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Design of a Majorana trijunction
by Juan Daniel Torres Luna, Sathish R. Kuppuswamy, Anton R. Akhmerov
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Submission summary
Authors (as registered SciPost users):  Anton Akhmerov · Juan Daniel Torres Luna 
Submission information  

Preprint Link:  scipost_202308_00031v1 (pdf) 
Code repository:  https://zenodo.org/record/8121655 
Data repository:  https://zenodo.org/record/8121655 
Date submitted:  20230822 11:17 
Submitted by:  Torres Luna, Juan Daniel 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Computational 
Abstract
Braiding of Majorana states demonstrates their nonAbelian exchange statistics. One implementation of braiding requires control of the pairwise couplings between all Majorana states in a trijunction device. In order to have adiabaticity, a trijunction device requires the desired pair coupling to be sufficiently large and the undesired couplings to vanish. In this work, we design and simulate of a trijunction device in a twodimensional electron gas with a focus on the normal region that connects three Majorana states. We use an optimization approach to find the operational regime of the device in a multidimensional voltage space. Using the optimization results, we simulate a braiding experiment by adiabatically coupling different pairs of Majorana states without closing the topological gap. We then evaluate the feasibility of braiding in a trijunction device for different shapes and disorder strengths.
Current status:
Reports on this Submission
Report #2 by Jay Sau (Referee 2) on 20231011 (Invited Report)
 Cite as: Jay Sau, Report on arXiv:scipost_202308_00031v1, delivered 20231011, doi: 10.21468/SciPost.Report.7930
Report
Dear Editor,
The manuscript titled "Design of a Majorana trijunction" analyzes a trijunction braiding device in a
2DEG, which is in a configuration that would be closest to the nanowire devices recently studied by
Microsoft [7]. The authors, for the first time as far as I am aware, determines the tunnel couplings in a
trijunction, which is a crucial component for braiding in nanowires, using a model that accounts for
the actual geometry of the Y junction beyond a tunneling approximation. The only other work that goes
beyond the tunneling limit that I am aware of is Phys. Rev. B 85, 144501 (2012), though this work
is unrelated to the braiding schemes of current interest. The trijunction of nanowires constructed
out of a 2DEG would be a crucial component of braiding that is needed for either the nanowire,
planar Josephson junction or quantum dot approach to braiding. The calculation and optimization
of this tunneling in a Yjunction can be considered a breakthrough needed in this pathway.
The authors specifically optimize over the length, width and
angle of the junction in the geometry shown in Fig. 1b. The optimized Majorana splitting relative to the
residual Majorana coupling and the topological gap as a function of device geometry as well as
the range in voltages over which the optimized value can be stable are shown in Fig. 5, which
summarizes the bulk of the conclusion of the manuscript. Sec 6. of the manuscript also considers
the effect of electrostatic disorder and estimates that devices would have to be rather clean
with a disorder scale on the order of 10^10/cm^2, consistent with what has been estimated
for a robust topological phase in the nanowires using a 3D nanowire model
(Ahn et al Phys. Rev. Materials 5, 124602 2021). I think this is very nice and timely work,
which should be seriously considered for Scipost. I just have a few reservations that are related
to the presentation of technical details being hard to follow, even for someone who considers this
to be an area of expertise.
In summary, I think this is a nice and important work spelling out in detail how well and when tunneling through
a trijunction, which is a key ingredient of braiding proposals would work. However, I think in its current form,
the manuscript is a bit of a difficult read, partly because of the complexity of describing a braiding device
in a reasonably compact way. Therefore, I cannot recommend publication in its present form until the issues listed as requested changes are addressed. It is possible that the authors decide that some of the issues are too detailed to clutter the
main text. It would be okay for such issues to be addressed in an appendix.
Requested changes
I list these below:
(1) While it may be obvious to the authors, I think they should elaborate on the setup of a
"depleted trijunction" they mention at the beginning of Sec 3. I think in their computation they actually
have nanowires radiating out from the trijunction and the Majoranas seen in the trijunction are
essentially the end Majoranas of the trijunction. These 1.5 micron nanowires are mentioned in the beginning
of Sec 2 but then never again. It would help to mention that the calculation in Sec 3 is focusing on the
trijunction region of a larger simulation that includes the 1.5 micron ideal nanowire device.
Another detail: How close to the center of the junction does
the SC pairing Delta go? This could become important in the case with a phase difference.
(2) The derivation and motivation of Eq. 4 is a bit confusing. The authors admit this part to be
"heuristic"  but they should probably explain (a) that the motivation is to extract the coupling matrix
elements Gamma_{ij} (b) why we need Gamma to characterize the junction given that ultimately one could
expect that the goal is to maintain a large gap at the trijunction and still couple the Majoranas at the
far ends of the wire. While this is partly explained in Eq. 5  Eq.5 seems to assume that only a single
pair of Majoranas are coupled. This seems contrary to Fig 4a as well as the list of requirements in
Sec 4 which seems to include a state where all 3 Majoranas are coupled. It would be good if Sec 3 could
explain the context of Eq. 4 i.e. the need for defining the Gamma_{ij} as well as the motivation
for the parameters delta_{+} better. For example should Eq 5 have absolute value functions on them?
