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Computing the eigenstate localisation length at very low energies from Localisation Landscape Theory
by Sophie S. Shamailov, Dylan J. Brown, Thomas A. Haase, Maarten D. Hoogerland
- Published as SciPost Phys. Core 4, 017 (2021)
|As Contributors:||Sophie Shamailov|
|Arxiv Link:||https://arxiv.org/abs/2008.05442v3 (pdf)|
|Date submitted:||2021-05-18 06:42|
|Submitted by:||Shamailov, Sophie|
|Submitted to:||SciPost Physics Core|
While Anderson localisation is largely well-understood, its description has traditionally been rather cumbersome. A recently-developed theory -- Localisation Landscape Theory (LLT) -- has unparalleled strengths and advantages, both computational and conceptual, over alternative methods. To begin with, we demonstrate that the localisation length cannot be conveniently computed starting directly from the exact eigenstates, thus motivating the need for the LLT approach. Then, we confirm that the Hamiltonian with the effective potential of LLT has very similar low energy eigenstates to that with the physical potential, justifying the crucial role the effective potential plays in our new method. We proceed to use LLT to calculate the localisation length for very low-energy, maximally localised eigenstates, as defined by the length-scale of exponential decay of the eigenstates, (manually) testing our findings against exact diagonalisation. We then describe several mechanisms by which the eigenstates spread out at higher energies where the tunnelling-in-the-effective-potential picture breaks down, and explicitly demonstrate that our method is no longer applicable in this regime. We place our computational scheme in context by explaining the connection to the more general problem of multidimensional tunnelling and discussing the approximations involved. Our method of calculating the localisation length can be applied to (nearly) arbitrary disordered, continuous potentials at very low energies.
Published as SciPost Phys. Core 4, 017 (2021)
Author comments upon resubmission
Editor: Many thanks for pointing out the paper [Phys. Rev. B, 101, 22, 220201(R) (2020)]. We have added a short discussion of it at the end of section 6, as well as a related publication [Phys. Rev. B, 101, 8, 081405 (2020)]. We have also created a Zenodo repository for the code relevant for the work in this paper, which is linked to the resubmitted manuscript.
Referee 1: We thank the Referee for his/her feedback. We would like to comment that while our method does require visualising the eigenstates to determine the range of energies where it is applicable, it is still “useful” in that it computes the localisation length, which cannot be done efficiently from the eigenstates alone.
Regarding the missing entry in Table 1, we thought it was “fair” to acknowledge that such things can happen at the given accuracy, as this is the precision we have used for all presented calculations. However, taking the the Referee’s advice, we have increased the precision and attempted to find the true semiclassical path again. We found that the accuracy had nothing to do with it: the formal minimal path is simply very difficult to find (certainly a pathological example – this does not happen often), and we were again unsuccessful in this task. The reason for the missing entry has been updated in the paper. Many thanks for the inquiry.
Referee 2: We thank the Referee for his/her comments. We have added a quantitative comparison between LLT and time-dependent simulations, as suggested, and agree that it improves the paper.
List of changes
1) Added a short discussion of [Phys. Rev. B, 101, 22, 220201(R) (2020)] and [Phys. Rev. B, 101, 8, 081405 (2020)] in the last paragraph of section 6, addressing generalisations of LLT, including the ability to handle higher energies.
2) Linked a code repository to the paper upon resubmission.
3) Removed mention of accuracy/precision in the reason for the missing entry in Table 1 given in the caption.
4) Included a description of the system geometry used for time-dependent simulations, a short note on the numerical method employed for this purpose, as well as the functional form of the initial condition at the end of section 2.
5) Added the formula for the energy distribution of the above-mentioned initial condition [the new eqn (11)].
6) Added a quantitative comparison between LLT and time-dependent simulations for two examples, at the end of sections 5.2 and 6.
7) Administrative change: added a current address for the second author (D.J.B.).
Submission & Refereeing History
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