Dissipation-Induced Steady States in Topological Superconductors: Mechanisms and Design Principles

\textbf{Q1.} “Even in the Hamiltonian case, Majorana modes are not required to be exact zero modes, but are considered protected when their eigenvalues are exponentially small. Although this is hinted at in some places in the manuscript, it would be good if the protection criteria in this manuscript are formulated more explicitly in this way.”

\textbf{A1.} We agree with the Referee’s remark on this matter. The core finding of our work—the algebraic relation (43) between the dimensions of zero mode subspaces in isolated and open systems—is mathematically exact only for strictly zero modes. When the zero modes of the isolated system are of Majorana type and are affected by weak disorder, interactions, or finite size effects, it is well known that the corresponding subspace undergoes an exponentially small splitting. In such a situation, the robustness of our result hinges on how the ZKMs subspace responds to this slight splitting of the subspace of MMs.

To formulate the applicability conditions, robustness, and physical implications of our main result more explicitly, we have significantly restructured Section 4. Specifically, the principal result is now presented in Section 4.1 as a rigorous mathematical statement, Theorem 1, together with its corollaries. The mathematical details of the proof of Theorem 1 have been moved to Appendix B. We have further expanded the discussion in Sections 4.2 and 4.3, where we analyze the stability of our conclusions with respect to disorder and interaction effects (Section 4.2) and to non-Markovian features of the external leads (Section 4.3). The discussion in Section 4.2 is directly related to our response to the Referee’s next question.

\textbf{Q2.} “An important question is the robustness of the remaining zero modes to small local perturbations in the Hamiltonian or dissipative fields. It seems that in Sec. 4.4, case (c) should be robust in this sense while cases (a) or (b) might not. It would be useful to state this clearly and provide some representative examples”

\textbf{A2.} We thank the Referee for this important question. Given that the primary effect of perturbing the Hamiltonian is the modification of its spectrum, in the revised manuscript, we have carefully analyzed the robustness of the ZKMs with respect to small variations in the energies of subgap excitations for the single-dissipator case $N_B = 1$. Specifically, in Appendix C, by analyzing the characteristic equation for the kinetic modes, we show that if the isolated subgap energies, including Majorana modes (MMs), are small compared to the spectral gap, then the characteristic splitting scale of the robust ZKMs will be of the same order. A similar general analysis for weak ZKMs is more complex; however, an examination of the special case $N_B = N_M = 1$ (see Eqs. (115)-(116) and the accompanying discussion) reveals that weak ZKMs are also robust against to small variations in the isolated modes energies, their wave functions, and the applied dissipative fields.

To verify these analytical findings, we performed numerical calculations of the length dependence for the subgap mode energies in both isolated and open systems, as shown in Fig. 5. We considered scenarios where the gapless MM subspace is split solely by finite-size effects (panels (a) and (b)) as well as by on-site disorder (panel (c)). For all cases, we considered random realizations of the linear dissipative fields and averaged the results over 500 such configurations. As evident from Fig.5, the characteristic splitting scales of the MM and ZKM subspaces are comparable across all investigated conditions.

This analysis has led us to assume a general robustness of ZKMs against small perturbations of the Hamiltonian
and the applied dissipative fields. This point has been elaborated upon in Sec.4.2 of the revised manuscript.

\textbf{Q3.} “Some of the works cited in the Manuscript considered the case where the Hamiltonian is trivial and the dissipation gives rise to the Majorana modes. In the language of the current work one could say that these previous studies often start from the limit of zero Hamiltonian, in which all modes are zero modes, but none is protected, and use coupling to baths to gap out most of them except for a few spatially-seperated ones. This is hinted in the manuscript, but an explicit discussion could be very useful.”

