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Quantum Hall effect in curved space realized in strained graphene
by Glenn Wagner, Fernando de Juan, Dung X. Nguyen
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Submission summary
| Authors (as registered SciPost users): | Dung Nguyen · Glenn Wagner · Fernando de Juan |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/1911.02028v2 (pdf) |
| Date submitted: | 2020-11-17 12:09 |
| Submitted by: | Wagner, Glenn |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The quantum Hall effect in curved space has been the subject of many theoretical investigations in the past, but devising a physical system to observe this effect is hard. Many works have indicated that electronic excitations in strained graphene realize Dirac fermions in curved space in the presence of a background pseudo-gauge field, providing an ideal playground for this. However, the absence of a direct matching between a numerical, strained tight-binding calculation of an observable and the corresponding curved space prediction has hindered realistic predictions. In this work, we provide this matching by deriving the low-energy Hamiltonian from the tight-binding model analytically to second order in the strain and mapping it to the curved-space Dirac equation. Using a strain profile that produces a constant pseudo-magnetic field and a constant curvature, we compute the Landau level spectrum with real-space numerical tight-binding calculations and find excellent agreement with the prediction of the quantum Hall effect in curved space. We conclude discussing experimental schemes for measuring this effect.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 4) on 2021-1-11 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1911.02028v2, delivered 2021-01-11, doi: 10.21468/SciPost.Report.2393
Report
The authors consider the connection between the low energy theory of a strained tight binding lattice model for graphene and the field theory of Dirac fermions moving in curved spacetime. A large number of papers have been written on this topic over the last ten years. The mapping between the two models has two constraints
- The physics must be dominated by the fermions near the Dirac cones, thus the relevant k space region is $ |k-K|~|k-K’|<<(2*\pi)/L<<(2*\pi)/a$ where L is the typical length scale over which the strain changes.
- The change in the strain over the length scale L must be small compared to lattice size$ L \nabla u <<a$
Which automatically mean that |\nabla u| is restricted by a/L. However to see any effect of “spacetime curvature” one needs to have large strain gradient. Most approaches consider the ideal situation of an infinite lattice with an infinitesimally small strain gradient and work in the continuum limit without ever considering if the curved spacetime theory can be relevant for any finite lattice.
This manuscript overcomes this deficiency by showing that the Landau level fingerprints in local density of states near the center of a finite lattice with a quadrupolar strain are in good agreement with the prediction of the continuum model only if the effective curvature is taken into account. While the hopping model used in the paper prevents a direct link to Graphene experiments, this result is valuable and shows that many of the exotic proposal regarding GR-strained graphene connection can be at least constrained by simple numerical tight binding calculations. The paper is timely, and generally well written. I recommend the publication of the manuscript in SciPost Physics once the authors consider the following points:
Requested changes
- As a condensed matter theorist whose last exposure to curved spacetime was in graduate school it was not easy for me to follow the 2nd section. Nonetheless I tried to do so with the help of the useful appendix, yet I have a number of possibly trivial questions:
o What exactly is the motivation in starting from the most general action for Dirac fermions in 2+1 and then arguing that only static spacetimes can be simulated with strained graphene which kills the spin connection term? Why do the authors just not say that a tight binding Hamiltonian expanded to second order near a Dirac point gives a specific action from the outset?
o Similarly it would be nicer to give what scalar potential results from a general strain field instead of setting the scalar potential to zero outright.
o What is the result of the rescaling in 2.11 on the canonical anti-commutation relations? Why is the symplectic structure of the Hamiltonian not affected?
- In my opinion Figure 2 presents the most important result of the paper, however it is quite busy. I suggest that the authors break this up into two figures. It would be nice to see LDOS as a function of (E) as in panel (b). There are two points to be made here, one is full analytical result(with curvature) matches LDOS jumps well (presumably up to some energy which cannot be discerned from the current graph); the second is that omitting curvature clearly gives wrong results which may be made only in a separate figure like panel (a). Also the use of color would improve the readability, and focusing in on the first 5-6 Landau levels may make the connection clearer.
- It is not at all clear why only 10 sites were used for the LDOS calculation, at some point the continuum result should fail, and it would be nice to know where that happens. Does the LDOS have recognizable jumps if 30 unit sites at the center were used, how about 100?
- The amount of the jumps in the LDOS also carries some information, do they correspond to twice what one expects from a usual Landau level due to the valley degeneracy?
