Computing the eigenstate localisation length at very low energies from Localisation Landscape Theory

We would like to deeply thank both Referees for their insightful questions and comments, which have greatly helped us in improving our paper. Below, we address these methodically.

Referee 1:

The Referee suggests that we should compare the approximate Agmon distance obtained via our LLT-based method to the exact value, arising from solving the semiclassical equations. We have done this for a few examples and the results are shown in Table 1 of the new version of the manuscript. The approximate value is only slightly greater than the true minimal decay cost, and the approximate paths are very close to the exact, classical ones, indicating that our method is performing well.

In the process of answering this question by the Referee, we have spotted an oversight in the previous version of the manuscript, which has now been corrected: the decay coefficient that was used in the LLT calculation of the localisation length was not taken as the smallest integral over all candidate paths arising from LLT, but as an average over these. It was also this average quantity that was plotted in Fig. 8 of the previous version of the manuscript. This was done because the Agmon distance itself underestimates the true decay coefficient, being a formal lower bound. On the other hand, we found that by averaging over all the paths connecting the two domains through the saddle points, we could in fact approximate the real decay coefficient much better. The first author apologises and takes full responsibility for the lapse in memory which led to this inaccuracy in the description given in the previous version in the manuscript. The new version both corrects this issue and comments on it from a physical point of view. We thank the Referee for helping us find and correct this mistake.

Second, the Referee proposes that we should compare our calculations of the localisation length to established methods, pointing out in particular the Transfer Matrix Method (TMM) which is exact and has been used successfully for discrete systems in all dimensions. Making such a comparison is indeed crucial, but we cannot use the TMM because we are not aware of any way to generalise it to 2D continuous potentials. In fact, we do not know of any exact (or at least fairly accurate and reliable) methods that could tackle the problem at hand, except for exact diagonalisation and direct integration of the time-dependent Schrödinger equation.

We have therefore performed time-dependent simulations, initiating a translating Gaussian wavepacket outside the disorder, choosing a large enough system to observe exponential decay, and extracted the associated length scale from the density profiles at long times. This was done in a parameter regime and at energies where the LLT localisation length was smooth and monotonically increasing, not at all affected by the noise seen at high energies (see inset of Fig. 9 in the previous version of the manuscript). We found that the localisation length from time-dependent simulations was considerably greater than our LLT value, by up to as much as an order of magnitude. Being rather surprised by this, we investigated the underlining cause by carefully inspecting the eigenstates, and discovered that our method is limited to much smaller energies than we previously believed. We have adjusted the entire paper to reflect this, as well as added a new section (section 6 in the new version of the manuscript) that described how and why the LLT calculation fails at higher energies. In short, the eigenstates are not governed by pure decay anymore: the quantum tunnelling picture in the effective potential breaks down and the Agmon inequality [equation (7) in the new version of the paper] ceases to be relevant. In this regime, it is best to think of the eigenstate localisation as arising directly from quantum interference effects. Here, we see several mechanisms come into effect that increase the localisation length beyond the prediction based on the clean tunnelling picture. We thank the Referee for asking this question, which helped us correct a serious misconception regarding the applicability regime of the new LLT method.

In terms of providing a comparison of LLT to a different approach at very low energies where the former is applicable, we have performed simulations with very strong noise such that the energy range over which the LLT calculation is correct is sufficiently wide to fit in a slowly moving Gaussian wavepacket. In this case, we have likewise found that time-dependent simulations yielded a larger localisation length than LLT. Having investigated this phenomenon, we realised that when one adds empty “reservoirs” on either side of the noisy region (at x=0 and x=L), the valley network is modified at these edges of the system. In particular, localisation is weaker here compared to the interior of the system due to the modified boundary conditions. When the disorder is very strong and the localisation length is smaller than the typical domain size, these edge effects strongly influence the outcome of the simulation, as the wavefunction only samples these very edges as it rapidly decays exponentially on its way through the noise.