\textbf{A3.} We agree with the reviewer’s comment on this point. At the end of Section 4.1 in the revised manuscript, we have added an explicit discussion of the limiting case with no unitary dynamics, which has been studied in considerable detail in the literature [ S. Diehl, E. Rico, M.A. Baranov, and P. Zoller, Nat. Phys. 7, 971–977 (2011); C.-E. Bardyn, M. A. Baranov, E. Rico, A. Imamoglu, P. Zoller, and S. Diehl, Phys. Rev. Lett. 109, 130402 (2012); C.-E. Bardyn, M. A. Baranov, C. V. Kraus, E. Rico, A. Imamoglu, P. Zoller, and S. Diehl, New J. Phys. 15, 085001 (2013) ] in the context of engineering nontrivial topological phases and Majorana modes in one-dimensional systems through purely dissipative effects. Our discussion clarifies that this regime emerges as a specific consequence of Theorem 1 when $\dim \ker A$ coincides with the dimension of the subspace spanned by the eigenmodes of the isolated Hamiltonian.

\textbf{Q4.} “A final small comment: When deriving the Lindblad equation in the Appendix, the Authors refrain from using the rotating wave approximation. As is well known in the field of open quantum systems, this gives rise to the Redfield equation, which, as opposed to the Lindblad equation, is not guaranteed to preserve the positivity of the density matrix. The way this issue might arise in the Appendix is because the matrix $\gamma$, while Hermitian by construction, might not be positive definite, implying an instability. While the rest of the manuscript does not depend on this issue, the Authors should of course still check this.”

\textbf{A4.} We agree with the Referee and thank them for this precise observation. Indeed, the main goal of Appendix A was to demonstrate the possibility of describing external Markovian reservoirs using linear dissipative fields. Similar question concerning single-band Fermi leads that probe the system has been discussed in our related work [S.V. Aksenov, M.S. Shustin, I.S. Burmistrov, arXiv:2511.07196 (2025)], where dissipative effects of reservoirs were treated within a quantum-field-theoretic framework. We cite this work in Section 4.3 of the revised manuscript, within the analysis of ZKM robustness against non-Markovian effects.

Nevertheless, the reviewer is correct to note that since the derivation of the linear dissipative fields in Eq.(87) was performed without invoking the standard rotating wave approximation, the matrix $\gamma$ may not be positive-definite, and one must monitor the regimes where positive definiteness breaks down. In the revised manuscript, we explicitly mention this nuance. However, we believe that a detailed study of the properties of the $\gamma$ matrix and the specific regimes where it loses positive definiteness lies beyond the scope of our study.

\textbf{Q1.} “First, the algebraic counting relation could be stated with greater rigor: its assumptions, scope, and precise conditions of validity should be formulated explicitly (ideally as a theorem with proof in an appendix).”

\textbf{A1.} We are grateful to the Referee for this suggestion. In the revised manuscript, the central result of our work — an algebraic relation that connects the number of Majorana modes (MMs) and zero kinetic modes (ZKMs) — is now presented as a formal theorem (see Sec. 4.1). This formulation allows us to state the necessary assumptions about the system with full rigor and to discuss the principal physical implications resulting from the theorem. These implications include: a natural classification of ZKMs into robust and weak ones; the physical meaning behind the stability of robust zeros; strategies for engineering weak ZKMs; a mechanism for maximizing the number of robust zeros; and the possibility of inducing weak ZKMs in dissipative systems (including the case without unitary dynamics). Section 4.1 also contextualizes these findings by discussing their connections to prior research in the field. The details of the mathematical proof of Theorem 1, which were previously included in the main text, have been moved to Appendix B in the updated version of the manuscript.

\textbf{Q2.} “Second, robustness issues are only lightly touched upon. Since realistic systems inevitably involve interactions, disorder, and non-Markovian baths, the authors should analyse or at least discuss stability under such perturbations.”

\textbf{A2.} We sincerely thank the reviewer for raising this significant and timely point. Addressing these questions formed the core of the revisions in the updated manuscript. First, recognizing that Theorem 1 presupposes strictly zero modes in an isolated system, we have investigated the robustness of our findings against small perturbations in subgap excitation energies. Such perturbations in topological superconductors can stem from electron-electron interactions, disorder, finite-size effects, or their combinations. A central element of our analysis was to examine how the subspace of degenerate ZKMs splits under these perturbations. In Appendix C of the revised manuscript, a detailed analysis of the characteristic equation for a dissipative system with a single bath ($N_B = 1$) demonstrates that the characteristic splitting of the ZKM subspace is of the same order as the typical energy splitting of the subgap modes in the isolated system. We have further corroborated these analytical insights with numerical calculations, considering the dependence of MM and ZKM energies on the length of extended Kitaev chain across different topological phases, configurations of dissipative fields, and levels of on-site disorder (see Fig. 5 and the corresponding discussion).

