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

1. Comprehensive analysis of when the analogy between black hole and space-dependent tilt breaks down.
2. Unlike other theoretical works, this work features a comparison between semiclassical treatment and tight-binding calculations.
3. Explicit analysis of the regime of validity of semiclassics.
4. Discussion of chirality depending trajectories which can lead to interesting chiral separation effects.
5. Timely subject: one the one hand lensing and focusing Dirac and Weyl fermions has a renewed interest as devices get better. On the other hand, gravitational analogues are being pushed.

1. The focus is largely towards semiclassics: the reasons tight binding deviates from semiclassics are discussed only superficially and most are left for future work. Admittedly, these are often challenging to compare.
2. I lacked a bit more context and connection to other works, particularly those discussing anomalous effects. I touch upon this point further in my report.
3. While applicable to other papers on the topic as well, it is hard to justify how this simple picture might be realizable in a real 3D solid-state material.

The paper is timely and clearly written. It provides a link between lensing effects more familiar in the context of black holes and that of space dependent parameters of Weyl semimetals. In my opinion, one a key contribution of this paper is to establish where these appealing analogies break down, and does not oversell the results. Specifically, I really appreciate the comparison to tight-binding simulations, even if the origin of the deviations are not entirely well understood.

The results are sound and the analysis shows transparent and explicit connections between the mentioned fields, with an honest analysis of their regime of applicability. The choice and description of the title in the main text are appropriate. Therefore I am willing to recommend publication. There are a few points that I would like the authors to discuss further before doing so:

1. It is not clear to me what specific phenomenon would a space dependent tilt result in that would not be possible with a space dependent fermi velocity or Weyl node separation. As the authors mention, there are previous papers that use a curved metric to interpret the results (e.g. ref. 18). I believe the manuscript would benefit from a discussion of the differentiating elements of the lenses discussed in previous works and those discussed here. In particular, is there any differentiating experiment (even if it is a gedanken experiment) when one could differentiate curved trajectories coming from space dependent tilts versus space dependent Weyl node separation?

2. Additionally, I found interesting that a tilt variation combined with the anomalous (Berry curvature) term gives rise to a chiral separation effect, where one chirality moves upwards and the other one downwards (as shown in Fig. 10). This behaviour is very reminiscent of anomalous currents that distinguish left and right chiralities. Can the authors link these results with anomalous currents? For example there are frame effects that can lead to seemingly anomalous currents.

Connected to the above points I would encourage the authors to:

1. Contextualize further their lensing results from previous lensing proposals.
2. Discuss the differences between lensing arising from Weyl node separation variation versus tilt variation.
3. Discuss the connection with anomalous currents, if possible.

1. Clear presentation
2. Systematic comparison of the different effective theories
3. Insightful classification of different kinetic regimes

1. Optimistic assumptions regarding solid-state materials

The authors consider Weyl materials with a spatially-varying tilt profiles and make connection to the particle kinetics near a black hole. This connection arises due to the presence of tilted Weyl cones in the effective long-wavelength theory. Studies similar in spirit are numerous, if one also counts studies in 2d materials. However, the present paper makes a number of significant contributions to the field of emergent curved-space physics in Weyl and Dirac materials.

The authors study particle kinetics in three increasingly realistic frameworks: the long-wavelength semiclassics, the lattice-regulated semiclassics and the fully microscopic scattering theory. I am not aware of previous works that would systematically cover all the three levels of description. The long-wavelength theory is particularly insightful regarding the curved-space kinetics, which is essential hidden in more accurate formulations. Here the paper makes its first important contribution in identifying different length scales in the problem of an axisymmetric "black hole" where the event horizon is marked by the interface between the 1st and 2nd type of Weyl dispersions. The equations of motion at this level predict the existence of separatrix which distinguishes the escaping and in-falling trajectories. While the conclusions of the linearized semiclassics cannot be taken as a face value, the authors go on to argue their meaning in more accurate descriptions. Working with a particular lattice model, the authors write down the semiclassical equations and solve them. This no longer admits analytical progress, and the connection to curved-space kinetics is obscured. However, the resulting trajectories are very well in agreement with the long-wavelength theory where expected, and indicate where the long-wavelength theory is no longer applicable. Finally, the predictions of semiclassics are compared to fully quantum-mechanical scattering calculations by comparing the trajectories of the former to the electron density obtained from the latter. While this is limited to 2d motion, this is very exciting since I have not seen similar comparison before. It lends a lot of credibility to the semiclassical treatments and qualitatively motivates a large number of previous works employing such methods. It also allows the authors to pinpoint the breakdown of the semiclassics in the regime where the gradient suggests an event horizon. Granted that the semiclassics cannot accurately describe extreme textures, it still seems as surprisingly accurate tool to understand the particle kinetics qualitatively. I think that the painstaking comparison of the different descriptions and justification of the semiclassics away from the overtilt regime is the main contribution of this work.

