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Quantum dynamics in 1D lattice models with synthetic horizons
by Corentin Morice, Dmitry Chernyavsky, Jasper van Wezel, Jeroen van den Brink, Ali G. Moghaddam
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|Authors (as registered SciPost users):
|Ali G. Moghaddam · Corentin Morice · Jasper van Wezel
We investigate the wave packet dynamics and eigenstate localization in recently proposed generalized lattice models whose low-energy dynamics mimic a quantum field theory in (1+1)D curved spacetime with the aim of creating systems analogous to black holes. We identify a critical slowdown of zero-energy wave packets in a family of 1D tight-binding models with power-law variation of the hopping parameter, indicating the presence of a horizon. Remarkably, wave packets with non-zero energies bounce back and reverse direction before reaching the horizon. We additionally observe a power-law localization of all eigenstates, each bordering a region of exponential suppression. These forbidden regions dictate the closest possible approach to the horizon of states with any given energy. These numerical findings are supported by a semiclassical description of the wave packet trajectories, which are shown to coincide with the geodesics expected for the effective metric emerging from the considered lattice models in the continuum limit.
Submission & Refereeing History
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Reports on this Submission
- Cite as: Anonymous, Report on arXiv:scipost_202201_00020v1, delivered 2022-02-14, doi: 10.21468/SciPost.Report.4417
- Comprehensive discussion of the physics of free 1d tight-binding models with power-law dependent hoppings
- Highlighted correspondence with the continuum massless Dirac equation in an external gravitational field and a detailed discussion of limitations thereof
- Multiple different approaches to tackle the problem
- Lack of clarity about a large overlap in results with the authors' previous publication (PHYSICAL REVIEW RESEARCH 3, L022022 (2021))
In the present paper, a rather detailed discussion of the salient phenomenology of free 1d tight-binding quantum models with position-dependent hoppings is presented. The authors' main motivation stems from an attempt to mimic the massless 1d Dirac equation in an external dilaton gravitational background. They put this idea forward recently in PHYSICAL REVIEW RESEARCH 3, L022022 (2021).
Here are my comments on the manuscript:
-I fully share the concern of another referee about the overlap of this paper with the previous publication (PHYSICAL REVIEW RESEARCH 3, L022022 (2021)). It appears that the authors decided to write a long follow-up on their original letter. In itself, I see no problem with that, but I think the current version of the manuscript does not reflect that properly. I encourage the authors to explain in the introduction section that it is indeed a follow-up paper and present clearly what new results are worked out here.
- In my opinion, the transformation (2) should not be called a particle-hole symmetry. Any particle-hole symmetry should swap particles and holes of a given vacuum and thus should be anti-unitary. In contrast, the transformation (2) is a unitary transformation, so I would not call it a particle-hole transformation. Also, following another referee, I am curious about the importance of the present symmetry in the manuscript.
After these comments are taken care of, I am happy to recommend the paper for publication in SciPost Core.
- Cite as: Anonymous, Report on arXiv:scipost_202201_00020v1, delivered 2022-01-30, doi: 10.21468/SciPost.Report.4278
- The paper approaches the problem from several angles, using anayltical and numerical techniques which show excellent agreement and provide a compelling picture.
- The results point to an interesting connection to one-dimensional horizons in black hole physics.
- The paper is clearly written.
- The paper is a clear follow-up of a previous work, and it does not discuss what is new and what is not, and what is the purpose of writing a new paper.
- The paper appears to ignore critical graphene literature on the subject.
This paper presents an analysis of 1D tight binding models with power lay decaying hoppings, which are mapped to a 1D dirac equation in the presence of power law decaying velocity. Different analytical and numerical approaches are used to show the emergence of an effective horizon for particle propagation, in analogy to a black hole.
The main problem with this work is that it is a follow-up of a previous work by the same authors (PRR 3, L022022 (2021), Ref.  in the manuscript). The current work appears to present very little advancement compared to the previous one. The main conclusions are the same, and even some figures are repeated (Fig 1a is Fig. 2 in Ref. . Fig. 3 is Fig. 3 of Ref. , Fig 6a is similar to Fig. 4 of Ref. ). Both papers study wave packet dynamics and spectral properties, and different values of gamma in v(x) = x^gamma. The only difference appears to be the presence of more details in the derivations. The motivation for writing a new paper is not clear. In my opinion, this paper cannot meet the threshold for SciPost Phys. because of this.
Disregarding the issues with Ref. , in general I find these results interesting, and their interpretation sound. The agreement between numerics and semiclassical results is excellent. The emergence of a horizon is clearly established, making the connection with gravitational physics appealing. I believe it could be suitable for SciPost Phys. Core, answering to some questions an recommendations I outline below.
- In my opinion, the authors should state very clearly in the introduction what was done before in Ref.  and what was the motivation to write this paper. It is critical that the authors do this honestly. Bear the readers in mind: if a reader read Ref. , what will be learnt by reading this paper afterwards? The discussion section would also benefit from this. The conclusions of Ref.  need not be repeated. The authors may rather discuss what is learnt from the new results.
- The authors might not be surprised to learn that velocity profiles for the Dirac equation have been extensively considered in the graphene literature. In particular, it would be worthwhile to compare the results of this paper with J. Phys. Condens. Matter 21 095501 (2009), where the gamma=1 case is solved in the continuum (sec. 3.4) and a slightly different tight binding model is presented (appendix A). In this reference, the analytical solution for gamma=1 predicts that wave-packets are always transmitted (when there is a regular lattice on both sides of the v(x) = x region). This appears inconsistent with the formation of a horizon described here. Is the horizon related to the presence of open boundary conditions more than to the power law hopping?
- Given the several analytical solutions presented, in particular for p =pi/2, is there any physical insight to explain why gamma=1 is the critical case for the emergence of an horizon? This result does not appear characteristic of Dirac fermions, but rather of the metric considered, and its geodesics.
- Another interesting point is the analytical solution for the eigenstates for gamma=1 and 1/2. Given Ref.  presented the DOS for these cases, can one understand the exact form of the DOS peak in the gamma=1 case? Does this analytical solution provide insight on why gamma=1 develops a singularity?
- A minor point: the effective metric in the continuum Dirac equation in Ref.  is different from the one derived here. Footnote  explains the coordinate change needed to bring this metric to the diagonal form in the current work. Again, for the benefit of the reader comparing the two works it would be worth mentioning this coordinate change and the relation with the anti de Sitter metric here as well.
- The authors state that the Dirac equation with v(x) represents a Dirac field in a background metric given by Eq. 7, without providing any derivation or citation. This is a subtle point, as Dirac fermions do not couple to the metric, but to the tetrads or vielbein (see standard book “Quantum fields in curved space” by Birrel and Davis for example). Since a lot of the motivation of this paper is the comparsion with gravitational physics, I think this derivation could be fleshed out in more detail.
- Why is particle-hole symmetry discussed in Eq. 2? It is not mentioned at all in the rest of the paper.
See Report above.