Computing the eigenstate localisation length from Localisation Landscape Theory

1-A potentially interesting and powerful way of calculating the localization length from the localization landscape is provided

1-Missing more direct comparison with earlier studies of localization lengths with more traditional methods

2-Some conclusions and discussions could be more quantitative, while at the moment a qualitative discussion bases on a handful of data curves is given

3-The motivation for much of section 4 is unclear

4-The paper is not self-contained

Before discussing the contents of this paper, I have to say that the origin of this paper is somewhat unusual. The authors have posted a long (111 page) paper on arXiv (2003.00149), which they call a report, and state there that they plan to submit the contents of that report as five shorter articles to SciPost. It is unclear if they plan to, or are in the process of publishing the longer report, in which case it would seem that the shorter articles could not be published as original and new work.

This way of writing up their work in two steps is clearly felt in this article. In several places, the manuscript refers to the longer work (Ref. [52]) for details, often details that are essential for reproducibility and understanding of what is done in this paper, and are otherwise not contained in the paper. Sentences like "Details about our computational techniques and the numerical methods employed can be found in the the appendices of [52]," "a more detailed description of the system can be found in [52]," and "see appendix C of [52] for details" are found throughout the text. This makes the paper not self-contained, as it is not possible to even know what has been done without having to refer back to the longer report. There are more signs of this being taken from a longer report: before Eq. (6) the authors write "we repeat here the definition of the energy-dependent quantity known as the Agmon distance," despite this quantity never having been given earlier in the manuscript. I assume it was given somewhere earlier in Ref. 52. For a paper to be publishable in SciPost it needs to contain new results and to be self-contained so that one can read the paper and understand what was done without having to read another paper. It is not just details that we are talking about here, it is, for example, essential information on how the data being presented was obtained that one can not extract directly from this paper.

This brings me to the actual contents of the paper. The authors start by giving arguments, based on their exact digaonalization data, for why it can be hard to extract the localization length from exact eigenstates. I don't think there is anything that is really new in this part as it is known that extraction of localization length can be a nontrivial task, though in some cases there are good methods for doing this. This, however, motivates the problem that they want to address, namely using the localization landscape to extract a localization length.

In section 4 they discuss the effective potential $W_E = 1/u$ with $u$ the localization landscape obtained by solving $Hu = 1$. I think this section is supposed to be here to motivate that $W_E$ is a good potential to describe localization, something that has been done in several papers before. However, I fail to understand the logic and motivation for the authors approach. In order to try to show this to be the case, the authors decide to solve the Schrödinger equation with $W_E$ as a potential and compare the eigenstates and eigenenergies with those obtained with the potential $V$. They don't really give any motivation for why this should be a meaningful comparison. In my understanding, $W_E$ works nicely to describe localised states because somehow in the construction of it, quantum interference and resonance effects are taken into account. $W_E$ can then be used essentially as a classical or semi-classical potential. It doesn't make sense, to me at least, to reinsert this into the Schrödinger equation, since that is a completely different problem from the original one, and it will add quantum interference effects and resonances into the problem that where not originally there. And if one is solving the Schrödinger equation, then surely one would just solve the one with the actual potential $V$. I may have missed something here, or things have not been explained clearly, but if not I fail to understand what one can actually learn from this calculation.

Section 5 is the most interesting section of the paper. This is based on the Agmon distance given in Eq. (6) that was already introduces in Refs. 47 and 48, where it was shown to "control the decay of the wave function in regions where $E < W$." That is, the exponential decay of the wave functions comes only from regions where the effective potential is higher than the energy. Here the authors use this to extract a localization length. In order to achieve this the make the simplifying approximation that it is sufficient to consider only nearest neighbour domains in the effective potential and to estimate the localization length from the decay of the wave function, as obtained from the Agmon distance between these two domains, via Eq. (8). This seems like a rather strong approximation that one would really only expect to hold for states that are very localised and at most have significant support on two domains. The authors say as much, and relate that to the energy of the eigenstates, which they claim should be small in order for this to work. It is an interesting question, not solved in this paper, how one would use the Agmon distance for more general eigenstates to extract localization lengths. The authors discuss in the last section the connection of the problem to multidimensional tunnelling, though without any suggestions on how to go beyond what they have done.