Thus, a meaningful comparison of the LLT method to time-dependent simulations remains elusive (we have included the discussion above in the paper to clarify the situation). However, the comparison to eigenstates at low energies, where the tunnelling picture applies, is not meaningless. Since the domain “diameter” calculation is very transparent at these low energies, and the decay coefficient has been shown to reflect the behaviour in the eigenstates well, we see no reason to question that the ratio of the two indeed gives the localisation length correctly.

Next, the Referee raises concerns over the similarity of V and W_E in the regime where the substitution of the former by the latter works well. This has motivated us to directly compare the two potentials and clarify the connection between them. The effective potential has peak ranges in the same locations where the physical potential has scatterers. The difference is that the effective potential is a “smoothed out” version of the physical one. Whereas V has clear gaps between the scatterers (at reasonable fill factors and scatterer widths), W_E has closed potential ranges encircling domains, which allows for classical trapping in these potential minima. The effective potential also has lower peaks than the physical, as well as a roughly constant background value away from the scatterers, resulting from the smoothing procedure. This insight has already been provided in one of the earlier LLT publications (Ref. [48] in the new version), but we agree that including this information and discussion in the paper is essential and thank the Referee for the idea.

The Referee further suggests that we should try to compute the localisation length (following our LLT prescription) with V in place of W_E, to check whether LLT provides us with any advantage at all. This is a good idea, but because the domains are classically connected in V (due to the gaps between the scatterers) such that the Agmon distance is always zero, even at vanishing energy, a semiclassical tunnelling picture would predict no exponential decay in V. Moreover, one needs the localisation landscape u to find the domains in the first place, as an analogous calculation with 1/V would not give closed domains in the valley network (c.f. gaps between the scatterers). This discussion has been added to the paper.

We have followed the Referees advice of illustrating V, side by side with W_E, and agree that it is constructive to the paper. Many thanks, once again.

Referee 2:

First, we have made the paper self-contained by including very brief descriptions concerning the algorithms used. We agree that it improves the readability considerably and thank the Referee for the suggestion. Please note that we do not plan to publish the long report on arXiv as an original article, because it is far too long for peer review.

Next, the Referee inquires as to the logic behind section 4, in particular pointing out that W_E is also a disordered potential and will induce quantum interference effects of its own. Since W_E is different to V, these could be expected to be different, in which case one wouldn’t expect the substitution to work well. Instead, W_E should be thought of as a classical potential.

We essentially agree with the Referee’s comments, outlined above. However, we believe that the eigenstate comparison is a meaningful exercise, because later in the paper, we apply a semiclassical approximation to tunnelling in the landscape W_E, which is derived by starting from the Schrödinger equation with W_E as the potential. It seems reasonable to us to first ensure that the exact eigenstates are similar before proceeding with the approximation. This has been made clear in the current version of the article. As for the time-evolution in the two potentials, we now know that the tunnelling picture breaks down at quite low energies (see response to Referee 1), well below the energies of the wavepackets used in the previous version of the manuscript. This comparison therefore indeed does not serve any purpose, and has been removed. As a final remark, having realised that beyond some fairly low energy cut-off, the tunnelling picture breaks down and the localisation of these higher eigenstates should be attributed directly to Anderson localisation, it is interesting that the first handful of these beyond-tunnelling eigenstates are actually similar between V and W_E. In other words, quantum interference effects are also similar at low energies, simply due to the fact that W_E bears close resemblance to V, in particular regarding the peak positions. This comment has also been added to the paper. Many thanks to the Referee for the question.

Next, the Referee comments on the approximation of considering tunnelling only between nearest-neighbour domains when computing the localisation length via our new method. We would like to point out that as far as we are aware, the reduction of the problem to adjacent domains does not introduce an extra level of approximation. In the regime where the tunnelling picture applies, such a decomposition will allow to extract the average cost of crossing a single domain wall, and decay over longer distances can then be composed of several such events. In this sense, our calculation is local in nature, and largely independent of system size (more is said about this below). A discussion to this effect has been added to the paper.