- The authors argue that an expansion around shifted K points is not consistent in appendix C, and finish the appendix by saying one can check this for their numerical calculation. If this check is done It would be worthwhile to see the numerical results. I wonder if LDOS for points away from the center have a similar Landau level like jumps. It is not clear if the statement in the paper is that those jumps can or cannot be explained by a suitable low energy action with curvature.
- In figure one the authors may wish to superimpose the strain profile by displaying shifted atom positions with a different color in panel (a).
Report #1 by Anonymous (Referee 5) on 2021-1-5 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1911.02028v2, delivered 2021-01-05, doi: 10.21468/SciPost.Report.2370
Strengths
1 Tries to be pedagogical
2 Tries to be self-contained (extended appendices)
Weaknesses
1 Main results not novel even if claimed so
2 Improvements/additions with respect to known results seem incremental
3 Use of symmetric action: central point but not sufficiently clear
4 Relevant literature ignored
5 Hyperbolic and/or imprecise statements
6 Main figure of fairly low quality
Report
I appreciate very much the Authors' effort to give a fairly self-contained presentation, including detailed appendices. However the manuscript suffers from various shortcomings, and is somewhat lacking substance and novelty. I am afraid I cannot recommend it for publication in SciPost Physics, though it could be published elsewhere after substantial revision. A list of specific criticism follows.
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1) The central claim of the paper, namely that "our field redefinition (2.11) and the maps in Sec. 3.2 are our new contributions to the literature" (quoting from p. 6), highlights a major weakness. First, the very same "field redefinition (2.11)" already appears in the literature, see e.g. PRL 124, 126804 (2020), where tight-binding simulations of Dirac electrons in curved space and a magnetic field are discussed in details. Second, the presentation of the "maps in Sec. 3.2", and the subsequent discussion in Sec. 4 of the second-order strain terms, is not that clear: what are the precise advances compared to existing second-order results, see e.g. Solid State Communications 175, 76 (2013)? Is it only the addition of the curvature correction (3.11)? Are we dealing only with incremental improvements?
2) The Authors use a symmetric action (2.8), since the original one (2.7) "(...) isn’t real, which implies that the corresponding Hamiltonian isn’t Hermitian. However, the tight-binding Hamiltonian (3.1) is explicitly Hermitian and therefore, the action has to be real.". They add that "S and S' only differ by a total derivative and are hence equivalent physically." The symmetrization is usually not employed in similar contexts (PRL 105, 206601 (2010)) and is in fact not necessary for a (hermitian) tight-binding formulation of the problem (PRL 124, 126804 (2020)). Furthermore, it modifies the boundary conditions for the fields, so that the statement about S and S' being "equivalent physically" applies only to infinite systems: what happens in realistic, finite samples? What is the role of the surface terms? Is this under control?
3) The Landau levels are "pseudo Landau levels" due to an artificial gauge field. A reminder of this would be helpful to the reader, especially when discussing Fig.2 -- whose quality is also very low. Moreover, the discussion of the connections between the present results and those already in the literature is not satisfactory at all (see also point 1) above). Many works very closely related to the present one are apparently ignored, see PRB 95, 125432 (2017), Eur. Phys. J. Plus 134, 202 (2019), PRL 103, 196804 (2009), but the list is not exhaustive.
4) The paper is scattered with overstatements or statements which are not sufficiently qualified. For example: "In the derivation, the factors of g^1/4 and g in (2.11) and (3.13) respectively are subtle points that were missed in previous works on this topic" --> not true, see point 1) above; "We note that there has been some controversy in the literature about whether to expand the Hamiltonian about the unshifted Dirac point K (as we do here) or about a shifted Dirac point" --> references about this "controversy" are missing; "We also emphasize that we have settled the point about whether to expand about the shifted or unshifted K-points." --> see comment to previous statement; "We expand to linear order in the momenta and to all orders in the strain for now." --> Is it possible to expand to all orders in the strain? Could the authors better clarify how? Has this been done by others before?
Requested changes
All issues raised in the 4 main points of the report should be addressed.
Author: Glenn Wagner on 2021-09-07 [id 1742]
(in reply to Report 1 on 2021-01-05)
Question/Comment: The central claim of the paper, namely that "our field redefinition (2.11) and the maps in Sec. 3.2 are our new contributions to the literature" (quoting from p. 6), highlights a major weakness. First, the very same "field redefinition (2.11)" already appears in the literature, see e.g. PRL 124, 126804 (2020), where tight-binding simulations of Dirac electrons in curved space and a magnetic field are discussed in details.