I do not find much to criticise in this work. However, there are some looming questions in this field that applies to all the studies. While it does not invalidate the results on the manuscript in anyway, one still feels a bit pessimistic about realizing the required textures in solid-state materials. I do not doubt that the discussed effects can be observed in highly idealized quantum simulators but arranging the needed strain or doping textures in solids seems difficult. Moreover, there are always some concentration of defects and artificial material manipulations are likely to increase them. Thus, the purely ballistic particle motion seems improbable. Of course, theorists need to start somewhere and including all the real-world complications is not a logical starting place. Still, acknowledging their problematic nature prominently is a good idea. I can imagine how much of the discussed physics would survive even in the presence of impurities, so I share the overall optimism regarding these studies.

Rather than requested changes, I would propose a short discussion of the effects of impurity scattering and the related hierarchy of length scales that the system should satisfy in order to preserve/wipe out the studied effects.

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

1 precise connection to gravity and black hole physics unclear (see below)
2 in parts: re-branding known physics (see below)

The authors discuss electron transport in Weyl semimetals with an inhomogeneous tilt profile of the Weyl cones. The discussion is timely, generalising several recent works on the emergence of analogue black holes in inhomogeneously tilted Weyl semimetals, and applying these to the practically relevant setting of two and three dimensional Weyl semimetals. Finally, it addresses the electron transport in various relevant regimes, including those with non-trivial Berry curvature effects, and it makes a case for the possibility of extending the current approach to the development of tilt-tronics.

All of this renders the work relevant and interesting to a broad community of physicists. The paper is also clear, self-contained, and well-written. Based on this, I do believe the current work warrants publication in SciPost Physics.
However, there are a few points in the discussion that remained unclear to me, which I would like to ask the authors to address:

1) Although the presented work is motivated by previous work on analogue black holes, I do not see any direct use of analogue gravity in the present paper. The tilt profiles used by the authors are inspired by black holes, but no black hole physics is discussed in the present work, the tilt profiles and state evolution are not matched to gravitational physics (see also below), and the majority of the analysis focusses on relatively high energies, where the correspondence to any gravitational system breaks down.
Altogether, the paper reads like a thorough and useful exposition of tilt-tronics and trajectory-engineering, but the link to gravitational physics appears tenuous.

2) The correspondence between the specific tilt profile that is the main focus of the present work and a gravitational black hole, is unclear in several ways:
— the trajectories presented in fig 1 and the effective potential presented in fig 2 contain large regions in which the semiclassical particle is *repelled* from the origin. Clearly this does not correspond to a situation with a (universally attractive) gravitational mass concentrated at the origin.
— the effective potential depicted in fig 2 does not contain a singularity at any point.
— the correspondence of the semiclassical trajectories to geodesic solutions of any gravitational metric is never made. Appendix A shows a recipe for finding such a correspondence in general, but does not reproduce the semiclassical trajectories in the main text (including ones accelerating away from the black hole) starting from a metric and using Einstein’s equations.
— No horizon is identified in the presented semiclassical dynamics or potential.

3) It is not made clear in the present text how the (semiclassical and fully quantum) trajectories connect to the specific tilt profile. As the authors explain, the “horizon” in their construction coincides with the point where a type-I Weyl semimetal transitions into a type-II Weyl semimetal. At this point, the group velocity in the Weyl cone dispersion along the radial direction is zero. The fact that semiclassical trajectories transition through the horizon, and that fully quantum trajectories can even exit the horizon from the inside out, are in apparent contradiction with the naive observation of zero velocity indicating a black hole horizon. Clearly both types of behaviour must be caused by a combination of band curvature and scattering events. How this “high energy” physics enters the effective description of the (semiclassical or fully quantum) trajectories is not discussed in the present manuscript.

4) I find the discussion of a “mirage” appearing in the title and several places in the manuscript very misleading. The low-energy effective description that is argued to mimic gravitational dynamics is found to break down upon approaching the horizon. All this means however, is that the low-energy description breaks down when it is probed at short length scales or high energies. This is not surprising at all, as the low-energy description was only ever an emergent (effective) description. Saying that the emergent long-wavelength physics is a “mirage” because it disappears when probed at short (lattice) length scales, is like saying that a table is not really rigid because its rigidity disappears when probed at the atomic scale.
I think this is an unfair rebranding of the already well-known limits of emergence.

5) It is unclear at the moment why trajectories in the simulations stop at the origin, or why in section 3.2 it is mentioned that wave pakets should accumulate at the origin. From the semiclassical dynamics, the trajectories reach the origin with non-zero velocity, and there is thus no reason for them to suddenly stop there, rather than continuing, bouncing, or scattering. In the fully quantum dynamics, wave packet evolution is unitary, and accumulation at the origin is fundamentally disallowed.

6) minor point: figure 4 appears to have a wrong title on the bottom right graph.

I suggest the authors either include additional details and discussion addressing the points above, or rephrase some of the claims they make in order to avoid these issues.