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Engineering spectral properties of non-interacting lattice Hamiltonians
by Ali G. Moghaddam, Dmitry Chernyavsky, Corentin Morice, Jasper van Wezel, Jeroen van den Brink
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Submission summary
| Authors (as registered SciPost users): | Ali G. Moghaddam · Corentin Morice · Jasper van Wezel |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202107_00075v1 (pdf) |
| Date submitted: | 2021-07-28 19:44 |
| Submitted by: | Moghaddam, Ali G. |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We investigate the spectral properties of one-dimensional lattices with position-dependent hopping amplitudes and on-site potentials that are smooth bounded functions of position. We find an exact integral form for the density of states (DOS) in the limit of an infinite number of sites, which we derive using a mixed Bloch-Wannier basis consisting of piecewise Wannier functions. Next, we provide an exact solution for the inverse problem of constructing the position-dependence of hopping in a lattice model yielding a given DOS. We confirm analytic results by comparing them to numerics obtained by exact diagonalization for various incarnations of position-dependent hoppings and on-site potentials. Finally, we generalize the DOS integral form to multi-orbital tight-binding models with longer-range hoppings and in higher dimensions.
Author comments upon resubmission
we would like to thank all referees for their reports and constructive criticism. We have addressed all the points in our replies and have modified the manuscript accordingly. We believe that thanks to the referees' suggestions we were able to further improve the quality of the manuscript. For convenience, we have highlighted all the changes in blue.
Best regards,
The Authors
Current status:
Reports on this Submission
Anonymous Report 2 on 2021-8-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202107_00075v1, delivered 2021-08-23, doi: 10.21468/SciPost.Report.3432
Report
With their revised version, the authors have addressed some of the previous issues. Consequently, the manuscript has been improved and one might proceed towards publication in one of the SciPost Physics journals.
Nevertheless, my concern about the partial Wannier functions (PWFs) remains unchanged. Indeed, the sketch in Fig. 1(a) is highly suggestive of damping at the boundaries of the PWFs, something that is not present in the explicit form of Eq. (4). Some related questions or comments are:
1- Why should one impose the usual quantisation condition on the $\vartheta$s in Eq. (4) if the periodic property is not used at all?
2- Accuracy is controlled by $1/M_c$, i.e., the number of boundary terms. The example shown in Fig. 4(c) looks Ok, but one should nevertheless note that the authors needed to push things to $M_c=40$ to get to this level of accuracy. Other choices of PWFs might be more efficient.
3- I am not sure if the comment just before Eq. (43) about the corrections scaling as $M_c^{-1}$ also in higher dimensions is correct. From the surface-to-volume ratio, I would estimate $M_c^{-1/d}$ scaling of the error, which could be quite bad in $d=3$, or even already for $d=2$.
Requested changes
1- Make "Partial Wannier" sketch in Fig. 1(a) consistent with Eq. (4), or at the very least add a caveat about absence of damping in Eq. (4) - preferably in the caption of Fig. 1.
2- Double-check, and if necessary correct, comment about scaling of corrections before Eq. (43).
3- In the first sentence of the Introduction (second line), I think that there is a noun missing, e.g., "particles" or "systems" after "interacting".
4- At the beginning of section 6, the justification of the statement "Such a power-law form has been motivated previously by the fact that the resulting low-energy (long-wavelength continuum) physics corresponds to a 1D Dirac equation subjected to a gravitational background that possesses a horizon for $\gamma \ge 1$" is unclear. In particular, does "previously" refer to a previous statement in the manuscript or to previous literature? In any case, I think that it would be safer to add a reference.
5- Figs. 2-4 may have been scaled down too much. In particular in my printouts, tick labels are very difficult to read. Maybe this can be fixed during production, but if the authors have another look at their manuscript anyway, they might also have a look at this issue.
6- Ref. [17]: if the authors wish to cite themselves, they should provide a link to their manuscript. I suspect that this is Phys. Rev. Research 3, L022022 – Published 9 June 2021, but it is up to the authors to confirm.
Anonymous Report 1 on 2021-8-2 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202107_00075v1, delivered 2021-08-02, doi: 10.21468/SciPost.Report.3330
Report
The authors have given a satisfactory response to my questions/comments. But there are two statements in the manuscript that I believe are still misleading, and should be corrected before publication.
The first is on p. 9: "The notion of higher-order van Hove singularities, with power-law form ... opposed to the logarithmic behavior at an ordinary band-edge van Hove singularity." I believe the authors mean "band-center" instead of "band-edge" here, such as the $\omega=0$ logarithmic singularity in the DOS of the 2D nearest-neighbor hopping model on the square lattice.
The second confusing statement is on p. 12: "Equivalently, a random tight-binding model with a uniform distribution of all hoppings among all sites has a constant DOS." The statement of constant DOS contradicts the preceding sentence, where it is (correctly) pointed out that the DOS follows the Wigner semicircle distribution Eq. (37). I suspect they authors mean "has a finite DOS at $\omega=0$."
Requested changes
Correct the misleading statements on p. 9 and p. 12.
Author: Ali G. Moghaddam on 2021-08-09 [id 1646]
(in reply to Report 1 on 2021-08-02)
We thank the referee for their report and for pointing out the two statements which need to be corrected.
We agree with the referee and will fix them in the next revision.
Author: Ali G. Moghaddam on 2021-08-26 [id 1711]
(in reply to Report 2 on 2021-08-23)The referee writes:
Our response: We thank the referee for their report and constructive comments. We agree with the referee that the periodicity constraint is not essential to the present formulation. In principle, one can choose a different quantization for $\vartheta$'s. But the plane waves with PBC are a natural choice as they make the treatment easier. The referee writes:
Our response: It is true that the accuracy is determined by $M_c$, however, it should be noted that the corresponding term, Eq. (14) actually accounts for the correction of ignoring spatial variations inside each small chain. It is thus a bulk rather than boundary term and different choice for PWFs (like including damping at the boundaries) does not influence this correction term that much.
The referee writes:
Our response: Yes, the referee is absolutely right and we really appreciate it to make us aware of this issue. In fact, our statement would be correct when $M_c$ counts the number of blocks along a single dimension not their total number in d dimensions. In the revised version, we have fixed this problem. Now, related to the scaling, it is true that the numerical calculations would be more expensive in higher dimensions, but at the same time, it shows that the integral forms provided in our work can be even more beneficial in those cases.
In the end, we have also implemented all the requested changes in our revision.