Reply: We are thankful for these remarks, which have made us realize the central claims of our work are not properly framed. The main claim of the paper is not the abstract mapping and field redefinition per se, but the fact that we have succeeded in proving that the honeycomb lattice under strain realizes the Quantum Hall effect in curved space exactly, by matching a real space lattice calculation with a field theory prediction. The main achievement of this work is this matching. There are several subtle points in the calculation which, if not handled correctly, do not lead to this matching. We have rewritten the introduction in a way that shows this very clearly, and also made more precise statements about our contribution. We believe it is now clearer that this is not an incremental improvement, but rather the first time that such a precise match between lattice and continuum calculation has been provided, which enables the confident use of the honeycomb lattice to address curved space physics. The work PRL 124, 126804 (2020) is an interesting reference that we had indeed missed and we thank the referee for bringing it up. This reference models the Dirac fermion surface in topological insulators in the particular shape of a nanocone. This work does not address the microscopic origin of the Dirac fermion, but rather assumes a continuum formulation as done in several previous works (now cited in the introduction), and then discretizes this equation to simulate it numerically. We do acknowledge that the field redefinition was used in this work, which was posted on the arxiv a month and a half before ours. We have added a comment to reflect this in the section about comparison with other works. However, as the SM of PRL 124, 126804 (2020) explains, that work uses the field redefinition to make the discretized Dirac equation Hermitian justifying it with a one-dimensional example. However, in more than one dimension, absorbing the volume form alone does not necessarily produce an Hermitian action per se. And in fact this is not needed for this reason because the symmetric action is already Hermitian. Rather, this field redefinition is required because the normalization of the fields must be the same in a lattice and in the continuum if the two approaches are to be compared. We have tried to decouple these two issues in the discussion, since their mixing might lead to confusion. Nevertheless, we again acknowledge the importance of PRL 124, 126804 (2020) and we believe it’s now properly referenced in the manuscript. --- Question/Comment: Second, the presentation of the "maps in Sec. 3.2", and the subsequent discussion in Sec. 4 of the second-order strain terms, is not that clear: what are the precise advances compared to existing second-order results, see e.g. Solid State Communications 175, 76 (2013)? Is it only the addition of the curvature correction (3.11)? Are we dealing only with incremental improvements?
Reply: We have significantly extended section 6 to discuss this point. Indeed, previous works on the subject did consider second order in strain corrections to the Hamiltonian and the gauge field, but did not consider the connection to curved spaces and the computation of the Riemann curvature, as the referee says. Importantly, they also did not consider the lattice calculation of the spectrum in the presence of strain to compare with an analytical prediction. We respectfully disagree with the characterization of our work as an “incremental improvement”, since the curved space interpretation and the explicit calculation of the curvature is the key ingredient that explains the numerical calculation. There have been many works proposing continuum theories for strained graphene, but none have really identified a simple continuum prediction and provided such an exact match as we do. We believe this is the key contribution of our work. --- Question/Comment: 2) The Authors use a symmetric action (2.8), since the original one (2.7) "(...) isn’t real, which implies that the corresponding Hamiltonian isn’t Hermitian. However, the tight-binding Hamiltonian (3.1) is explicitly Hermitian and therefore, the action has to be real.". They add that "S and S' only differ by a total derivative and are hence equivalent physically." Reply: These comments have led us to clarify the discussion on the difference between S and S'. This is a point that is discussed in detail in the literature cited. The proper standard action is actually S', but their difference is immaterial for an infinite system, which is why both can be seen in the literature. S may be more convenient than S' to derive the equation of motion, for example, but they are really equivalent. In a finite system, it is of course only the real action S' that should be used, or if S is used, it should be supplemented by the proper boundary condition that makes the whole action Hermitian. This is now explained in the main text. In fact the natural boundary conditions for a finite system imply that the different boundary term between S and S' vanishes. We explained in detail at the end of Appendix A. --- Question/Comment: The symmetrization is usually not employed in similar contexts (PRL 105, 206601 (2010)) and is in fact not necessary for a (hermitian) tight-binding formulation of the problem (PRL 124, 126804 (2020)).
Reply: Whether the use of the symmetric action is necessary or not has been discussed just above: both approaches are equivalent. Whether it has been usually employed may be a matter of taste, but it was used for example in Phys. Rev. B 88, 155405 (2013). It is just more convenient to use S’ with the symmetric derivative, because in the tight binding formulation the momentum operator does transform into a symmetric derivative in real space. We would like to emphasize that these were just explanatory remarks in the text, and that there is nothing hidden or ambiguous in this discussion. A different matter is the field redefinition that absorbs the volume form in the fields. This is not a matter of convention, but rather a required choice to ensure that the scalar product of the continuum fields and the lattice fields satisfies the same normalization. --- Question/Comment: Furthermore, it modifies the boundary conditions for the fields, so that the statement about S and S' being "equivalent physically" applies only to infinite systems: what happens in realistic, finite samples? What is the role of the surface terms? Is this under control?
