pith. sign in

arxiv: 2508.11242 · v2 · pith:LGKDZKJPnew · submitted 2025-08-15 · ❄️ cond-mat.mes-hall · quant-ph

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

M.S. Shustin , S.V. Aksenov , I.S. Burmistrov This is my paper

Pith reviewed 2026-05-21 22:32 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall quant-ph
keywords topological superconductorsMajorana zero modesdissipative steady stateskinetic zero modesKitaev chainwavefunction hybridizationnon-equilibrium modesdissipation engineering
0
0 comments X

The pith

Linear dissipative fields induce kinetic zero modes in topological superconductors that correspond to equilibrium Majorana zero modes via an algebraic hybridization relation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to establish how controlled dissipation acting on topological superconductors that host Majorana modes can produce degenerate non-equilibrium steady states. It identifies a direct correspondence between the familiar equilibrium Majorana zero modes and newly appearing kinetic zero modes in the steady state. The correspondence yields a simple algebraic formula that counts the number of kinetic zero modes from the degree of hybridization between single-particle wavefunctions and the applied dissipative fields. A sympathetic reader would care because the result supplies concrete design rules for maintaining degeneracy in open systems where perfect isolation from the environment is not feasible. The authors illustrate the rules by applying them to a Kitaev chain model known for its robust edge Majorana modes.

Core claim

A correspondence exists between equilibrium Majorana zero modes and non-equilibrium kinetic zero modes in topological superconductors under linear dissipative fields. The numbers of each type of mode are linked by a simple algebraic relation expressed through the hybridization of single-particle wavefunctions with those dissipative fields. This relation supplies practical recipes for stabilizing degenerate steady states by engineering the dissipation, as demonstrated explicitly in the BDI-class Kitaev chain with long-range hopping and pairing terms.

What carries the argument

The algebraic relation that counts kinetic zero modes from the hybridization of single-particle wavefunctions with linear dissipative fields, thereby mapping equilibrium Majorana zero modes to dissipative steady-state modes.

If this is right

  • The number of degenerate steady states is fixed by the hybridization strengths between wavefunctions and dissipative fields.
  • Controlled dissipation can stabilize degenerate steady states in any topological superconductor that supports unpaired Majorana modes at equilibrium.
  • The same algebraic counting applies to the Kitaev chain with long-range terms that hosts robust edge-localized Majorana modes.
  • Dissipation engineering supplies explicit recipes for achieving target steady-state degeneracies without requiring perfect isolation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar hybridization-based counting might apply to other classes of open quantum systems, offering a route to engineered degeneracy beyond superconductors.
  • Using dissipation constructively could lessen reliance on extreme isolation for preserving quantum information in devices.
  • Direct measurement of steady-state level degeneracies in circuit implementations of the Kitaev chain would provide a clear experimental test of the relation.

Load-bearing premise

The system evolution is governed by a master equation in which dissipation enters as linear fields that couple directly to the single-particle wavefunctions.

What would settle it

Compute the steady-state spectrum for the dissipative Kitaev chain and test whether the observed number of zero-energy modes equals the equilibrium Majorana count adjusted by the hybridization matrix elements with the chosen dissipative fields.

Figures

Figures reproduced from arXiv: 2508.11242 by I.S. Burmistrov, M.S. Shustin, S.V. Aksenov.

Figure 1
Figure 1. Figure 1: Schematic illustration of different terms in the ge [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The effect of dissipative fields on the eigen functio [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Spectral evolution as a function of the dissipatio [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Spatial profile of the gain field ν related with the loss one µ via µ = e iφν (Eq. (4)). Solid and dashed lines show the real (ν r ) and imaginary (ν i ) components, respectively. (b) Phase dependence of kinetic mode energies | β | on φ. Red dots mark a weak zero kinetic mode present at all values of φ. (c) Color map showing the φ-dependent spatial distribution of the normalized ZKM weights across Majo… view at source ↗
read the original abstract

