Design of a Majorana trijunction

Dear Editor,

The manuscript titled "Design of a Majorana tri-junction" 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 Y-junction 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.

I list these below:
(1) While it may be obvious to the authors, I think they should elaborate on the set-up 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 anti-symmetric to maintain PHS.


(3) As apparent from the end of the last comment. Part of the issue is that the particle-hole
symmetry of the effective Hamiltonian is not clear. The RHS of Eq. 4 is clearly particle-hole symmetric,
assuming Gamma is an
antisymmetric matrix. On the other hand, the LHS of Eq. 4 doesn't appear particle-hole 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 "left-top" and "right-top" 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.

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 state-of-the 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. (1-3). 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. (10-11)? 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