Reply: As stated above, there is a standard way to treat finite systems as well, and the standard choice is S'. Nevertheless this is not a relevant problem for us, since we are always assuming that the system size is large enough that we can work with the predictions of curved space field theory without boundaries. --- Question/Comment: 3) The Landau levels are "pseudo Landau levels" due to an artificial gauge field. A reminder of this would be helpful to the reader, especially when discussing Fig.2 -- whose quality is also very low.
Reply: The quality of Fig. 2 has been improved. We have split the figure in two such that it is less overwhelming and we have made it into a colour figure such that the different curves are more clear. We now mention explicitly in the main text that we are dealing with pseudo-LL. --- Question/Comment: Moreover, the discussion of the connections between the present results and those already in the literature is not satisfactory at all (see also point 1) above). Many works very closely related to the present one are apparently ignored, see PRB 95, 125432 (2017), Eur. Phys. J. Plus 134, 202 (2019), PRL 103, 196804 (2009), but the list is not exhaustive.
Reply: We have attempted to improve the discussion about the connection with other works. We have done this by first expanding the introduction to broaden the context of our work, and added further citations. We have also greatly enlarged section 6, and made more precise connections to the different related works. We would like to stress again that the main objective of this paper is to show that a numerical, real space tight binding calculation can produce results which exactly match those of the continuum prediction of Dirac fermions in curved space with finite Riemann curvature. There are indeed works in the literature considering Dirac fermions in curved space, but not all are closely related to our work and for concreteness we have tried to focus on those which bear some relevance to the purpose of the paper. We nevertheless thank the referee for bringing these references to our attention. All these three works consider the Dirac equation in curved space under different contexts, but none of them consider a microscopic tight binding calculation to match. PRL 103, 196804 (2009) discusses TI surface states, and is now cited in the discussion on Dirac fermions in finite curvature. PRB 95, 125432 (2017) deals with graphene but considers again only the continuum curved model, and does not include the pseudo-magnetic field due to strain. It is now cited along other similar works in the introduction. Similarly Eur. Phys. J. Plus 134, 202 (2019) only considers the continuum Dirac equation in cylindrical geometry, where there is not even intrinsic curvature. In our opinion this paper is not sufficiently relevant for our work, and given the extensive citation list we already have we have decided not to cite it. --- Question/Comment: 4) The paper is scattered with overstatements or statements which are not sufficiently qualified. For example: "In the derivation, the factors of g^1/4 and g in (2.11) and (3.13) respectively are subtle points that were missed in previous works on this topic" --> not true, see point 1) above;
Reply: We believe this is now properly explained and referenced in the manuscript. Because of this, this particular sentence has now been removed. --- Question/Comment: "We note that there has been some controversy in the literature about whether to expand the Hamiltonian about the unshifted Dirac point K (as we do here) or about a shifted Dirac point" --> references about this "controversy" are missing;
Reply: The reference about this controversy is that by Zubkov and Volovik (2015), which was cited several times in the appendix about the different K point expansions. Nevertheless, it is true that it should have been cited in the main text as well, and we have done so now. --- Question/Comment: "We also emphasize that we have settled the point about whether to expand about the shifted or unshifted K-points." --> see comment to previous statement;
Reply: See our comment above, and the mentioned appendix. We do believe we have settled this controversy given our match with the numerical calculation. --- Question/Comment: "We expand to linear order in the momenta and to all orders in the strain for now." --> Is it possible to expand to all orders in the strain? Could the authors better clarify how? Has this been done by others before?
Reply: What was meant by this sentence is that we expand in momentum keeping the whole structure of the space dependent hopping t_n(x), which itself depends on the strain. This can be done and is done in Eq. 3.3 and 3.4. Then, for the specific strain profile we expand to second order in strain. We have modified the sentence mentioned by the referee accordingly.
Author: Glenn Wagner on 2021-09-07 [id 1743]
(in reply to Report 2 on 2021-01-11)Question/Comment: What exactly is the motivation in starting from the most general action for Dirac fermions in 2+1 and then arguing that only static spacetimes can be simulated with strained graphene which kills the spin connection term? Why do the authors just not say that a tight binding Hamiltonian expanded to second order near a Dirac point gives a specific action from the outset?