Gamma_{ij} cannot be positive, because they have to be antisymmetric to maintain PHS.
(3) As apparent from the end of the last comment. Part of the issue is that the particlehole
symmetry of the effective Hamiltonian is not clear. The RHS of Eq. 4 is clearly particlehole symmetric,
assuming Gamma is an
antisymmetric matrix. On the other hand, the LHS of Eq. 4 doesn't appear particlehole symmetric unless
one of E0,E1, E2 is zero and the other two have the same magnitude and opposite signs. There are a few other
confusing aspects to the notation. The overlap matrix is defined in terms of psi_j>, where this was not
defined till this point. Assuming this is a typo and was meant to be phi_j>, S appears to be a 3X3 matrix from Eq. 4.
On the other hand, the beginning of Sec 3 talks about the index j describing 6 states. One can make a reasonable guess
based on the text that what is being talked about are the j corresponding to the three lowest states. But this
becomes confusing because these are the lowest states excluding the Majoranas at the ends of the extended nanowire
(top red dot in Fig 4a), which are presumably at exactly zero energy if we assume the Majoranas are well separated.
These are presuably left out when considering E0, E1 and E2.
(4) Moving on the Sec 4, it is not clear that the state with "all three Majoranas coupled" can be described in
terms of delta_{+} i.e. it feels like this would have a very bad ratio of delta_+/delta_. But this is
probably because this is not of concern here for reasons that need to be addressed a la point 1.
A minor notational issue is that the text refers to "lefttop" and "righttop" which feels inconsistent with
the R, L , T notation already introduced in Fig. 1a.
(5) In the sentence "linearly interpolates between the points where two and all Majorana
states are coupled" in Sec. 4, it would be helpful to specify explicitly which state i.e. couplings are being referred
to, maybe simply by referring to Fig 4c.
(6) A somewhat technical point is that one imagines that conductance through a trijunction that leads to reasonable
Majorana gaps might interfere with charging energy (first ingredient listed in Sec 4) because of screening
through tunnel coupling as described in Phys. Rev. Lett. 122, 016801 (2019). Since this is a key requirement of a
trijunction to avoid this screening, it would be good for the authors
to comment on this even if this is something beyond the scope of the analysis.
Report #1 by Anonymous (Referee 1) on 2023108 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202308_00031v1, delivered 20231008, doi: 10.21468/SciPost.Report.7913
Report
Luna, Kuppuswamy, and Ahkmerov numerically investigate the optimal design of Majorana trijunctions by using a 3D electostatic model and then optimising Majorana overlaps to ensure desired strong/weak Majorana couplings. The authors then assess the stability of the optimal trijunctions against shape variations and electrostatic disorder. Ultimately the authors conclude that such trijunctions are unlikely to work, even in devices with purportedly very stateofthe art disorder levels.
I find that the manuscript is clear, mostly well written (there are a few typos, see below), and the conclusions reasonable. I will likely recommend publication after the authors have answered/clarified the following points:
1) The authors have a confusing mix of physical and absolute units when defining the Hamiltonian/setup in Eq. (13). I think it would be useful if the authors had a table, or equivalent, outlining all the physical parameters they utilise in their simulations. (I apologise if I have missed these listed somewhere in the text, but even if they are somewhere in the text I still feel it would be useful to have them centralised since I cannot find them).
In particular:
 The value of the dielectric constant is not listed. It would also be useful to discuss what influence this will have on e.g. the disorder discussion at the end.
 The sizes and lengths of the wires are listed, but not the coherence length. This is especially important to understand the results in Fig. 5 (see also below).
 In the final lines of section 2 there are 3 pairing potentials: \Delta_0, \Delta, and \Delta_t. From the description it appears that these are all (probably) the same. If they are not then it should be made clear how they are related.
 Essentially none of the parameters for Eq. (3) are provided e.g. the value of Zeeman energy is not listed, only that the topological phase is reached. Mass, SOI strength are also not given. This is important to be sure the authors used realistic parameters.
2) Is there any firmer justification for the thresholds outlined in Eqs. (1011)? It would be useful to understand what these thresholds (roughly) mean in terms of operational temperatures/timescales. At the moment they just appear to be rather arbitrarily chosen.
3) As stated above, Fig. 5 would be greatly improved by understanding the size of the coherence length in comparison with the junction geometry. However, I also do not really understand the physics behind the statement “smaller geometries perform better”. Is there any way of understanding this finding? At first sight it is rather surprising given that one would expect unwanted couplings to be larger in smaller junctions. (I am also not sure I agree that ~ 10^11 cm^2 is really achieved in Microsoft’s experiment).
Minor remarks/typos:
a) Last sentence of Sec. 4 “wavevunctions”
b) Full stop missing before last sentence in Fig. 5 caption
c) Many typos due to lack of capitalisation in references
Author: Juan Daniel Torres Luna on 20240116 [id 4248]
(in reply to Report 1 on 20231008)
Reply to the Referee 1
We thank the referee for reading our manuscript and commenting on possible points of improvement. In the updated version we have addressed all points raised by the referee. We attach a pdf with the highlighted changes.