The Referee then insightfully comments that he/she expects the method to only work for very low energies and very strongly localised eigenstates. We now know that this is indeed the case, and that at higher energies, the classical trapping in the effective potential picture is no longer helpful. Section 6 in the new version of the manuscript is dedicated to this discussion. We are, however, not aware of any way one could use the Agmon distance to obtain the localisation length in the regime where the mechanisms of section 6 come into effect, primarily because we do not think the eigenstates can be captured by tunnelling in W_E any longer and it is best to think of the physics as quantum interference directly. We greatly thank the Referee for these questions and remarks.

Regarding the new top panel of Fig. 7 (the old Fig. 8), indeed the scatter is strong, but we perform extensive averaging during the calculation of the localisation length, so we do not think this is problematic. The sufficiency of the degree of averaging can be checked by confirming that ξ_E varies smoothly and monotonically with parameters.

We agree that it would have been excellent if we could compare our method to another, established calculation, but as discussed in the reply to Referee 1, to the best of our knowledge, there is nothing else we can do. We have also searched the literature for continuous 2D systems where accurate results for the localisation length are known so that we could apply our method to these as a test, but have not found such examples. Instead, we have added this idea to the future work list, in case an example is uncovered at a later time.

Next, the Referee raises the excellent question of system size compared to the localisation length, and whether the results are reliable if the latter is of the same order or even larger than the former. In principle, as mentioned above, our calculation is local: we only need the system to be large enough to fit in a few domain pairs, such that we can extract several decay coefficients and domain areas. In other words, averaging over many small systems is equivalent to averaging over fewer larger systems. In practice, there are finite size effects that change the structure of the valley network when the system size is comparable to or smaller than the average valley line spacing (studied in Refs. [53,65] in the new version of the manuscript), and we must keep in mind that the system size determines when the noise in the ξ_E(E) curve begins (see Fig. 9 of the previous version of the paper). Importantly, however, we now know that our method stops working long before we reach the system-size-limited noise. Therefore, we have replaced the old Figs. 9 and 10 by the new Fig. 8, which shows only E=0 data, confirmed as meaningful by exact diagonalisation. We have clarified these issues in the article and thank the Referee for the questions.

The Referee points out that the old Fig. 7 (now Fig. 6) shows a systems with only a handful of domains. We chose to use this example so that the details of the network can be clearly seen by eye, but we agree with the Referee that if we increase the strength of the disorder (and thus decrease the localisation length), the validity range of our computation grows, which indeed makes such a regime more useful to study.

As for the comparison of our data with the analytical formula (10) [previously (9)], we have now fitted the LLT data points shown in Fig. 8 and added the curves to the plot. However, we can no longer check the energy dependence of the coefficients in equation (11) [previously (10)] because we now know that our LLT calculation is only valid at very low energies. In addition, we have realised that it is the Boltzmann mean free path which enters equation (10) [previously (9)], and not the scattering mean free path, which means that we would need to compute the scattering cross section of the Gaussian scatterers – not a simple task. Since the formula is very approximate and accurate LLT results are not available above a fairly low-energy cut-off anyway, we decided against developing this further.

As for the remaining (minor) suggestions, points 5-15 on the list of changes in the report, we are grateful to the Referee for bringing these to our attention. We have accepted the Referee’s advice on all points (where possible). The ones that require an answer and have not already been discussed above are addressed presently:

