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Black hole mirages: electron lensing and Berry curvature effects in inhomogeneously tilted Weyl semimetals

by Andreas Haller, Suraj Hegde, Chen Xu, Christophe De Beule, Thomas L. Schmidt, Tobias Meng

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Submission summary

Authors (as registered SciPost users): Andreas Haller · Thomas Schmidt
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-02-01 15:13
Submitted by: Haller, Andreas
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Gravitation, Cosmology and Astroparticle Physics


We study electronic transport in Weyl semimetals with spatially varying nodal tilt profiles. We find that the flow of electrons can be guided precisely by judiciously chosen tilt profiles. In a broad regime of parameters, we show that electron flow is described well by semiclassical equations of motion similar to the ones governing gravitational attraction. This analogy provides a physically transparent tool for designing tiltronic devices like electronic lenses. The analogy to gravity circumvents the notoriously difficult full-fledged description of inhomogeneous solids. A comparison to microscopic lattice simulations shows that it is only valid for trajectories sufficiently far from analogue black holes. We finally comment on the Berry curvature-driven transverse motion and relate the latter to spin precession physics.

Current status:
Has been resubmitted

Author comments upon resubmission

Dear Editor,

Thank you for sending us the reports of the Referees. We are very grateful for their careful, positive and constructive reports that helped us improve the manuscript. We have added technical and logical steps and clarified some parts in the new version of the paper, complying with the requests of all Referees. Please find our more detailed reply in the sections below. In view of the already positive reports in the last round, we hope that our work can now be accepted for publication on SciPost Physics.

On behalf of all authors,
Kind regards,
Andreas Haller

List of changes

Sec. 1 and 2:
- Included a paragraph on the discussion of the effective potential
- Added references to the introduction

Sec. 3:
- Avoided the notion of "accumulation at the tilt center".

Sec. 4:
- Extended and revised Sec. 4.1 and 4.3

Sec. 5:
- Added paragraph on "anomalous currents"

- Added conclusion from the new extension of Sec. 4

Reports on this Submission

Anonymous Report 2 on 2023-2-10 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2210.16254v3, delivered 2023-02-10, doi: 10.21468/SciPost.Report.6714


I would like to thank the authors for their concise remarks on my comments.
Specifically thanks to their answer of point 3.2 it is now clearer what differences can be expected between a space dependent tilt and a space-dependent Fermi velocity.

Regarding point 3.1 it is true that they mention several references in the introduction. The goal of my comment was not to add more references, but rather elaborate further on what concretely differentiates their predictions from earlier works. Thanks to their reply to point 3.2 it is possible to guess somewhat the differences and I will not insist more on this comment.

Their reply to 3.3 is rather short and not very concrete, but I guess this attests to the difficulty of the question. Since I raised it as optional, I won't dwell either on it.

Therefore, I recommend publication in SciPost.

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

Author:  Andreas Haller  on 2023-02-17  [id 3370]

(in reply to Report 2 on 2023-02-10)

We thank the Referee for his/her report, and like to add the following remarks on point 3.1 and 3.2. In Schwarzschild coordinates, the matrix representation of the metric $g_{\mu\nu}$ reads
g =
1-\frac{u(r)^2}{c^2} & 0 & 0 & 0 \\
0 & -\frac{1}{1-\frac{u(r)^2}{c^2}} & 0 & 0 \\
0 & 0 & -r^2 & 0 \\
0 & 0 & 0 & -r^2 \sin^2(\theta )
This leads to the Hamiltonian
\mathcal H =
\frac{1}{2} \left(\frac{k_1^2 u(r)^2}{c^2}+\frac{E^2}{c^2-u(r)^2}-k_1^2-\frac{k_2^2}{r^2}-\frac{k_3^2 \csc^2(\theta )}{r^2}\right),
with a dispersion relation on the $(x,y)$ plane, assuming $r>r_t$
E_\pm = \pm c\sqrt{1-\frac{u(r)^2}{c^2}}\sqrt{k_1^2+\frac{L_z^2}{r^2}}
and therefore a different dynamics parametrized by the temporal coordinate. We note that the analog setup could be naively realized by a spatially modulated Weyl cone of the form $E_\pm=\pm v(r) k$ (using $k_z=0$, like in our manuscript, such that contributions to the dynamics by anomalous terms vanish). The associated Hamiltonian is given by $\mathcal H = \chi\,v(r)\,{\vec k}\cdot {\vec\sigma}$, where the tilt is translated into an isotropic velocity field $v(r)=\sqrt{c^2-u(r)^2}$. Note that, contrary to Gullstrand-Painlevé coordinates, Schwarzschild coordinates only make sense for $r>r_t$ because the metric is divergent at $r=r_t$, and the dispersion becomes complex at $r<r_t$. Despite these differences, we find by straightforward evaluation that the effective potential identified from $\dot r(\tau)$ for $r>r_t$ is equal in both coordinate systems.