(1) The authors have a confusing mix of physical and absolute units when defining the Hamiltonian/setup in Eq. (13). I think it would be useful if the authors had a table, or equivalent, outlining all the physical parameters they utilise in their simulations. (I apologise if I have missed these listed somewhere in the text, but even if they are somewhere in the text I still feel it would be useful to have them centralised since I cannot find them).
In order to make it easier to find all the parameters that we used in the simulations, we added a corresponding appendix.
The value of the dielectric constant is not listed. It would also be useful to discuss what influence this will have on e.g. the disorder discussion at the end.
We have added the value of the dielectric constant to the parameter listing in the appendix. At this point we do not have concrete conclusions regarding optimal material choices and believe that a proper analysis extends beyond comparing the dielectric constants.
In the final lines of section 2 there are 3 pairing potentials: $\Delta_0$, $\Delta$, and $\Delta_t$. From the description it appears that these are all (probably) the same. If they are not then it should be made clear how they are related.
The different Delta's are detailed in the last few sentences of the last paragraph of Sec. 2.
(2) Is there any firmer justification for the thresholds outlined in Eqs. (1011)? It would be useful to understand what these thresholds (roughly) mean in terms of operational temperatures/timescales. At the moment they just appear to be rather arbitrarily chosen.
In the new version of the manuscript we explain the reasoning behind Eqs. (10, 11) and why these requirements must be satisfied to achieve braiding. We agree that the values of the thresholds are somewhat arbitrary. These values roughly correspond to minimal requirements for braiding. We now state this in the updated manuscript.
(3) As stated above, Fig. 5 would be greatly improved by understanding the size of the coherence length in comparison with the junction geometry. However, I also do not really understand the physics behind the statement “smaller geometries perform better”. Is there any way of understanding this finding? At first sight it is rather surprising given that one would expect unwanted couplings to be larger in smaller junctions.
We have explained in the manuscript that the performance of the trijunction depends on the length scales of the normal region rather than on the length scales of the superconductor. We have added the coherence length and the Majorana localization length to the appendix for completeness.
(I am also not sure I agree that ~ $10^{11}$ cm$^{2}$ is really achieved in Microsoft’s experiment).
We have reworded the citation to use a more neutral stance on the Microsoft claim ("reported" instead of "achieved").
Author: Juan Daniel Torres Luna on 20240116 [id 4247]
(in reply to Report 2 by Jay Sau on 20231011)Reply to Referee 2
We thank the referee for reading our manuscript and raising points for improvement. In the updated version we address all points raised by the referee. We attach a pdf with the highlighted changes.
We agree with the referee that the full setup of our simulation was not clear. To address this point we added a panel to Figure 1 where we show the entire simulation domain and added a corresponding clarification in section 2.
This is explicitly stated in the text. Specifically, after Eq. (3) we write "The superconducting pairing is absent in the normal region". The "normal region" is defined above Eq. (1) by referencing Fig. 1 (c).
We agree with the referee that the motivation of the effective Hamiltonian derivation was not clear. Because the goal of optimizing the couplings $\Gamma_{ij}$ is to implement the braiding sequence, we have moved the braiding sequence to Fig. 1 and its corresponding explanation to the beginning of section 3. Furthermore, we have also explained that we focus on the microcopic coupling of Majoranas closest to the middle region, and therefore we project out the far Majoranas.
We thank the referee for this suggestion. We have expanded the motivation of Eq. 4 and $\Gamma_{ij}$ at the beginning of section 3. We believe that the role of $\delta_{+}$ as a tool for the optimization procedure is clear, however, we have extended the explanation of its role in the loss function right before Eq. 6.
We agree with the referee, and we have updated $delta_{+}$ accordingly.
We agree with the referee that the particlehole symmetry of Eq. (4) is not clear and that the energies should be as described. In the new version we address this point by explicitly stating the particlehole symmetry of the Hamiltonian, and the antisymmetry of the $\Gamma_{ij}$.
We thank the referee for pointing out the confusing aspect of the notation. In this case $\psi_j\rangle$ refers to a general eigenstate of three strongly coupled Majoranas which we decompose as a linear combination of three individual Majoranas. We have properly defined $\psi_j\rangle$ referencing Fig 2 (b) where a coupled eigenstate is shown.
In the updated manuscript we explain that we study only the suitability of the trjunction for braiding, and Sec. 3 we state that we project away the eigenstates not localized in the trijunction.
We have clarified this aspect by renaming the sections and explaining that we use a different loss function that does not rely on $\delta_+$ or $\delta_$ for tuning to the triple coupled point in Eq. 9.
We have changed the notation to L, R and T everywhere.
We now label each step of the braiding protocol, and in the new version we refer specifically to the steps between which we interpolate the voltages.
We agree with the referee. We have modified Sec. 3 to clarify that the competition between Coulomb and trijunction mediated couplings is beyond the scope of our work.
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