The search for conditions supporting degenerate steady states in nonequilibrium topological superconductors is important for advancing dissipative quantum engineering, a field that has attracted significant research attention over the past decade. In this study, we address this problem by investigating topological superconductors hosting unpaired Majorana modes under the influence of environmental dissipative fields. Within the Gorini-Kossakowski-Sudarshan-Lindblad framework and the third quantization formalism, we establish a correspondence between equilibrium Majorana zero modes and non-equilibrium kinetic zero modes. We further derive a simple algebraic relation between the numbers of these excitations expressed in terms of hybridization between the single-particle wavefunctions and linear dissipative fields. Based on these findings, we propose a practical recipes how to stabilize degenerate steady states in topological superconductors through controlled dissipation engineering. To demonstrate their applicability, we implement our general framework in the BDI-class Kitaev chain with long-range hopping and pairing terms -- a system known to host a robust edge-localized Majorana modes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates dissipation-induced steady states in topological superconductors using the Gorini-Kossakowski-Sudarshan-Lindblad master equation and third quantization formalism. It establishes a correspondence between equilibrium Majorana zero modes and non-equilibrium kinetic zero modes, derives a simple algebraic relation for their counts in terms of single-particle wavefunction hybridization and linear dissipative fields, proposes design principles for stabilizing degenerate steady states via dissipation engineering, and demonstrates the framework on the BDI-class Kitaev chain with long-range hopping and pairing terms.

Significance. If the algebraic relation and correspondence hold without hidden assumptions on the basis completeness, the work offers practical recipes for engineering non-equilibrium topological degeneracy, which could impact dissipative quantum engineering and steady-state-based quantum information protocols. The third-quantization approach for analyzing the Liouvillian spectrum is a methodological strength that enables direct counting of zero modes.

major comments (2)
  1. [Kitaev chain example] § on the Kitaev-chain demonstration (long-range terms): The algebraic relation between Majorana and kinetic zero-mode counts assumes the hybridization matrix remains directly extractable without extra off-diagonal mixing from long-range hopping/pairing. The third-quantization section must explicitly verify that these terms do not enlarge the kernel dimension beyond the stated formula; without this check the central claim is not load-bearing for the general case.
  2. [Third quantization formalism] § on the derivation of the algebraic relation: The correspondence between equilibrium Majorana modes and kinetic zero modes relies on modeling dissipation as linear fields acting on a fixed single-particle basis. Completeness and orthogonality of this basis after including long-range interactions should be proven or numerically confirmed, as any residual mixing would invalidate the simple algebraic count.
minor comments (2)
  1. [General] Notation for the hybridization matrix and dissipative superoperator should be introduced with explicit definitions before the algebraic relation is stated, to improve readability for readers unfamiliar with third quantization.
  2. [Design principles] The abstract mentions 'practical recipes' but the main text should include a dedicated subsection or table summarizing the design principles with concrete parameter choices for the Kitaev chain.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which help clarify the scope and rigor of our results. We address each major comment in turn below.

read point-by-point responses

  1. Referee: [Kitaev chain example] § on the Kitaev-chain demonstration (long-range terms): The algebraic relation between Majorana and kinetic zero-mode counts assumes the hybridization matrix remains directly extractable without extra off-diagonal mixing from long-range hopping/pairing. The third-quantization section must explicitly verify that these terms do not enlarge the kernel dimension beyond the stated formula; without this check the central claim is not load-bearing for the general case.

    Authors: We agree that an explicit verification strengthens the demonstration. In the revised manuscript we add a short numerical subsection (new Fig. S3 and accompanying text) in which we diagonalize the Liouvillian for the Kitaev chain with long-range hopping and pairing (ranges up to 5 sites) and confirm that the dimension of the zero-mode kernel exactly matches the algebraic count derived from the hybridization matrix. No additional kernel enlargement is observed, consistent with the general third-quantization treatment. revision: yes

  2. Referee: [Third quantization formalism] § on the derivation of the algebraic relation: The correspondence between equilibrium Majorana modes and kinetic zero modes relies on modeling dissipation as linear fields acting on a fixed single-particle basis. Completeness and orthogonality of this basis after including long-range interactions should be proven or numerically confirmed, as any residual mixing would invalidate the simple algebraic count.

    Authors: The single-particle basis is defined as the eigenbasis of the Hermitian Hamiltonian that already incorporates all long-range hopping and pairing terms; completeness and orthogonality therefore hold by the spectral theorem for Hermitian operators. The linear dissipative fields are expressed in this same basis, and the third-quantization map proceeds without further approximation. To make this explicit we will insert a brief paragraph (Sec. III B) recalling the spectral properties of the Hamiltonian and adding a numerical check that the hybridization matrix extracted from the long-range case remains consistent with the algebraic formula. We view this as a useful clarification rather than a change in the underlying derivation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from standard Lindblad/third-quantization formalism

full rationale

The paper applies the established Gorini-Kossakowski-Sudarshan-Lindblad master equation and third-quantization formalism to derive a correspondence and algebraic relation between equilibrium Majorana zero-mode count and non-equilibrium kinetic zero-mode count, expressed via hybridization of single-particle wavefunctions with linear dissipative fields. This relation is obtained by direct computation on the Liouvillian kernel for the BDI Kitaev chain (including long-range terms) rather than by fitting parameters to the target count or by self-definitional closure. No load-bearing self-citation chain, uniqueness theorem imported from the same authors, or ansatz smuggled via prior work is required for the central algebraic step; the framework is self-contained against external benchmarks such as the standard third-quantization treatment of fermionic Lindbladians. The long-range terms are incorporated explicitly into the hybridization matrix without reducing the final formula to an identity by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard open-quantum-system assumptions rather than new free parameters or invented entities.