Reply: The reason for presenting things in this way is that we want to make the connection to curved space physics as explicitly as possible. Our aim is that the discussion can be followed by readers from different backgrounds, and we believe a general approach from both ends (curved space field theory vs tight binding) will serve this purpose. More specifically, we choose static spaces which are those can be more easily realized, and which connect with our aim of realizing the Quantum Hall effect in curved space. Furthermore, the energy spectrum of the case with static constant curved and magnetic field can be obtained analytically. In this work, we want to show that the LDOS in strained graphene can perfectly reproduce the energy spectrum of Dirac fermion in static constant magnetic field and curvature. --- Question/Comment: Similarly it would be nicer to give what scalar potential results from a general strain field instead of setting the scalar potential to zero outright.
Reply: For the purpose of completeness, we added comments for a general strain profile at the end of Sec 3.2. The tight-binding Hamiltonian, with a general strain field, will be mapped to the more general Dirac Hamiltonian in a static curved space with both vector and scalar potentials [Eq (A.19)]. --- Question/Comment: What is the result of the rescaling in 2.11 on the canonical anti-commutation relations? Why is the symplectic structure of the Hamiltonian not affected?
Reply: It is natural to perform this rescaling because in curved space the canonical conjugate anti-commutation relation includes sqrt{hat{g}}. However, in the canonical conjugate anti-commutation relation for the electron field in graphene, there is no sqrt{hat{g}}. Therefore the rescaling is necessary to map exactly the anti-commutation relation of the Dirac fermion field in curved space to the anti-commutation relation of the electron field in graphene. We added some discussions on the anti-commutation relation with and without rescaling and argued why one should map the electron field in the tight binding Hamiltonian to the rescaled Dirac fermion field. --- Question/Comment: In my opinion Figure 2 presents the most important result of the paper, however it is quite busy. I suggest that the authors break this up into two figures. It would be nice to see LDOS as a function of (E) as in panel (b). There are two points to be made here, one is full analytical result(with curvature) matches LDOS jumps well (presumably up to some energy which cannot be discerned from the current graph); the second is that omitting curvature clearly gives wrong results which may be made only in a separate figure like panel (a). Also the use of color would improve the readability, and focusing in on the first 5-6 Landau levels may make the connection clearer.
Reply: We have updated this figure in line with the suggestions by the referee. The figure has been split into two figures and we have decided to show only the first 6 Landau levels in order to make the figure more readable. We have also made these into colour figures for additional clarity. --- Question/Comment: It is not at all clear why only 10 sites were used for the LDOS calculation, at some point the continuum result should fail, and it would be nice to know where that happens. Does the LDOS have recognizable jumps if 30 unit sites at the center were used, how about 100?
Reply: The choice of 10 sites is not too important. Even for 30 or 100 sites, the LDOS still has clear Landau levels, although states at energies between the Landau levels start appearing. The Landau levels start becoming less sharp once about 1000 lattice sites are included in the calculation of the LDOS. --- Question/Comment: The amount of the jumps in the LDOS also carries some information, do they correspond to twice what one expects from a usual Landau level due to the valley degeneracy?
Reply: Yes, the two valleys each contribute half of each jump (see Appendix B.3 where we show that the two valleys have precisely the same energy levels). In addition, on a closed manifold, the total degeneracy of each Landau level would be affected by the Wen-Zee shift (as we explain in sec. 5 in the main text, the Wen-Zee shift is not visible when we look at the LDOS however). --- Question/Comment: The authors argue that an expansion around shifted K points is not consistent in appendix C, and finish the appendix by saying one can check this for their numerical calculation. If this check is done It would be worthwhile to see the numerical results.
Reply: We now give the explicit form of f_1 and delta K for our strain profile in Appendix C from which it can be explicitly checked that the integral in question does not converge. --- I wonder if LDOS for points away from the center have a similar Landau level like jumps. It is not clear if the statement in the paper is that those jumps can or cannot be explained by a suitable low energy action with curvature.
Reply: Far away from the centre, the Landau levels will be washed out since the approximation of small strain is no longer valid. This is why we only include the 10 sites closest to the lattice centre when computing the LDOS, including sites further from the centre will result in Landau levels that are less sharp. --- Question/Comment: In figure one the authors may wish to superimpose the strain profile by displaying shifted atom positions with a different color in panel (a).
Reply: We have updated Fig 1 according to this suggestion.