These results allow us to argue in Section 4.2 that the implications of Theorem 1, outlined in Section 4.1, are not merely ideal mathematical constructions. Instead, they represent robust physical features of topological superconductors, stable against interaction effects, disorder, and finite-size effects — provided the system are in a topological phase with $N_M$ pairs of Majorana modes. Finally, Section 4.3 examines the validity of Theorem 1 when considering non-Markovian effects from external baths, specifically wide-band Fermi reservoirs. This section synthesizes relevant findings from our recent related work [ S. V. Aksenov, M. S. Shustin, I. S. Burmistrov, \textit{Controlling Quantum Transport in a Superconducting Device via Dissipative Baths}, arXiv:2511.0719 ]. That work showed that modeling tunnel coupling to multiple Fermi reservoirs is spectrally equivalent to introducing into the GKSL equation additional linear dissipative fields, governed by the tunneling amplitudes, see Eqs.(47)-(48). As Theorem 1 places no constraints on the physical origin of the dissipative fields, its conclusions regarding the number and structure of ZKMs remain fully applicable even in the presence of such non-Markovian, wide-band leads.

In summary, the newly added Sections 4.2 and 4.3, along with Appendix C, provide, in our view, a comprehensive discussion of the robustness of Theorem 1’s results against interactions and disorder in topological superconductors, as well as their extension to systems with non-Markovian reservoirs.

\textbf{Q3.} “Third, the connection to prior literature, particularly “no-go” results on dissipative topology, should be sharpened: the manuscript must explicitly delineate how the present approach circumvents or complements those limitations.”

\textbf{A3} We agree with the reviewer’s comment. In the updated version of the manuscript, we have provided a more detailed description of the connection between our findings and the results of previous studies.

In Section 4.1, we specifically highlighted a limiting case of Theorem 1 where the dissipative system lacks unitary dynamics (i.e., the GKSL equation contains no Hamiltonian). This scenario has been extensively studied in prior works [ S. Diehl, E. Rico, M.A. Baranov, and P. Zoller, Nat. Phys. 7, 971–977 (2011); C.-E. Bardyn, M. A. Baranov, E. Rico, A. Imamoglu, P. Zoller, and S. Diehl, Phys. Rev. Lett. 109, 130402 (2012); C.-E. Bardyn, M. A. Baranov, C. V. Kraus, E. Rico, A. Imamoglu, P. Zoller, and S. Diehl, New J. Phys. 15, 085001 (2013)] in the context of engineering nontrivial topological phases and Majorana modes in one-dimensional systems through purely dissipative effects. We demonstrate that the results of Theorem 1 can, in a sense, be viewed as a generalization of these earlier constructions.

However, we subsequently note that while Theorem 1 provides information about the number and structure of ZKMs, it does not inform us about the purity or topological properties of the density matrix corresponding to the nonequilibrium steady state (NESS). Here, we specifically reference the known no-go theorem from the literature, which states that it is impossible to realize pure and topologically nontrivial NESS in systems of free fermions with spatial dimension $d \geq 2$, under some relatively weak assumptions about the properties of the Liouvillian [ M. Goldstein, SciPost Physics 7 (2019) ]. Our results are in no way at odds with these findings. Finally, in Section 4.4, we explicitly demonstrate that the conditions of Theorem 1 for a nonzero number of ZKMs (a nonzero kernel of the hybridization matrix B, see Eq. (44)) do not contradict the known theorems regarding the conditions for a unique NESS in an open system. Specifically, we explicitly construct a strong symmetry operator, see Eq. (53), whose nontrivial structure is consistent with the condition $\dim \ker B > 0$, while at the same time it violates the conditions for NESS uniqueness theorems of Refs.[A. Frigerio, Lett. Math. Phys. 2, 79 (1977); A. Frigerio, Commun. Math. Phys. 63, 269–276 (1978); H. Yoshida, Phys. Rev. A 109, 022218 (2024) ].