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Quantum Scars and Caustics in Majorana Billiards

by R. Johanna Zijderveld, A. Mert Bozkurt, Michael Wimmer, İnanç Adagideli

Submission summary

Authors (as registered SciPost users): Inanc Adagideli · A. Mert Bozkurt · Michael Wimmer · Johanna Zijderveld
Submission information
Preprint Link:  (pdf)
Code repository:
Date submitted: 2024-01-22 14:10
Submitted by: Zijderveld, Johanna
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approaches: Theoretical, Computational


We demonstrate that the classical dynamics influence the localization behaviour of Majorana wavefunctions in Majorana billiards. By using a connection between Majorana wavefunctions and eigenfunctions of a normal state Hamiltonian, we show that Majorana wavefunctions in both p-wave and s-wave topological superconductors inherit the properties of the underlying normal state eigenfunctions. As an example, we demonstrate that Majorana wavefunctions in topological superconductors with chaotic shapes feature quantum scarring. Furthermore, we show a way to manipulate a localized Majorana wavefunction by altering the underlying classical dynamics using a local potential away from the localization region. Finally, in the presence of chiral symmetry breaking, we find that the Majorana wavefunction in convex-shaped Majorana billiards exhibits caustics formation, reminiscent of a normal state system with magnetic field.

Current status:
Awaiting resubmission

Reports on this Submission

Anonymous Report 2 on 2024-4-15 (Invited Report)


1- Detailed discussion of the mapping between the wavefunctions of topological superconductor and normal state eigenfunctions.

2- Study of the effect of the confinement one the properties of superconductor wavefunctions

3- The paper is clearly written


1- The results for the localization of superconductor wavefunctions often lack quantitative analysis.

2- The fact that what is seen on Fig 5 and 6 are caustic associated with the "effective magnetic field " of Eq (9) is not demonstrated.

3- The importance of these results for the physics of topological superconductor might be discussed further.


In this paper, the authors make use of the mapping between the wave function of topological superconductors at parity switch and normal state Hamiltonian to study the localization properties of Majorana wavefunctions. The question addressed is interesting and the paper clearly written.

The authors however often stop their analysis to this mapping. For instance in Fig 2, it is clear that (b) is deduced from (a) through the mapping (5) and one can recognize visually the trace of the periodic orbit shown on (a). But does this really imply localization ? One would like to see a quantitative comparison of, for instance, inverse participation ratio, to see how much of the localization implied by scarring transfer to the Majorana through (5).

In the same way, the fact that a stopper placed in a location where the Majorana wavefunction is almost zero affect this latter is very reminiscent of Aharonov-Bohm effect, and maybe more discussion about how much this analogy is valid could be useful.

Also, although it is true that Berry discuss in [32] the consequence of caustic for the "magnetic Green function", this is the free Green function which is discussed there (without confining potential). This of course would also apply qualitatively to a billiard if the cyclotron radius is much smaller than the typical size of the system. However the regime considered here (weak chiral symmetry breaking) seems to imply weak effective magnetic field, and I am afraid that what we see might just be the caustics of the unperturbed dynamics. This in any case has to be clarified.

Finally, as non-specialist of topological superconductor, I would have appreciated more in depth discussion about the importance, for topological superconductor physics, of the kind of localization discussed in the paper.

Requested changes

1-Study more quantitatively the localization properties of the scarred wavefunctions, by comparing for instance their inverse participation ration to the average one.

2- Provide information about the cyclotron radius for the effective magnetic field in Fig 5 and 6 and discuss how much this effective magnetic field modifies the classical dynamics.
More generally show which classical trajectories (and thus which caustics) are involved in these figure.

3- Discuss the connection (or absence of connection) between the phenomenology observed in Fig. 3 & 4 and Aharonov-Bohm effect.

4- Provide more discussion about the importance, for topological superconductor physics, of the kind of localization discussed in the paper.

Point of details :

a- As the Majorana wavefunctions are 1/2 spinors, it might be useful to specify to what correspond the scalar functions plotted in the various figures.

b- Figure 2c is a rather trivial consequence of the the mapping (5). Is this really useful ? In any case putting it next to Fig 2a&b could give the wrong impression that it's a check of the localization of the wavefunction (due to scarring), and is thus slightly misleading.

c- Just before Eq (9) : why could we set \Delta_x = \Delta_y= \Delta "without loss of generality" ?


Ask for major revision

  • validity: ok
  • significance: good
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: good

Anonymous Report 1 on 2024-3-18 (Invited Report)


The manuscript presents a novel localization phenomenon in Majorana billiards that is explained by connecting the scarred Majorana states to the eigenstates of the corresponding normal state system.


The main weakness of the manuscript is that some parts of text would benefit more depth explanations.


The authors study scarring in various Majorana billiards that they connect to the scarring in the normal state counterpart, even demonstrating a way to manipulate these states with a localized perturbation. However, some parts of the text would benefit of further clarification. In conclusion, I can recommend the publication of this work in SciPost if the manuscript was adequately revised according to at least some of the mentioned issues.

Requested changes

I recommend the following clarifications/additions to the manuscript:

(1) The authors mention the Heller type scarring, and also refer to the many-body-scars. However, in this context, the authors could consider the perturbation-induced scarring [Phys.
Rev. Lett. 123, 214101 (2019); J. Phys. Condens. Matter 31, 105301 (2019); Phys. Rev. B 96, 094204 (2017)] as well. As I see, there should be a very straightforward way to generalize the Majorana description to this type of scarring by considering locally perturbed p-type superconductors.

(2) The Hamiltonian for a topological superconductor is given, but it would be beneficial if the context behind this Hamiltonian would be described in more detailed, at least to give a citation.

(3) The authors refer to the potential of the system, but it is not defined clearly. What I have deduced is that they consider various billiard systems with hard wall boundaries where the potential is zero.

(4) It is unclear what the authors mean by the semiclassical limit. As far I see, all the simulations and analysis are fully quantum, and close to the ground state. Furthermore, the authors emphasize the topological nature of the superconductor, but not clarify if it has an impact the observed scarring, for instance whether the scars are chiral also (based on my conclusion, they are not).

(5) The authors consider how a hard disk stopper affects the scarred states, and mention this as a possible experimental avenue. However, a realistic STM nanotip would instead produce a soft bump. Nevertheless, I don't see this modification to change their conclusion.

(6) Finally, the manuscript mention multiple times the chaotic behavior of the system. However, no studies is carried out or shown, such as Poincare's surface of sections, or level statistics. In particular, I suspect the system to be highly mixed in the case when the artificial vector potential is present. For example, the candy wrap shape seen in Fig. 7 is not an periodic in a hard wall stadium, but it does appear in a smooth stadium that appear more like an elliptical oscillator.

  • validity: high
  • significance: good
  • originality: high
  • clarity: good
  • formatting: excellent
  • grammar: excellent

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