I am not fully convinced about how successful this method has been for the authors in the example they have studied. Figure 8 compares the localization length from exact eigenstates and from the localisation landscape. A solid line representing when they agree is drawn through the data and the authors claim that there is good correspondence between the two. I don't know, there is some correlation but the variance is also rather high. It would have been good to have some systematic way of estimating more quantitatively how well their method works. Either by applying their method to systems where some more results are known, or by having a way to improve on the approximation they make in extracting the Agmon distance, for example by going to further domains as well.

I also worry a bit about the system sizes that are being looked at and the localization lengths being extracted. If I look at Fig. 9 and Fig. 10 I notice that the localisation length is a significant fraction of the system size, and for much of the data actually even larger than system size. It's hard to imagine one can get reliable localization lengths in this case. If I look at Fig. 7, it seems that going across the system one passed by only two or three domains in each direction. Would it not have been useful to study system sizes and or disorder strengths where the localization length $\xi \ll L$?

Overall, the data presented in section 5 is at most qualitative, it shows the one can extract some length which one can define as the localization length, and this length is somewhat correlated with what they get form exact diagonalization. When one tries to compare this with some expectations from localisation theory, such as Eqs. (9) and (10), then again the comparison is only qualitative (and in this case, data is not even shown). Perhaps this is all one could expect to have, but it would be good to have a more detailed discussion of this.

With all the above in mind, I feel that the paper would need to go through a major revision before it can be considered for publication in SciPost. In the requested changes section below I give some more detailed comments and questions.

General requests:
1-Make the paper self-contained. Give details that are essential for understanding what is done in this paper.

2-Provide a motivation and justification for approach taken in Section 4, reinserting $W_E$ into the Schrödinger equation.

3-If possible, provide a discussion of how one could improve the method beyond only nearest neighbor domains in a systematic way.

4-Is it possible to have a more direct comparison with earlier results on localization length?

Further comments/suggestions/questions:

5-On page 2, the Anderson model is said to be "also known as the tight-biding Hamiltonian." It is true that it is "a" tight binding Hamiltonian, but I have never heard it being called "the" tight-binding Hamiltonian.

6-At start of section 2, they describe the system in words and then refer to [52]. The system should be fully described here, and I think it might be useful to write down explicitly the Hamiltonian they use and the form of the disorder potential.

7-In section 3: "Increasing the width of the scatterers $\sigma$ also leads to stronger localisation (not illustrated)." Maybe I'm missing how exactly they are varying $\sigma$ but if $\sigma \rightarrow \infty$ wouldn't they get a constant potential with no localization? So at least at some point one would expect the localization to get weaker with larger $\sigma$?

8-In Fig. 1, it would be useful to add a label on the color bar, and it would also be useful if they plotted the disorder potential for more direct comparison with wave function.

9-In Fig. 2, is the disorder realisation (position of Gaussian potentials) kept fixed while the other parameters are varied? It's not obvious that they are but this would make most sense in the comparison being made.

10-Second paragrap in section 4: "...it appears that $W_E$ may, to a good approximation, be able to replace $V$ in the real Schrödinger equation, directly in the Hamiltonian. " As written above in the main report, I really don't understand this. Why would one want to do this? Why should it be true?

11-In section 4 there is extensive discussion of the "valley lines of $u$" . This is never explicitly defined, but should be.

12-In Figs. 5 and 6, ticks are not visible.

13-In Fig. 7 "candidate paths of least cost" are plotted as green and black lines. Is there any significance to the color?