Report 1 by Jasper van Wezel on 2023-2-7 (Invited Report)

  • Cite as: Jasper van Wezel, Report on arXiv:2210.16254v3, delivered 2023-02-07, doi: 10.21468/SciPost.Report.6695


1 well-written
2 thorough analysis on multiple scales
3 novel and timely
4 opens up an area of “tilt-tronics”.


Precise connection to gravity and black hole physics remains unclear on one point (see below).


I would like to thank the authors for their careful and thorough consideration of my earlier remarks.

The responses to questions 1.2, 1.3, 1.4, and 1.5 are convincing, and I thank the authors for including several clarifications and a more nuanced discussion of the "mirage" effect in their manuscript.

I am not really convinced by the response of the authors to question 1.1, however.

In particular, the reference mentioned by the authors in 1.1.1 focusses on a so-called Kiselev black hole (including "quintessence") rather than a Schwarzschild black hole. In passing, it does also show the effective potential for the pure Schwarzschild black hole (their eq 14), but that result in fact clearly shows that V_eff is proportional to the angular momentum L^2. For radial geodesics such as the red line in Figure 1 of the current manuscript, the effective potential is negative everywhere and corresponds to the uniform attraction expected for a Schwarzschild black hole (as shown more explicitly for example in figure 3 of arxiv:1911.04305). This contradicts the outward acceleration of the radial red line shown in figure 1 of the current manuscript.
I am therefore not convinced the situation described by the authors really does represent a Schwarzschild black hole, and I encourage the authors to clarify this inconsistency.

Notice also that the caption of Figure 2 in the current manuscript does not indicate which value for the angular momentum is used to generate the curves in the left panel.

Furthermore, in appendix A the authors derive geodesics in a Schwarzschild metric, and they show that the tilt profile for Weyl cones considered in their manuscript matches the local tilts of light cones in the Painlevee-Gullstrand coordinate description of the Schwarzschild metric. However, they do not show that the worldlines of semiclassical wavepackets subjected to their specific Weyl cone tilt profile follow the geodesics of freefalling particles in the gravitational metric.
My reason for asking the authors to explicitly demonstrate this, is that the outward radial acceleration of the red line in fig 1 does not seem to agree with the naive expectation for geodesics in the gravitational setting (see also 1.1.1). If there was no such conflicting observation, I would agree with the authors that further elaboration on the correspondence is superfluous. As it stands, however, I do not see how the results of appendix A agree with the red trajectory shown in Fig 1.

Please note that the observation of trajectories escaping the black hole in Sec 4 (which clearly do not correspond to gravitational black hole geodesics) is sufficiently explained in the revised manuscript by the added discussion of the effects of lattice dispersion.

If the authors can address this final point of confusion, I am happy to recommend publication in SciPost Physics.

Requested changes

I suggest the authors address the one remaining point of confusion identified above.

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

Author:  Andreas Haller  on 2023-02-17  [id 3372]

(in reply to Report 1 by Jasper van Wezel on 2023-02-07)

We are indebted to Mr. van Wezel for being persistent on his last doubts. Before we submit a revised version, we would like to eliminate the last point of confusion.

We believe the main reason for his doubt is rooted in the fact that radial trajectories are accelerated during their outwards movement, such that we are focussing on this discussion in more detail here. Consider initial coordinates $x>\rho_t$, $y=z=0$ and $k_x>0$, $k_y=k_z=0$, such that the radial trajectory is along the $x$-axis. The equations of motion can then be solved for the momentum coordinate
k_x(t)=k_x(0)^2\exp\left(-\int_0^t u'(x(\tau))d\tau\right).
From the solution of $k_x$, we recognize $\dot x(t) = u(x) + v_F>0$. One can also show that the acceleration is $\ddot x(t) = u'(x)(u(x) + v_F)>0$. Also, since $L_z=0$, the effective potential vanishes, and $\dot x^2(t) = E_+^2/k^2=E_+^2\exp(2\int_0^t u'(x(\tau))d\tau)/k_x(0)^2$.

