like long wavelength limit, or 2D limit.
5. I found it very difficult to read the color dots in Fig 1, indicating where the spectra were taken.
6. In Fig 1f, the authors use “( )” and in the paper “{ }.” I would suggest to keep this consistent, and in the text introduce what this notation means as it is just suddenly used. Can they relate {x,y} also to the crystallographic axes?
7. I fail to understand why the tight binding parameters contradict the experiment. Can the authors give more insight as to why, and in what manner this could be checked or reconciled for any future calculations for follow-up work?
8. Why is dz/dV more sensitive than dI/dV (bottom page 4)? Can this explain why the authors don’t see standing waves in the 1D structures? I’m not sure I agree with this sentence, especially if the argument is that this is just a normalized dI/dV curve?
9. A helpful suggestion: dI/dZ(V) has also been used in the past to measure band onsets. Maybe this can help in future measurements of the larger structures?

1. Paper presents a systematic fabrication of artificial lattices in atomic precision and solidifies previous work in the same system
2. The experimental results look very reliable and indicate that defect-free lattices can be fabricated relatively routinely
3. Simple theoretical modelling in term of a next-nearest-neighbor tight-binding model and an energy-level broadening seems to capture the essential features of observations
4. The paper is very clearly written for most parts and

1. The artificial lattice engineering, demonstrated in a systematic manner in the present paper, has been demonstrated previously in the same system ( albeit not in as detailed manner).
2. While the artificial lattices and their properties are demonstrated in a convincing manner, the lattice states are located much higher than the Fermi energy of the substrate. Therefore the artificial lattice remains completely unpopulated and it is not clear whether this system can be, even in principle, employed in nanoelectronics applications. This is not a shortcoming of the manuscript but rather the studied system.

The manuscript of J. Girovsky and collaborators demonstrate systematic fabrication of finite 1d and 2d artificial lattices by STM methods. These systems are realized by hybridized vacancy sites on chlorine monolayer on a Cu(100) surface. The authors can explain their findings by relatively simple tight-binding model attributing one electronic orbital per vacancy site. The same system has recently been employed as a platform for topological and flatband engineering and the results of the present manuscript seem to be largely in agreement with the previous work. While the basic band-formation mechanism has been demonstrated previously, the main contribution of the manuscript is the systematic approach to different 1d and 2d lattice geometries, further solidifying the engineering of artificial lattices with atomic precision.

The manuscript is clearly written with high-quality experimental results and appropriate theoretical modeling. However, before I suggest publication of the manuscript, I would like to see clarification/modification of certain aspects of the conclusions of the paper.

1) It is stated that both the checkerboard and stripe lattices can be accurately modeled by the same tight-binding parameters taking into account the first and the second neighbor hopping while simple first neighbor approximation proves insufficient. Does this conclusion hold universally for the considered 1d and 2d lattices geometries and a collection of few sites? If so, then one should make a stronger case and declare that the tight-binding model with the same parameters provides a good universal description of these systems (modulo broadening which can be added by hand). If the same description is not accurate for all cases (meaning that for same separations one has to use different hopping parameters), then the reason for that should be pondered/identified.

2) Fig. 4 and the paragraph above it explains how to make the connection to the momentum dispersion of the lattice states. Especially, the authors extract the energy as a function of the average <k^2>. This information, in turn, is employed in extracting the effective mass. The lattices are anisotropic so one would expect different masses in different directions. How is this anisotropy averaged for <k^2> and how is it reflected in the FFT of dz/dV maps for stipe lattice (only checkerboard is plotted)?

3) Regarding the comparison of effective masses for stripe lattice (sl) and checkerboard lattice (cl), I do not think it is necessary “counterintuitive” that the mass of cl is higher than sl lattice. The average mass is essentially proportional to average 1/(t*a^2), where t is a hopping element and a is the lattice constant. The masses then depend on the product on t and a^2 and their size is not a priori clear from the geometry. Furthermore, it should be calculated from the tight-binding model with the extracted fitting parameters for hoppings. I would like to see the data of Fig 4 f compared to “theoretical” value from the tight-binding model with NN and NNN hopping. It could turn out that the simple comparison to the tight-binding model reproduces the values of averaged effective masses. If it does not do so, then perhaps there is something counterintuitive in the situation but before doing that there is no way to tell. I would like the authors to complement the comparison of masses with the tb model and rephrase their findings about the masses if they follow from the above simple argument.

1 - Makes an important contribution to a rapidly developing, potentially ground-breaking topic: quantum simulations of condensed matter systems.

2 - Overall, the manuscript is clearly written and the arguments are easy to follow.

1 - the manuscript could/should provide some more details on various procedures.

The manuscript by Jan Girovsky et al. describes the use of Cl vacancies in a monolayer of chlorine on Cu(001) to generate electronic lattices of varies geometries. To realize this, the authors cleverly combine two facts 1: there is an electronic state associated with each vacancy and 2. the position of these vacancies can be controlled using the tip of a scanning tunneling microscope. The authors exploit the atomic scale precision by which lattices can be made, to study some very fundamental aspects, such as the influence of the magnitude of the coupling between neighbors on the electronic structure. The manuscript is clearly written and overall the arguments are easy to follow.

These type of experiments can be regarded as quantum simulations of condensed matter systems. These have the potential to elucidate long-standing issues in condensed matter physics. As such, I believe this manuscript should be published, pending some minor corrections detailed below.

1 - On page 2, in the second paragraph, first the authors state that a monolayer of chlorine atoms on Cu(100) leads to a shift in the substrate’s work function of 1.35 eV If I am not mistaken, Figure 3b in the Supporting material of Ref 19, indicates that the shift is 1.25 eV (not 1.35 eV).

2 - On page 2, last paragraph: It may be beneficial to some readers if the authors clarify what the numbers in brackets mean.

3 - Page 2, last paragraph: Please describe how the energetic position of the conductance band minimum was determined.

4 - Page 5, second paragraph: I found it difficult to follow the arguments here. The clarity of the manuscript would be significantly improved if the authors would also show (some of) the maps simulated without surface interactions in Figure 3 (currently in Supplementary Figure 2). In addition, I suggest the authors highlight the discrepancies between the experimental maps and the maps simulated without surface interaction in the figure.

5 - Page 6, last paragraph: please describe the procedure how <k^2> was calculated.

6 - Page 6, last paragraph: Typo: 'Fig. 4e shows .....' I believe it should be 'Fig. 4f shows .....'