14- On page 19, the authors mention that "once $E$ exceeds all saddle points, $\xi_E$ diverges to infinity and ceases to exist, at which point our curves must terminate." Since the authors are considering spineless fermions in 2D, all states should strictly speaking be localised in the thermodynamic limit. I guess this means that the above statement would be reflecting finite size effects. It's however not clear to me how one would recover localization here. Can one give some argument for how localization would appear in cases with large energy or will the method fail to detect localized states with energy larger than the amplitude of disorder?

15-I do believe it's important to make comparison with earlier predictions of localization theory where possible, as they authors do with Eqs. (9) and (10). However, no data is provided and only a qualitative discussion. Would it make sense to provide the data?

1 - calculation of localization length based on localization landscape theory
2 - proposed simple algorithm to compute approximate Agmon distance
3 - good literature review

1 - lack of comparison of the estimate of Agmon distance with its real value
2 - lack of comparison of obtained localization length with established methods
3 - lack of comparison of localization length obtained from $W_E$ and $V$

In this paper authors discuss how the effective potential $W_E$ obtained from the localization landscape $u=H^{-1}\mathbf{1}$ can be used in place of the real disordered potential $V$ in finding the properties of solutions to the Schrodinger equation. The main new result of the paper is a method of computing localization length based on the effective potential $W_E$. While the idea that $W_E$ can govern the decay of eigenstates through associated with it Agmon distance $\rho_E$ was already proposed in one of the original LLT papers [47], this is, to my knowledge, the first attempt to use this idea to compute localization length in practice. Authors present a convenient way of finding the approximate Agmon distance, which otherwise would require finding minimum over infinite number of possible paths. They also propose a new formula to calculate an estimate of localization length based on the Agmon distance computed in such a way.

While these are interesting results, the paper mostly lacks in comparing them with other established methods.

Firstly, authors only compare the approximate Agmon distance between two basins of $W_E$ with the decay of wavefunction between these basins (Fig. 8). While these two result show a good correlation, which is in favor of the proposed method, whether the estimate of Agmon distance is a good one still remains in doubt. It should be shown that computed estimate agrees well with the exact value. This could be done, for instance, for several values of energy $E$ and a couple of pairs of minima of $W_E$. Authors mention in section 6 that this is impractical when the calculation has to be performed so many times, however I think it should still be feasible for the sake of such demonstration.

Secondly, the computed values of localization length are not compared with results obtained with any established method, such as based on transmission coefficient as in Ref. [26]. This comparison is necessary to determine, whether the proposed method is actually useful.

Thirdly, as authors notice in section 4, replacing $W_E$ with $V$ works better as the strength of disorder increases. However, as the strength of disorder tends to infinity we have $W_E \approx V$. This raises a concern that the authors might be working in the regime where $W_E \approx V$, when computing quantities based on $W_E$ gives no advantage over simply using $V$. This concern would be avoided if authors also provided results of localization length based on $V$ as well as $W_E$, which would hopefully show that those based on $V$ are meaningless. Paper would also benefit if authors included a figure presenting a typical realization of disorder $V$ and the corresponding effective potential $W_E$ to demonstrate that the two are in fact qualitatively different.

In my opinion, this work has the potential to constitute an important development in the field of localization landscape theory, by making use of its ideas to calculate meaningful physical quantities. At this point, however, it is not clear whether this calculation is actually valid, as I expressed in the three points above. If these concerns are addressed, and the results hold up to such scrutiny, I would recommend this paper being published in SciPost Physics. Otherwise, it would significantly diminish interest of this work.

1 - Exact value of Agmon distance should be calculated for a couple of examples and shown whether it agrees with the proposed estimate
2 - Localization length should be calculated using established methods, such as bases on transmission coefficient (e.g. Ref [26]) and compared with the results obtained using the method proposed by the authors
3 - Localization length should be computed using Agmon distance based on $V$ (or it should be shown that such calculation makes no sense), to determine whether there is an advantage of using $W_E$ over $V$
4 - A typical realization of disorder $V$ and the corresponding effective potential $W_E$ should be presented to determine whether they are qualitatively different