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Geodesic geometry of 2+1-D Dirac materials subject to artificial, quenched gravitational singularities

by S. M. Davis, M. S. Foster

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Submission summary

Authors (as registered SciPost users): Seth Davis · Matthew Foster
Submission information
Preprint Link: scipost_202107_00036v1  (pdf)
Date submitted: 2021-07-20 18:33
Submitted by: Davis, Seth
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


The spatial modulation of the Fermi velocity for gapless Dirac electrons in quantum materials is mathematically equivalent to the problem of massless fermions on a certain class of curved spacetime manifolds. We study null geodesic lensing through these manifolds, which are dominated by curvature singularities, such as nematic singularity walls (where the Dirac cone flattens along one direction). Null geodesics lens across these walls, but do so by perfectly collimating to a local transit angle. Nevertheless, nematic walls can trap null geodesics into stable or metastable orbits characterized by repeated transits. We speculate about the role of induced one-dimensionality for such bound orbits in 2D dirty d-wave superconductivity.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2021-11-1 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202107_00036v1, delivered 2021-11-01, doi: 10.21468/SciPost.Report.3769


During the last years, analogies between the behavior of electrons in quantum materials and those in curved spacetimes have been extensively investigated. The presented manuscript is devoted to the problem along these lines. The authors rely on a mathematical correspondence between electrons in Dirac materials and the problem of particles’ propagation in the gravity context. The manifolds under consideration possess singularities classified into two types. The so-called nematic singularities are of particular interest due to the geodesic collimation effect reminiscent of some features known in BCS theory.

Although the primary motivation for the paper was dictated by the potential applicability of the results in the context of quantum materials, it could be of interest in the gravity context as well. For instance, the authors could have analyzed whether some known metrics (e.g., for black holes, cosmological horizons, etc.) belong to the class considered in the paper, providing thus new links between gravity and condensed matter theory. Perhaps, it will be done and published elsewhere.

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Report 1 by Seyed Akbar Jafari on 2021-10-9 (Invited Report)

  • Cite as: Seyed Akbar Jafari, Report on arXiv:scipost_202107_00036v1, delivered 2021-10-09, doi: 10.21468/SciPost.Report.3632


1- Introduction of the paper is nicely written and the basic message of the paper is clearly communicated.
2- Content of equations are illustrated with informative figures
3- The language of geometry ("gravitation") adopted in this paper is a new lens through which intuitive picture of classical dynamics can be constructed.


1- The eventual goal of any mathematical modeling is to compute a measurable quantity. The (null) geodesics studied in this paper are not directly measurable. A simple estimate of conductivity and the role of attractors and collimator singularities in conductivity is missing.

2- As pointed out in the introduction, the null geodesics play a role in shaping the quantum mechanical wave function. Authors have meticulously computed null geodesics for interesting solvable toy models. But there is no comment/thoughts about possible roles of these (classical) solutions for the corresponding quantum mechanical problem.


Summary of the paper:
The paper looks at the disorder in high temperature superconducting compounds from a theoretically appealing angle and cosiders it as QGD that arises from the modulation of the velocity of Dirac electrons. Authors view the band flattening in twisted bilayer graphene as isotropic curvature singularity. Apart from the possible role that the null geodesics play in shaping the quantum mechanical wave function in the curved spacetime, the paper provides adequately appealing semiclassical picture of the physics of disorder in the above compounds. Models for anisotropic velocity fluctuations as well as nematic fluctuations are presented and solved for their geodesics. The authors demonstrate that nematic curvature singularities collimate the geodesics, while the in the case of isotropic fluctuations, the geodesics are captures by the singularities. The fact that such singularities are like absorbers of
geodesics is nicely demonstrated by a more generic solvable models.

The message of the paper is clearly communicated with a nice introduction and informative pictures. Geometric approaches are powerful and can shed new light on the physics of disorder in Dirac electrons. So I like the paper quite a lot and I am happy to recommend it for publication in SciPost. However, I do have the following comments/suggestions, the incorporation of which I hope will instructive for the paper.

Requested changes

Scientific comments/suggestions:
- In Eq. (27), the conservation of E/m is associated with the global Killing vector (1,0,0)^T. Can it be explicitly derived by writing down the Killing equation? What I am wondering at this stage is that, are there any other Killing vectors in addition to the one that gives the conservation of E/m (energy)?

- Rewriting the equation of motion in terms of t, rather than the affine parameter s in Eq. (3), seems to obscure the fact that energy is conserved as there appear quadratic dissipation-looking terms. Can the authors help the reader and the present referee to understand how the dissipation-looking terms at the end do no harm to the conservation of E/m by adding a brief calculation of intuitive explanation?

- What is missing in this paper, is the implications of collimination and absorption of geodesics in transport properteis? What can be implied e.g. about the conductivity tensor for the nematic and isotropic singularities?

- Classically geodesics of isotropic singularities are absorbed by the singularities. Would that quantum mechanically correspond to possible bound state? If yes, does it imply that isotropic singularities are more efficient in forming Anderson insulator?

- How essential is the temporal-flatness condition in collimation property of nematic singularities and/or attractive property of isotropic singularities?

- The motivation for the geometrical models of velocity modulated Dirac equation comes from hight Tc superconducting compunds. Also there is a brief mention of graphene and twisted bilayer graphene. In these examples the gravitational (geometrical) coupling can be encoded into spatial-spatial compoents of the stress tensor.

It has been recently proposed in [arxiv:2108.08183] that in 8Pmmn borophene, substitution of boron atoms with carbon atoms substantially affects the tilt of the resulting Dirac cone. Since the tilting of the Dirac cone always induces a velocity anisotropy (like nemacity), how likely are these compounds to realize possible QGD in non-superconducting phase? This will correspond to modulation of the spatio-temporal components of the velocity. Does the random modulation of tilt velocity relax "temporal flatness" condition?

Styling comments:
- All over sectoin 1.1 the concept of "Null geodesics" has been used repeatedly, which is a crucial concept to understand the paper. Perhaps it will help the readers to define it right at the beginning.

- Fig. 9B seems too crowded to follow examples of geodesics that collide or do not collide with the singularities. Maybe it helps the readers to make one curve of each category bolder than the others to assist the readers to follow at lest two bold geodesics.

  • validity: top
  • significance: high
  • originality: top
  • clarity: top
  • formatting: excellent
  • grammar: perfect

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