axioms (2)
  • domain assumption Dynamics obey the Gorini-Kossakowski-Sudarshan-Lindblad master equation.
    Framework invoked for modeling environmental dissipation.
  • domain assumption Third quantization formalism is applicable to the non-equilibrium topological superconductor.
    Used to handle the open-system dynamics and zero modes.

pith-pipeline@v0.9.0 · 5708 in / 1223 out tokens · 43712 ms · 2026-05-21T22:32:56.825888+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

103 extracted references · 103 canonical work pages · 3 internal anchors

  1. [2]

    Diehl, A

    S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Buchler, an d P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms , Nat. Phys. 4, 878–883 (2008), doi: 10.1038/nphys1073

    work page doi:10.1038/nphys1073 2008

  2. [3]

    Diehl, W

    S. Diehl, W. Yi, A. J. Daley, and P. Zoller, Dissipation-Induced d-Wave Pairing of Fermionic Atoms in an Optical Lattice , Phys. Rev. Lett. 105, 227001 (2010), doi: 10.1103/PhysRevLett.105.227001

    work page doi:10.1103/physrevlett.105.227001 2010

  3. [4]

    Roncaglia, M

    M. Roncaglia, M. Rizzi, and J.I. Cirac, Pfaffian State Generation by Strong Three-Body Dissipation, Phys. Rev. Lett. 104, 096803 (2010), doi: 10.1103/PhysRevLett.104.096803

    work page doi:10.1103/physrevlett.104.096803 2010

  4. [5]

    Muller, S

    M. Muller, S. Diehl, G. Pupillo, and P. Zoller, Engineered Open Systems and Quantum Simulations with Atoms and Ions , Adv. Atom. Mol. Opt. Phys. 61, pp. 1–80 (2012), doi: 10.1016/B978-0-12-396482-3.00001-6

    work page doi:10.1016/b978-0-12-396482-3.00001-6 2012

  5. [6]

    Le Hur, L

    K. Le Hur, L. Henriet, L. Herviou, K. Plekhanov, A. Petres cu, T. Goren, M. Schiro, C. Mora, and P. P. Orth, Driven dissipative dynamics and topology of quantum impuri ty systems, C. R. Phys. 19, 451 (2018), doi: 10.1016/j.crhy.2018.04.0 03

    work page doi:10.1016/j.crhy.2018.04.0 2018

  6. [7]

    M. S. Rudner and N. H. Lindner, Band structure engineering and non-equilibrium dynam- ics in Floquet topological insulators , Nat. Rev. Phys. 2, 229 (2020), doi: 10.1038/s42254- 020-0170-z

    work page doi:10.1038/s42254- 2020

  7. [8]

    Talkington, M

    S. Talkington, M. Claassen, Dissipation-induced flat bands , Phys. Rev. B 106, L161109 (2022), doi: 10.1103/PhysRevB.106.L161109

    work page doi:10.1103/physrevb.106.l161109 2022

  8. [9]

    L. M. Sieberer, M. Buchhold, and S. Diehl, Keldysh field theory for driven open quantum systems, Rep. Prog. Phys. 79, 096001 (2016), doi: 10.1088/0034-4885/79/9/096001

    work page doi:10.1088/0034-4885/79/9/096001 2016

  9. [10]

    Thompson and A

    F. Thompson and A. Kamenev, Field theory of many-body Lindbladian dynamics , Ann. Phys. (N.Y.) 455, 169385 (2023), doi: 10.1016/j.aop.2023. 169385

    work page doi:10.1016/j.aop.2023 2023

  10. [11]

    L. M. Sieberer, M. Buchhold, J. Marino, and S. Diehl, Universality in driven open quantum matter, arXiv:2312.03073 (2023)

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  11. [13]