7 – Indeed, increasing either f or σ excessively causes scatterer overlap and a decrease in randomness. A comment to this effect has been added to the paper. 8 – We are deeply sorry, but we genuinely do not understand what label the Referee would like to be added to the colour bar. Instead of plotting the potential V for comparison with the eigenstates (V is now shown in the new Fig. 4 for a different noise realisation), we have added a comment that describes the fact that the weight of the eigenstates lies inside the LLT domains, with the amplitude forced down at the valley lines. Since valley lines correspond to ridges in the effective potential, which is turn occur at the positions of the scatterers in V, we believe this conveys the same information. 9 – No, completely different noise realisations are used, as the fill factor changes between the panels as well. This has been clarified in the article. 12 – Thank you for pointing it out. These figures have now been removed, but we are afraid that there is nothing we can do about this for surface plots plotted in Matlab in general. 13 – The colours used for the valley lines (red and blue) and the candidate paths (green and black) do not carry any distinctive meaning. We simply use different colours to make it easier to see the structure of the network. This has been clarified in the relevant figure captions. 14 – The mobility edge predicted by LLT is indeed not a finite size effect (although it is a natural assumption, one that we ruled out by increasing system size and observing the effect). We have investigated it thoroughly in [53, 65] (numbers correspond to citations in the new version of the manuscript). We have concluded that LLT stops being useful at higher energies. Quantitatively, this happens beyond the narrow energy range where the eigenstates obey pure tunnelling in the effective potential. Qualitatively, as long as there are valley lines remaining, the eigenstates will be pushed down at their positions, so the valley network still provides at least some useful information. However, past the peaks of W_E (note: not V), there is no more eigenstate confinement predicted by LLT, which then loses its value. This does not mean that the eigenstates cannot be suppressed, as LLT theory says that if there are effective valley lines, then the eigenstates are suppressed, but it is not an “if and only if” statement (this was first pointed out to us by Jan Major, whom we thank for his contribution). 15 – We are now showing some limited data in the new version of the paper, but it is inconclusive in terms of supporting or negating the analytical formula (10) [numbering refers to the new version].

We would once again like to deeply thank the Referees for their time, effort and excellent questions and suggestions which we believe have helped to improve our article significantly.

Minor rephrasing (e.g. for better flow) etc. not listed if the content and meaning are not changed.

Title:
- Added “at very low energies” to reflect calculation applicability range.

Abstract:
- Made the description of section 4 more concrete, leaving only the eigenstate comparison.
- Added conditions of applicability of our method.
- Added a description of the new section 6, discussing the breakdown of our method at higher energies.

Introduction:
- Added the new Ref. [2] in several places.
- Added a short paragraph just before the introduction of LLT that highlights the unavailability of other, conventional methods that could accurately capture our system, apart from direct time-integration of the Schrödinger equation.
- Clarified in several places that our method is only effective at very low energies.
- Modified the description of what is done in section 4, sharpening the motivation and focusing on the eigenstate comparison. [this was done in two places]
- Explicitly wrote out the condition on the potential for LLT to apply.
- Replaced a reference to the Agmon distance by one to the decay coefficient (the two are not the same).
- Added a description of the new section 6 [in two places].
- Removed references to the Report on arXiv, background LLT citations, and external technical appendices, as these are no longer needed.

System of Interest:
- Gave a more complete description of the system, including writing out the Hamiltonian.

Exact diagonalisation:
- Added a note regarding the unavailability of other useful methods for tackling our problem as motivation for considering exact diagonalisation, with a mention of time-dependent simulations that are mostly left for another paper.
- Added a brief note on the algorithm used.
- Clarified that in Fig. 2, the noise realisations differ from panel to panel.
- Added a discussion of the limit of very large fill factor or scatterer width, in which case the randomness deceases as scatterers overlap, and localisation weakens.
- Removed the discussion of possible classical trapping in V, as a better version is now incorporated into the next section (see below).
- Figure 2 caption: added a note that the stripes seen in the right-hand-side panels are artefacts of the algorithm.
- Last paragraph of section 3: added another argument for why the variance should be treated with caution: the presence of secondary bumps in the eigenstates increases the variance, even if their size and decay rate are identical to the main bump.

The effective potential:
- Definition of ‘valley lines of u’ included upon first mention.
- Clarified the motivation behind the section.
- Added a note on the numerical algorithm used.
- Described the connection between V and W_E and discussed the (im)possibility of classical trapping in V, and the fact that WE will also give rise to Anderson localisation on account of being a random potential.
- Added a note relating the eigenstates in Fig. 1 to the valley lines, and through these, to the effective and physical potentials.
- Added the new Fig. 4 to illustrate V and W_E.
- Added some forward references to the new section 6.
- Removed the qualitative discussion justifying the existence of the energy shift seen in the new Fig. 5, and replaced with a citation to one of the original LLT papers where it is treated rigorously.
- Removed the old Figs. 5 & 6 and all discussion thereof, as wavepackets lie above the energy cut-off where our calculation stops being meaningful.
- Added a note to link the better transmission in the effective potential to the lower peaks it has compared to V.
- Modified the concluding paragraph to account for changes to the section, and attempted to make the logic of the exercise clearer.