We want to stress that the radial velocity and the effective potential in our manuscript are consistent with Eqs. (12), (13) [for $\sigma=0$] and Fig. 1 of Phys. Rev. D 92, 084042. The quantitative differences in the amplitude vs. distance of Fig. 1 compared to our Fig. 2 are caused by choosing dimensionless units $GM=c=1$, which means $r_t/2=v_F=1$ in our setting. The crucial difference to the dynamics we present in the main text is that we parametrize the differential equations using the temporal coordinate, which differs with respect to Eq. (12) by an additional factor $({dt}/{d\tau})^{-1}$ (for details, we refer to the revised App. A). Note that this factor is equal to $\pm c/k$ when $L=L_z$, which gives rise to a decreasing effective mass for radially escaping trajectories,
m_{\rm eff}(t)=2k^2=2k(0)^2\exp\left(-2\int_0^tu'(\rho(\tau))d\tau\right),
and the particle is accelerated away from the origin.

The geodesic equation is parametrized by an affine parameter instead, resulting in $\dot \rho(\tau)^2=\pm E_\pm^2$ (for radial trajectories with $L_z=0$), such that the velocity of particles escaping radially is constant in $\tau$. In conclusion, the "repulsion" of radially escaping particles is caused by the effective mass as a result from the temporal coordinate parametrization of the trajectories, not by the effective potential.

Compared to arXiv:1911.04305v1 mentioned by the Referee, note that the authors use a different notation, i.e. $h^2=L^2/(m\mu)$ and $E=kmc^2$ (see also the Wikipedia article on Schwarzschild geodesics, and our reply to the second report). Therefore, their effective potential in Eq. (3.21) is written in different units, which, upon multiplying the right hand side of Eq. (3.21) by $m$, followed by $m\rightarrow0,\mu\rightarrow1$ leads to the equations in our main text.

We admit that we could have given more details in App. A relating the geodesic equation parametrized by an affine parameter to the semiclassical equations of motion parametrized by the temporal coordinate, which we will incorporate if Mr. van Wezel agrees to the conclusion of this discussion.

Jasper van Wezel  on 2023-02-17  [id 3373]

(in reply to Andreas Haller on 2023-02-17 [id 3372])

Dear authors,

thank you very much for reaching out and for the detailed arguments.
I think a clarification of appendix A using affine parameters and the e.o.m. using temporal coordinates will be very useful.

My confusion however, is much more basic.

If you have x(0)>0 and d^2 x/dt^2 x>0, then the particle is accelerating away from the origin.
If at the same time you claim that this particle trajectory coincides with a geodesic found in Schwarzschild spacetime, then that means there needs to be a free-falling observer in the Schwarzschild spacetime who is accelerated away from the black hole.
I don't understand how that could be the case, as all (zero angular momentum) free-fall motion should be attracted towards the black hole, not repelled away from it.

If you could please clarify this one point, that would be great.

Anonymous on 2023-02-20  [id 3382]

(in reply to Jasper van Wezel on 2023-02-17 [id 3373])

Dear Mr. van Wezel,

We sincerely thank you for your question - digging into it has led us to some very enjoyable reading. On a basic level, it is straightforward to see that the velocity of an outwards particle, written in Gullstrand-Painlevé coordinates does feature a positive acceleration, if measured in coordinate time, i.e. a "repulsion". For a quick and dirty check, see this link. To add a bit of physics, consider a particle falling into a black hole. As you’re certainly aware, a standard textbook view is that an observer asymptotically far away from the black hole will see the particle become slower and slower as it approaches the black hole - gravity thus seems to decelerate or "repel". On a deeper level, we found PhysRevD.25.3191 and in particular the conclusion of arXiv:1606.08300v2 helpful. Overall, your question is very well posed, and as you can see reveals that GR is simply fun to work with.

On behalf of all authors Kind regards, Andreas Haller

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