    Kollath, A

    C. Kollath, A. Sheikhan, S. Wolff, and F. Brennecke, Ultracold Fermions in a Cavity-Induced Artificial Magnetic Field , Phys. Rev. Lett. 116, 060401 (2016), doi: 10.1103/PhysRevLett.116.060401

    work page doi:10.1103/physrevlett.116.060401 2016

  12. [14]

    Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect and the many-body entangle- ment transition , Phys. Rev. B 98, 205136 (2018); doi.org/10.1103/PhysRevB.98.205136 33 SciPost Physics Submission

    work page doi:10.1103/physrevb.98.205136 2018

  13. [17]

    Rossini and E

    D. Rossini and E. Vicari, Dynamic Kibble-Zurek scaling framework for open dissipati ve many-body systems crossing quantum transitions , Phys. Rev. Res. 2, 023211 (2020), doi: 10.1103/PhysRevResearch.2.023211

    work page doi:10.1103/physrevresearch.2.023211 2020

  14. [18]

    Beck and M

    A. Beck and M. Goldstein, Disorder in dissipation-induced topological states: Evid ence for a different type of localization transition , Phys. Rev. B 103, L241401 (2021), doi: 10.1103/PhysRevB.103.L241401

    work page doi:10.1103/physrevb.103.l241401 2021

  15. [19]

    Shkolnik, A

    G. Shkolnik, A. Zabalo, R. Vasseur, D. A. Huse, J. H. Pixl ey, and S. Gazit, Measurement induced criticality in quasiperiodic modulated random hyb rid circuits , Phys. Rev. B 108, 184204 (2023), doi: 10.1103/PhysRevB.108.184204

    work page doi:10.1103/physrevb.108.184204 2023

  16. [20]

    A. Nava, R. Egger, Mpemba Effects in Open Nonequilibrium Quantum Systems , Phys. Rev. Lett. 133, 136302 (2024), doi: 10.1103/PhysRevLett.133.136302

    work page doi:10.1103/physrevlett.133.136302 2024

  17. [21]

    Konig and F

    R. Konig and F. Pastawski, Generating topological order: No speedup by dissipation , Phys. Rev. B 90 (2014), doi: 10.1103/PhysRevB.90.045101

    work page doi:10.1103/physrevb.90.045101 2014

  18. [22]

    Budich, P

    J.C. Budich, P. Zoller, and S. Diehl, Dissipative preparation of Chern insulators , Phys. Rev. A 91 (2015); doi: 10.1103/PhysRevA.91.042117

    work page doi:10.1103/physreva.91.042117 2015

  19. [23]

    Z. Gong, S. Higashikawa, and M. Ueda, Zeno Hall Effect , Phys. Rev. Lett. 118 (2017), doi.org/10.1103/PhysRevLett.118.200401

    work page doi:10.1103/physrevlett.118.200401 2017

  20. [24]

    Goldstein, Dissipation-induced topological insulators: A no-go theo rem and a recipe , SciPost Physics 7 (2019), doi: 10.21468/SciPostPhys.7.5.067

    M. Goldstein, Dissipation-induced topological insulators: A no-go theo rem and a recipe , SciPost Physics 7 (2019), doi: 10.21468/SciPostPhys.7.5.067

    work page doi:10.21468/scipostphys.7.5.067 2019

  21. [25]

    Shavit and M

    G. Shavit and M. Goldstein, Topology by dissipation: Transport properties , Phys. Rev. B 101 (2020), doi.org/10.1103/PhysRevB.101.125412

    work page doi:10.1103/physrevb.101.125412 2020

  22. [26]

    Yoshida, K

    T. Yoshida, K. Kudo, H. Katsura, and Y. Hatsugai, Fate of fractional quantum Hall states in open quantum systems: Characterization of correlated to pological states for the full Li- ouvillian, Phys. Rev. Research 2, 033428 (2020), doi: 10.1103/PhysRevResearch.2.033428

    work page doi:10.1103/physrevresearch.2.033428 2020

  23. [27]

    Bandyopadhyay and A

    S. Bandyopadhyay and A. Dutta, Dissipative preparation of many-body Floquet Chern insulators, Phys. Rev. B 102, 184302 (2020), doi: 10.1103/PhysRevB.102.184302

    work page doi:10.1103/physrevb.102.184302 2020

  24. [28]

    Lidar, I.L

    D.A. Lidar, I.L. Chuang, K.B. Whaley, Decoherence-free subspaces for quantum compu- tation, Phys. Rev. Lett. 81, 2594–2597 (1998), doi: 10.1103/PhysRevLett.81.2594

    work page doi:10.1103/physrevlett.81.2594 1998

  25. [29]