Eigenstate localisation length:

Preamble:
- Clarified calculation only works at very low energies.
- Removed reference to technical appendix on arXiv.
- Added a discussion about the unavailability of other methods to compare our calculation to, and briefly described how far we can get with time-dependent simulations in this regard.

Outline of the LLT method:
- Re-emphasised that V will not allow for classical trapping under moderate levels of disorder.
- Added a note about the Agmon inequality, to highlight that it is only applicable in situations involving tunnelling, and that previous LLT work has demonstrated its high levels of performance in a 1D example.
- Modified the text to make it clear that the Agmon distance need not give the correct decay rate (as an a priori assumption).
- Included a comparison of our approximate calculation of the Agmon distance to the (numerically) exact number for a few examples, given in Table 1.
- Explained that considering only nearest-neighbour domains does not introduce an extra level of approximation, and that our calculation is local in nature.
- Described the test of the Agmon distance as a way of quantifying the decay coefficient between domains (shown in the bottom panel of Fig. 7 in the new version), finding that it underestimates the true value.
- Motivated and introduced the “mean” Agmon distance, taken as the average integral over all candidate paths from LLT, and provided a short physical discussion of it.
- Pointed out that an analogous calculation starting from V directly (avoiding LLT) is not possible.

Test of decay constants:
- Clarified that the calculation is only applicable at low energies for strongly localised states with pure, straight-forward decay only.
- Added a second panel to Fig. 7 to check the performance of the Agmon distance proper as an estimate of the decay coefficient, together with a description of this figure.
- Added a discussion of how one could use time-dependent simulations to extract the localisation length for comparison to our method, and described the outcome of such a comparison, referring the reader to the new section 6 for an explanation of the observations.
- Have corrected throughout instances where \bar{ρ}_E was previously erroneously labelled (and described) as ρ_E.

Effect of parameters:
- Removed the old Figs. 9 & 10, and discussion of these, because we now know that our calculation is not meaningful for most of the energy range shown, as is now explained in the text.
- Added the new Fig. 8 which focuses on zero-energy results and serves to demonstrate the effect of fill factor and scatterer height, while variation with system size is described in words.
- Replaced the extensive discussion of how our calculation becomes inaccurate when limited by system size by a brief description, as we now know that the computation ceases to be relevant at much lower energies.
- Energy dependence is no longer illustrated, but merely verbally described.
- Changed the way our results are compared to equation (10) [in the new version of the paper]. Since we can no longer do the fits for various values of the energy (the range of applicability is very small and changes with parameters), we only handle the E=0 case and display the best fit curves for it exclusively. Taking into account that we do not know (without performing additional calculations) the scattering cross section of a single Gaussian bump, we have no way of knowing the energy dependence of the Boltzmann mean free path. Moreover, we are not able to extend our calculation to non-zero energy reliably, so we cannot judge whether equation (10) is supported by the numerical data or not. We can only say that reasonable fits are possible at E=0 as a function of fill factor.

Breakdown at higher energies:
This entire section is new.

Multidimensional tunnelling:
- Added the differential equations in parametric form that need to be solved to obtain the true semiclassical minimal path.
- Elaborated on the need to guess the initial direction of the classical trajectory at least twice if the points of interest are separated by one or more turning surfaces.
- Added a short discussion of the fact that our “mean” Agmon distance is able to capture the true decay rate, rather than providing a lower bound.
- Removed the idea of applying the path decomposition method to our system as we do not think it would be practical.

Conclusions and future work:
- Emphasised that our calculation is limited to very low energies.
- Added a note on making use of the “mean” Agmon distance and the advantage it provides.
- Added a description of the new section 6.
- Removed mention of the comparison between time-evolution in V and W_E.
- Added the future work idea of perhaps repeating the LLT calculation for a system where the Green’s functions method would be applicable.