Majorana zero-modes in a dissipative Rashba nanowire

1 - Detailed analysis of Majorana zero modes in quantum wires subject to dissipation
2 - Intriguing and highly relevant findings: dissipation-induced Majorana zero modes, additional robust zero modes
3 - Clear presentation of the results

1 - The results have not been fully exploited, some questions that can probably be answered with what the authors have found remain open
2 - Implications for Majoranas as potential topological qubits unclear

In their manuscript, the authors discuss a 1D topological superconductor subject to dissipation. In particular, they analyze the presence of edge-localised zero modes in the spectrum of the Liouvillian.

They find several very interesting results. Majorana zero modes of the decoupled (hermitian) model remain robust under dissipation. Dissipation larger than a critical value can stabilize Majorana zero modes even in a topologically trivial regime (as indicated by a bulk gap closing and a change in winding number). Furthermore, additional non-topological zero modes can be induced in the trivial regime via exceptional points. Both Majorana and trivial zero modes are robust to on-site disorder.

These results are timely, interesting to researchers analyzing non-Hermitian topology, open quantum systems, and ones trying to create topological qubits. The presentation of the results is mostly very clear and easily readable. Given the novelty and interest of the results and the target group, I believe that the paper could eventually be published in SciPost Physics.

Before I can make that recommendation, however, I have a number of points that I would kindly ask the authors to consider and address.

1. Maybe most importantly, the interpretation of the results should be sharpened. More precisely, what do the zero modes mean physically? To elaborate: in a hermitian setting, Majorana zero modes (MZMs) for example encode a ground state degeneracy due to a spatially split complex fermionic state, which in turn can be used for potentially robust qubits. Indeed, the authors state that the MZMs acquire a finite lifetime set by the dissipation \gamma. While the authors do not explicitly comment on it, this probably implies all information stored in dissipative MZMs to be effectively washed out within the time scale set by \gamma.

However, there is probably more to the story that is worth mentioning. The MZMs relate to edge states of the damping matrix X (note that the spectrum of the Liouvillian is governed by the damping matrix). As the authors correctly point out, the Liouvillian governs the time-evolution of the density matrix. But what is the steady state that the system evolves to? Which physical observable can be connected to the edge Majoranas (or the „robust zero modes“ [RZMs]), and on which time scales (In the steady state? In the asymptotic approach towards the steady state? In an initial time range?)? The authors for example write at the end of Sec. 3.1 that the MZMs survive - yes, but what is the physical quantity related to them that survives?

Less important, but still to be discussed:

1. The loss specified in Eq. (4) is definitely not a generic loss: it can be seen as loss of electrons with eigenvalue +1 of spin-sigma_x. This is the same spin component to which the magnetic field couples. Which of the results obtained are special to this type of loss, which are generic? It would be nice to have some comment on other effect of other spin polarisations of the loss (although there remains so much to be explore about truly generic couplings to environments that additional future studies seem part of the full answer).
2. In Fig. 3 (a), the MZMs seem show a splitting at large disorder. Are they split at all disorder strengths (maybe just very weakly so)? Is the splitting merely due to an increased overlap of the Majorana wavefunctions with disorder?
3. In Sec. 3.2.1, disorder averages are performed. The authors should comment a bit more on what exactly they do. Should I think of the data shown in Fig. 5 as running different microscopic disorder configurations, then ordering the states in some form (e.g. the real part of their energy or so), and then averaging the eigenvalues with same ordering number over the disorder configurations? Also, it would have been easier for me to have the details of disorder averaging (e.g. 50 runs) in the main text, not the figure caption.
4. Maybe related to point 3, why do the averaged RZMs in Fig. 6(a) all have identical imaginary parts? If one looks at individual disorder runs, do these states still come in pairs with identical imaginary parts within the pair?
5. Grouping RZMs into pairs with identical imaginary parts, do both pairs have weight at both ends, or is one pair located at one end, and the other pair at the other end (if that were the case, why the asymmetry)?
6. Can the authors confirm that the disorder averaging converges after 50 configurations?
7. Do the authors have any idea as to why the peaks of the end states are sometimes not right-left-symmetric? Is that a feature that has converged w.r.t. disorder configurations? Is only the maximal peak height different, but the integrated weight per side remains the same (which would be an edge-dependent smearing out)? Is there a shift of weight from one side to the other?
8. In Fig. 2 (b), 4(c), 6(c), the dots for the MZMs are all red. Red is the end of the shown color scale for the imaginary part of the eigenvalue. Are the MZMs modes with maximal imaginary part, or are they just modes of „high“ imaginary part?
9. Can the authors say more about the robust zero modes (RZMs)? Could they for example identify their wave functions or energies analytically? Is there an analytical way to connect them to the exceptional points (the numerics are certainly quite convincing, but maybe that could help identify the reason for their robustness)?

Finally, I noticed a couple of minor issues - nothing dramatic, but let me just point them out.

1. The authors use the formulation that the density matrix decays - that is a bit ambiguous. The density matrix preserves its trace. It is correct, however, that it evolves towards its steady state expression with an exponential time-dependence.
2. In Eq. (1), the „+h.c.“ leads to a doubling of the hermitian terms (chemical potential, magnetic field). One could introduce an extra factor 1/2, or add the +h.c. only for the terms that need it.
3. To be overly picky, the formulation „we consider uniform loss i.e., L_i \neq o \forall i“ below Eq. (4) would be more on point if the loss amplitude in Eq. (4) would be \gamma_i, and then one could set \gamma_i=\gamma in the main text and \gamma_i spatially-dependent as chosen in the Appendix