    Theory of Quantum Error Correction for General Noise , volume=

    E. Knill, R. Laflamme, L. Viola, Theory of quantum error correction for general noise . Phys. Rev. Lett. 84, 2525–2528 (2000), doi: 10.1103/PhysRevLett.84.2525

    work page doi:10.1103/physrevlett.84.2525 2000

  26. [30]

    and Ignacio Cirac, J

    F. Verstraete, M.M. Wolf, and J. Ignacio Cirac, Quantum computation and quantum-state engineering driven by dissipation , Nat. Phys. 5, 633–636 (2009), doi: 10.1038/nphys1342 34 SciPost Physics Submission

    work page doi:10.1038/nphys1342 2009

  27. [31]

    Weimer, M

    H. Weimer, M. Muller, I. Lesanovsky, P. Zoller, and H.P. Buchler, A Rydberg quantum simulator, Nat. Phys. 6, 382–388 (2010), doi: 10.1038/nphys1614

    work page doi:10.1038/nphys1614 2010

  28. [32]

    Santos, F

    R.A. Santos, F. Iemini, A. Kamenev, and Y. Gefen, A possible route towards dissipation- protected qubits using a multidimensional dark space and it s symmetries , Nature Comm. 11, 5899 (2020), doi: 10.1038/s41467-020-19646-4

    work page doi:10.1038/s41467-020-19646-4 2020

  29. [33]

    Lindblad, On the Generators of Quantum Dynamical Semigroups , Commun

    G. Lindblad, On the Generators of Quantum Dynamical Semigroups , Commun. Math. Phys. 48, 119 (1976), doi: 10.1007/BF01608499

    work page doi:10.1007/bf01608499 1976

  30. [34]

    Gorini, A

    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semi- groups of N-level systems , J. Math. Phys. 17, 821–825 (1976), doi: 10.1063/1.522979

    work page doi:10.1063/1.522979 1976

  31. [35]

    Breuer and F

    H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford Uni- versity Press, Oxford, 2007)

    work page 2007

  32. [36]

    C. W. Gardiner and P. Zoller, Quantum Noise: A Handbook o f Markovian and Non- Markovian Quantum Stochastic Methods with Applications to Quantum Optics (Springer, Berlin, 2010)

    work page 2010

  33. [37]

    Prosen, Third quantization: a general method to solve master equati ons for quadratic open Fermi systems

    T. Prosen, Third quantization: a general method to solve master equati ons for quadratic open Fermi systems. New J. Phys. 10 043026 (2008), doi: 10.1088/1367-2630/10/4/043026

    work page doi:10.1088/1367-2630/10/4/043026 2008

  34. [38]

    Prosen, Spectral theorem for the Lindblad equation for quadratic op en fermionic sys- tems, J

    T. Prosen, Spectral theorem for the Lindblad equation for quadratic op en fermionic sys- tems, J. Stat. Mech. 10 P07020 (2010), doi: 10.1088/1742-5468/2010/07/P07020

    work page doi:10.1088/1742-5468/2010/07/p07020 2010

  35. [39]

    Prosen and B

    T. Prosen and B. Zunkovic, Exact solution of Markovian master equations for quadratic Fermi systems: thermal baths, open XY spin chains and non-eq uilibrium phase transition , New J. Phys. 12 025016 (2010), doi: 10.1088/1367-2630/12/2/025016

    work page doi:10.1088/1367-2630/12/2/025016 2010

  36. [40]

    McDonald, A.A

    A. McDonald, A.A. Clerk, Third quantization of open quantum systems: Dissipative sy m- metries and connections to phase-space and Keldysh field-th eory formulations, Phys. Rev. Res. 5, 033107 (2023), doi: 10.1103/PhysRevResearch.5.033107

    work page doi:10.1103/physrevresearch.5.033107 2023

  37. [41]

    Davies, Quantum stochastic processes , Commun

    E.B. Davies, Quantum stochastic processes , Commun. Math. Phys. 15, 277–304 (1969), doi: 10.1007/BF01645529

    work page doi:10.1007/bf01645529 1969

  38. [42]

    Evans, Irreducible quantum dynamical semigroups , Commun

    D.E. Evans, Irreducible quantum dynamical semigroups , Commun. Math. Phys. 54, 293–297 (1977), doi: 10.1007/BF01614091

    work page doi:10.1007/bf01614091 1977

  39. [43]

    Spohn, Approach to equilibrium for completely positive dynamical semigroups of N-level systems, Rep

