Localization versus incommemsurability for finite boson system in one-dimensional disordered lattice

1- general proofreading. If this cannot be done consistently by the Authors, ask a specialist in the field to read the article. The Manuscript is full of confusing sentences. Just reading the Abstract "covering the full interaction ranging" - rephrase "we apply .. Hartree" - Hartree is a person, probably Authors apply the "Hartree method" instead "is done by the measures", no, is done by "measurements", as "measure" is a quantity of something "observe Mott", Mott is a person, probably "Mott phase" is observed "fermionization limit when the pairing bosons fragment" - is badly written I do not continue with the main text, which as well is full of confusing and badly written sentences.

2 - "the incommensurate frequency ratio ... will be necessarily be approximated by a rational number due to finite numerical precision." I disagree that this is the limiting effect, for a "double" the rounding precision is $2^{-53} \approx 10^{-16}$, instead the real limitation comes from being able to simulate small that ten particles with this method.

3 - Authors use $\lambda$ to denote the coupling constant (explicitly introduced as a coefficient in front of the delta function in the interaction potential). I believe the more common notation is $g$ for the coupling constant. Instead, $\lambda$ is often used to denote eigenvalues of the one-body density matrix $\rho^{(1)}$, while Authors use $\rho_i$.

4- I find that the discussion around (6) is not complete, "to fragmented Mott state ($\Delta \to 1/S$)", what is $S$ here? I do not find where it was defined. Moreover, "\Delta\to 1" would require $\rho_0 = N$, that is full Bose-Eintein condensatio.n Instead, due to the Hohenberg-Mermin-Wagner theorem, fluctuations destroy Bose-Einstein condensation in one dimension, so it should be clarified how $\Delta\to 1$ condition can be satisfied

5 - Units .The Authors comment that the energies can be measured in some arbitrary unit $\hbar^2 / (m L^2)$. Later, Authors use recoil energy $E_r$ when giving the strength of interactions. At that point, $E_r$ serves as a unit of energy, so there is no point in introducing a second energy scale. The coupling constant $\lambda$ is given in units of energy $E_r$, which has incompatible units. I strongly advise checking all units, and reporting them explicitly, i.e. $x/L$ instead of $x$, ... On the other hand, $\lambda =0$ does not require writing $\lambda = 0 E_r$.

6 - figures. Some figures are not readable, i.e. Fig 1 should have 10 lines, although only 3 lines are clearly visible. Either a different scheme for presenting the results should be used, or if some lines overlap, this should be commented.

1. • figures. What is actually shown? Figure 3c says that the horizontal axis shows "disorder", what is meant by that? Figs 1,3, 5 show "The one-body density $\rho(x)$" which sometimes "overlaps with the density". At that point, I am confused, is that a density profile? Then what does it mean that it overlaps with the density? Is that density related to the eigenstates of the one-body density matrix? As Authors do natural orbital analysis, this might be the case. So make it clear what is actually shown.
2. • "When a single eigenvalue is macroscopic, the state is superfluid". I think this is the definition of a "Bose condensate" rather than of a superfluid.

3. Fig 6, I do not understand why the data shown in Fig a is not symmetric.

Reject

1) Unveils the correlated character of few-body states in bichromatic finite optical lattices 2) Provides an overview for different filling factors 3) Uses relevant observables to quantify many-body effects

1) Missing explanations and proper justification in some places 2) More details can be included about the analysis of beyond Hubbard Physics and the symmetry breaking at incommensurate fillings 3) The correlated disorder statement should be clarified 4) Needs improvement of the presentation 5) The Figures need to be improved

1) I suggest to shorten the abstract focusing on the most important results. Also, in the abstract the notation of primary lattice is currently unclear to non-experts. 2) In the Introduction, the statements “…exhibits incompressible Mott insulating phase, whereas for the incommensurate filling an extended superfluid…” and “…but non-coherent phase-Bose Glass phase” need to be accompanied by suitable references. 3) In the Introduction, the sentence “Whereas Ref. [29] presents the expected features…” needs to be rephrased. Currently, it is difficult to follow. 4) The references regarding the experimental realization of finite sized systems in the Introduction need to be updated. At the present stage they do not reflect relevant systems to the one studied. 5) On page 2, the statement “…is justified due to poor population” needs to be rephrased and properly justified. 6) I find that the motivation of the present work needs to be significantly revised and enhanced. Currently, revolves around the argument how disorder affects delocalization. However, to my opinion there are further outstanding questions such as how known phases like SF and MI are impacted, whether beyond Hubbard physics emerges, what are the emergent correlation patterns and so on. Moreover, in general, besides the motivation a large amount of relevant few-body references in finite one-dimensional optical lattices — some of them also reporting beyond Hubbard physics and a few of them using the MCTDH method — are missing. This is another very important issue missing from the manuscript (i.e., omission of relevant few-body references). I suggest to add such references both in the motivation of the work but also within the discussion in the main text. A note here is that to observe beyond Hubbard Physics one does not need to necessarily rely on strong interactions as it is currently implied in the text and has to be corrected. There are numerous such examples in the literature both in the weak interaction limit and in the dynamics. 7) I suggest to include in the Introduction a brief description of the basic states discussed in the text, i.e., SF, MI and fragmented MI. This will greatly help non-expert readers to follow the discussion. 8) To my understanding the used optical lattice is not disordered as it is claimed throughout the text but rather a bichromatic optical lattice. This has to be clarified. If the authors prefer to use the term disorder, this need to be properly and clearly justified. Along these lines, the statements “…correlated disorder on top of the periodic primary lattice.” and “correlated quasiperiodic disorder” are not clear and if used need clarification. What type of correlations the authors refer to here? 9) On page 3, the sentence “We repeat the computation by …” needs rephrasing to be easier accessible. 10) It would be good to have a reference for the spectral decomposition of the one-body reduced density given by Equation (5). 11) The origin of localization as Vd and \lambda increase does not become clear. Is there any energy argument for this behavior? Does the second lattice suppress hoping of specific sites? Please elaborate. Also, the configuration depicted in Figure 2(i) is not clear. Why there is still population of the most outer sites while inner ones are depleted and then only the three central ones are populated? 12) The origin of the symmetry breaking of the density in Figures 5 and 6 and the correlation functions in Figures 7 and 8 for systems with filling larger than unity when dimers are forming is not clear. In particular, I do not foresee a reason that symmetric wells do not exhibit the same population. Even in the case of delocalized particles the symmetry on the many-body wave function level should be preserved. Please explain.
13) The discussion of higher-band and hence beyond Bose-Hubbard physics is suppressed. How the authors identified the presence of higher-band states? Did they follow a specific analysis? To my understanding MCTDHB orbitals do not necessarily coincide with Wannier states. Please elaborate. 14) The colormaps of the one- and two-body reduced matrices in Figures 2, 4, 7, and 8 do not contain values. What are the lower and upper values of these heat maps? Along the same lines, the legends (a), (b), … in Figures 6 are scattered in different places in the different panels, while in Figures 1(a), 3(a), and 5 are missing. Please revise accordingly. 15) Which kind of basis functions were used for the grid in the MCTDH simulations? 16) There are many grammatical errors in the main text, e.g. “several exotic quantum phenomenon”, “areas in disordered lattice”, “onsite energies of the lattice, resulting lattice becomes”, “When bosons in the periodic lattice shows”, “lattice depth and different incommensurate setups are created”, “unseen features is observed” etc. These are only a few examples. Please check carefully the entire text and amend accordingly.

Ask for major revision