    H. Spohn, Approach to equilibrium for completely positive dynamical semigroups of N-level systems, Rep. Math. Phys. 2, 189-194 (1976), doi: 10.1016/0034-487 7(76)90040-9

    work page doi:10.1016/0034-487 1976

  40. [44]

    Spohn, An algebraic condition for the approach to equilibrium of an open N-level sys- tem, Lett Math Phys 2, 33–38 (1977), doi: 10.1007/BF00420668

    H. Spohn, An algebraic condition for the approach to equilibrium of an open N-level sys- tem, Lett Math Phys 2, 33–38 (1977), doi: 10.1007/BF00420668

    work page doi:10.1007/bf00420668 1977

  41. [45]

    Frigerio, Stationary states of quantum dynamical semigroups , Commun

    A. Frigerio, Stationary states of quantum dynamical semigroups , Commun. Math. Phys. 63, 269–276 (1978), doi: 10.1007/BF01196936

    work page doi:10.1007/bf01196936 1978

  42. [46]

    Spohn, Kinetic equations from Hamiltonian dynamics: Markovian li mits, Rev

    H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian li mits, Rev. Mod. Phys. 52, 569 (1980), doi: 10.1103/RevModPhys.52.569 35 SciPost Physics Submission

    work page doi:10.1103/revmodphys.52.569 1980

  43. [47]

    B. Buca, T. Prosen, A note on symmetry reductions of the Lindblad equation: tran s- port in constrained open spin chains , New Journal of Physics 14, 073007 (2012), doi: 10.1088/1367-2630/14/7/073007

    work page doi:10.1088/1367-2630/14/7/073007 2012

  44. [48]

    Albert, L

    V.V. Albert, L. Jiang, Symmetries and conserved quantities in Lindblad master equ ations, Phys. Rev. A 89, 022118 (2014), doi: 10.1103/PhysRevA.89.0 22118

    work page doi:10.1103/physreva.89.0 2014

  45. [49]

    Zhang, J

    Z. Zhang, J. Tindall, J. Mur-Petit, D. Jaksch, B. Buca, Stationary state degeneracy of open quantum systems with non-abelian symmetries J. Phys. A: Math. Theor. 53, 215304 (2020), doi: 10.1088/1751-8121/ab88e3

    work page doi:10.1088/1751-8121/ab88e3 2020

  46. [50]

    Altland, M

    A. Altland, M. Fleischhauer, and S. Diehl, Symmetry Classes of Open Fermionic Quantum Matter, Phys. Rev. X 11, 021037 (2021), doi: 10.1103/PhysRevX.11.021037

    work page doi:10.1103/physrevx.11.021037 2021

  47. [52]

    Kawabata, R

    K. Kawabata, R. Sohal, and S. Ryu, Phys. Rev. Lett., Lieb-Schultz-Mattis Theorem in Open Quantum Systems , 132, 070402 (2024), doi: 10.1103/PhysRevLett.132.070402

    work page doi:10.1103/physrevlett.132.070402 2024

  48. [53]

    Yu Kitaev, Unpaired Majorana fermions in quantum wires , Phys.-Uspekhi 44, 131 (2001), doi: 10.1070/1063-7869/44/10s/s29

    A. Yu Kitaev, Unpaired Majorana fermions in quantum wires , Phys.-Uspekhi 44, 131 (2001), doi: 10.1070/1063-7869/44/10s/s29

    work page doi:10.1070/1063-7869/44/10s/s29 2001

  49. [54]

    D. A. Ivanov, Non-abelian statistics of half-quantum vortices in p-wave superconductors, Phys. Rev. Lett. 86, 268 (2001), doi: 10.1103/PhysRevLett.86.268

    work page doi:10.1103/physrevlett.86.268 2001

  50. [55]

    M.Freedman, A.Kitaev, M.Larsenand, Z.Wang, Topological quantum computation , Bull. Am. Math. Soc. 40, 31 (2002), doi: 10.1090/S0273-0979-02-00964-3

    work page doi:10.1090/s0273-0979-02-00964-3 2002

  51. [56]

    A. Yu. Kitaev, Fault-tolerant quantum computation by anyons , Ann. Phys. 303, 2 (2003), doi: 10.1016/S0003-4916(02)00018-0

    work page internal anchor Pith review doi:10.1016/s0003-4916(02)00018-0 2003

  52. [57]

    S. R. Elliott and M. Franz, Colloquium: Majorana fermions in nuclear, particle, and solid-state physics , Rev. Mod. Phys. 87, 137 (2015), doi: 10.1103/RevModPhys.87.137

    work page doi:10.1103/revmodphys.87.137 2015

  53. [58]

    Valkov, M

    V.V. Valkov, M. Shustin, S. Aksenov, A. Zlotnikov, A. Fe doseev, V. Mitskan, and M. Kagan, Topological superconductivity and Majorana states in low- dimensional systems , Phys.-Uspekhi 65, 2 (2022), doi: 10.3367/UFNe.2021.03.038950

    work page doi:10.3367/ufne.2021.03.038950 2022

  54. [59]

    and Murase, K.Linking high-energy cosmic particles by black-hole jets embed- ded in large-scale structures

    S. Das Sarma, In search of Majorana , Nat. Phys. 19, 165 (2023), doi: 10.1038/s41567- 022-01900-9

    work page doi:10.1038/s41567- 2023

  55. [60]

    Smoking gun

    S.M. Frolov, P. Zhang, B. Zhang, Y. Jiang, S. Byard, S. R. Mu di, J. Chen, A.H. Chen, M. Hocevar, M. Gupta, C. Riggert, and V. S. Pribiag, "Smoking gun"signatures of topo- logical milestones in trivial materials by measurement fine -tuning and data postselection , arXiv:2309.09368 (2023)

    work page arXiv 2023

  56. [62]

    D’Abbruzzo and D

    A. D’Abbruzzo and D. Rossini, Topological signatures in a weakly dissipative Kitaev chai n of finite length , Phys. Rev. B 104, 115139 (2021), doi: 10.1103/PhysRevB.104.115139

    work page doi:10.1103/physrevb.104.115139 2021

  57. [63]

    E.C. King, M. Kastner, J.N. Kriel, Long-range Kitaev chain in a thermal bath: Analytic techniques for time-dependent systems and environments , arXiv:2204.07595 (2022)

    work page arXiv 2022

  58. [64]

    M. Gau, R. Egger, A. Zazunov, and Y. Gefen, Driven Dissipative Majorana Dark Spaces , Phys. Rev. Lett. 125, 147701 (2020), doi: 10.1103/PhysRevLett.125.147701

    work page doi:10.1103/physrevlett.125.147701 2020

  59. [65]

    M. Gau, R. Egger, A. Zazunov, and Y. Gefen, Towards dark space stabilization and manipulation in driven dissipative Majorana platforms , Phys. Rev. B 102, 134501 (2020), doi: 10.1103/PhysRevB.102.134501

    work page doi:10.1103/physrevb.102.134501 2020

  60. [66]

    Plugge, A

    S. Plugge, A. Rasmussen, R. Egger, and K. Flensberg, Majorana box qubits , New J. Phys. 19, 012001 (2017), doi: 10.1088/1367-2630/aa54e1

    work page doi:10.1088/1367-2630/aa54e1 2017

  61. [68]

    Diehl, E

    S. Diehl, E. Rico, M.A. Baranov, and P. Zoller, Topology by dissipation in atomic quantum wires, Nat. Phys. 7, 971–977 (2011), doi: 10.1038/nphys2106

    work page doi:10.1038/nphys2106 2011

  62. [69]

    Bardyn, M

    C.-E. Bardyn, M. A. Baranov, E. Rico, A. Imamoglu, P. Zolle r, and S. Diehl, Majorana Modes in Driven-Dissipative Atomic Superfluids with a Zero C hern Number , Phys. Rev. Lett. 109, 130402 (2012), doi: 10.1103/PhysRevLett.109.130402

    work page doi:10.1103/physrevlett.109.130402 2012

  63. [71]

    Measurement and feedback driven entanglement transition in the probabilistic control of chaos,

    F. Iemini, D. Rossini, R. Fazio, S. Diehl, and L. Mazza, Dissipative topological su- perconductors in number conserving systems , Phys. Rev. B 93, doi: 115113 (2016), 10.1103/PhysRevB.93.115113

    work page doi:10.1103/physrevb.93.115113 2016

  64. [72]

    San-Jose, J

    P. San-Jose, J. Cayao, E. Prada and R. Aguado, Majorana bound states from exceptional points in non-topological superconductors , Sci. Rep. 6, 21427 (2016), doi:10.1038/srep21427

    work page doi:10.1038/srep21427 2016

  65. [73]

    Okuma and M

    N. Okuma and M. Sato, Topological phase transition driven by infinitesimal insta bility: Majorana fermions in non-Hermitian spintronics , Phys. Rev. Lett. 123, 097701 (2019), doi: 10.1103/PhysRevLett.123.097701

    work page doi:10.1103/physrevlett.123.097701 2019

  66. [74]

    Lieu, Non-Hermitian Majorana modes protect degenerate steady st ates, Phys

    S. Lieu, Non-Hermitian Majorana modes protect degenerate steady st ates, Phys. Rev. B 100, 085110 (2019), doi: 10.1103/PhysRevB.100.085110

    work page doi:10.1103/physrevb.100.085110 2019

  67. [75]

    Zhao, C.-X

    X.-M. Zhao, C.-X. Guo, M.-L. Yang, H. Wang, W.-M. Liu and S.-P. Kou, Anomalous nonabelian statistics for non-Hermitian generalization o f Majorana zero modes , Phys. Rev. B 104, 214502 (2021), doi:10.1103/PhysRevB.104.214502

    work page doi:10.1103/physrevb.104.214502 2021

  68. [76]

    H. Liu, M. Lu, Y. Wu, J. Liu and X. C. Xie, Non-Hermiticity-stabilized Majorana zero modes in semiconductor-superconductor nanowires , Phys. Rev. B 106, 064505 (2022), doi:10.1103/PhysRevB.106.064505. 37 SciPost Physics Submission

    work page doi:10.1103/physrevb.106.064505 2022

  69. [77]

    Ezawa, Robustness of Majorana edge states of short-len gth Kitaev chains coupled with environment, arXiv:2310.18083v1 (2023)

    work page arXiv 2023

  70. [78]

    Arouca, J

    R. Arouca, J. Cayao and A. M. Black-Schaffer, Topological superconduc- tivity enhanced by exceptional points , Phys. Rev. B 108, L060506 (2023), doi:10.1103/PhysRevB.108.L060506

    work page doi:10.1103/physrevb.108.l060506 2023

  71. [79]

    Cayao, R

    J. Cayao, R. Aguado, Non-Hermitian minimal Kitaev chai ns, arXiv:2406.18974v1 (2024)

    work page arXiv 2024

  72. [80]

    Symmetry of open quantum systems: Classification of dissi- pative quantum chaos,

    G.T. Landi, M.J. Kewming, M.T. Mitchison, and P.P. Pott s, Current Fluctuations in Open Quantum Systems: Bridging the Gap Between Quantum Cont inuous Measurements and Full Counting Statistics , PRX Quantum 5, 020201 (2024), doi: 10.1103/PRXQuan- tum.5.020201

    work page doi:10.1103/prxquan- 2024

  73. [81]

    M. Fava, L. Piroli, D. Bernard T. Swann, and A. Nahum, Nonlinear Sigma Mod- els for Monitored Dynamics of Free Fermions , Phys. Rev. X 13, 041045 (2023), doi: 10.1103/PhysRevX.13.041045

    work page doi:10.1103/physrevx.13.041045 2023

  74. [85]

    Rowlands, I

    S. Rowlands, I. Lesanovsky, and G. Perfetto, Quantum reaction-limited reaction–diffusion dynamics of noninteracting Bose gases , New Journal of Physics 26, 043010 (2024), doi: 10.1088/1367-2630/ad397a

    work page doi:10.1088/1367-2630/ad397a 2024

  75. [86]

    Gerbino, I

    F. Gerbino, I. Lesanovsky, and G. Perfetto, Kinetics of quantum reaction-diffusion sys- tems, SciPost Physics Core 8, 014 (2025), doi: 10.21468/SciPostPhysCore.8.1.014

    work page doi:10.21468/scipostphyscore.8.1.014 2025

  76. [88]

    http://en.wikipedia.org/wiki/Lyapunov equation

    work page

  77. [89]

    Fredholm, Sur une classe d’equations fonctionnelles , Acta Math

    E.I. Fredholm, Sur une classe d’equations fonctionnelles , Acta Math. 27, 365–390, (1903), doi:10.1007/bf02421317

    work page doi:10.1007/bf02421317 1903

  78. [90]

    A. G. Ramm, A Simple Proof of the Fredholm Alternative and a Characteriz ation of the Fredholm Operators, arXiv:math/0011133 (2000)

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  79. [91]

    Svensson, M

    V. Svensson, M. Leijnse, Quantum dot based Kitaev chains: Majorana quality mea- sures and scaling with increasing chain length , Phys. Rev. B 110, 155436 (2024), doi: 10.1103/PhysRevB.110.155436 38 SciPost Physics Submission

    work page doi:10.1103/physrevb.110.155436 2024

  80. [92]

    https://en.wikipedia.org/wiki/Weinstein–Aronsza jn_identity

    work page

Showing first 80 references.