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Topological states of generalized dissipative Majorana wires

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Pith's one-line read Generalized dissipative Kitaev chains with next-nearest-neighbor jump terms realize steady states with topological winding number 4, while imbalanced couplings switch the winding number sign.

arxiv 2608.10556 v1 pith:P3SVU2LJ submitted 2026-08-11 cond-mat.mes-hall cond-mat.quant-gas

Farokhnaz Hosseinifar , Ali G. Moghaddam This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.quant-gas
keywords majoranaquantumstateswindingdampingdissipativefermionicgaps
verification ladder T0 review T1 audit T2 compute T3 formal

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The reading

The paper considers one-dimensional fermionic wires coupled to a bath, so that the system's dynamics is described by a Lindblad master equation. For a purely dissipative setup (no Hamiltonian), the steady state can be a pure 'dark state' with nontrivial topology. The standard dissipative Kitaev model has a winding number of 2, meaning two Majorana bound states at each end. The authors generalize this model in two ways: first, by making the excitation and de-excitation strengths to neighboring sites unequal (Model I), and second, by adding couplings to next-nearest neighbors (Model II). Using the adjoint Lindblad equation for two-point correlation functions, they compute the damping and purity spectra, and then evaluate the winding number of the steady state. They find that Model II supports a phase with winding number 4 when the next-nearest-neighbor coupling strength exceeds a threshold, and Model I allows switching between positive and negative winding numbers by tuning the imbalance. The calculations are analytic and produce phase diagrams for the two models. A caveat is that the paper relies on a mean-field replacement of the original number-conserving jump operators by phase-conserving linear operators, following earlier work, and this replacement is not re-derived for the generalized models. There is also an internal sign inconsistency in the definition of the d-vector that affects the sign of the winding number, although the relative sign-switching behavior may still hold.
Extended reading notes

Core claim

By inclusion of quantum jumps between next-nearest-neighbor sites, higher winding numbers and, equivalently, more Majorana bound states can be achieved, giving a phase with winding number ν=4 for |β|>1/2; imbalanced couplings allow switching between positive and negative winding numbers, with the sign determined by sgn(tanθ).

Load-bearing premise

The replacement of the original particle-conserving quartic Lindblad jump operators (Eq. 2) by quadratic phase-conserving operators (Eq. 3) via a mean-field/fixed-phase ensemble is assumed to hold for the generalized n.n.n. and imbalanced models, following Ref. [13], but is not re-derived; if this approximation is uncontrolled for these models, the correlation-matrix and winding-number analysis does not apply.

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Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard open-quantum-system assumptions and on the mean-field reduction taken from prior work. No fitted constants or new physical entities are introduced. The model parameters κ, θ, β are physical coupling constants of the Lindblad operators, not free parameters tuned to data.

assumptions (6)
  • domain assumption Born-Markov approximation and Lindblad form of the master equation
    Eq. (1) assumes Markovian, time-local dynamics for the reduced density matrix.
  • domain assumption Purely dissipative dynamics with H=0
    Section II sets H=0, justified by truncation to a ground-state manifold with large energy splitting.
  • domain assumption Mean-field replacement of quartic Lindblad operators by quadratic phase-conserving operators (Eq. 3)
    Adopted from Ref. [13] without re-derivation for the generalized n.n.n. and imbalanced models.
  • standard math Gaussian steady-state ansatz for the density matrix (Eq. 6)
    Follows from the quadratic form of the Lindblad operators; the entanglement Hamiltonian is free-fermionic.
  • standard math Correlation matrix encodes all information of Gaussian states
    Standard result from Refs. [38-40]; used to reduce the density matrix to C_k.
  • domain assumption Topological transitions occur only at closures of damping or purity gaps
    Cited to Ref. [13]; assumed to hold for the generalized models.

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Pith. "Pith review of Topological states of generalized dissipative Majorana wires." pith.science (2026). https://pith.science/paper/P3SVU2LJ

@misc{pith2026260810556,
  author       = {Pith},
  title        = {Pith review of: Topological states of generalized dissipative Majorana wires},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P3SVU2LJ}},
  note         = {Machine review of arXiv:2608.10556}
}
read the original abstract

We study the generalized one-dimensional (1D) quantum dissipative models corresponding to a Majorana wire which can possess more than one Majorana bound state at each end. The system consists of a 1D fermionic open quantum system whose dynamics is governed by a quadratic Lindblad equation. Using the adjoint Lindblad equation for the fermionic two-point correlations, we find the gaps in the damping and purity spectra of a generic 1D model. Then, using the symmetry-based classification, we show that a winding number as the topological invariant can be defined which distinguishes different steady states of the system in the presence of damping and purity gaps. Then we focus on certain models with different Lindblad quantum jump terms and explore their phase diagrams by calculating the damping and the purity gaps as well as the winding number. In particular, we show that by inclusion of quantum jumps between next-nearest-neighbor sites, higher winding numbers and, equivalently, more Majorana bound states can be achieved. Also, by introducing imbalanced couplings, we can switch between states with negative and positive winding numbers. Finally, we should mention that since our formulation is based on the fermionic correlations rather than the Majorana operators, it can be easily extended to the dissipative topological phases belonging to other symmetry classes.

Figures

Figures reproduced from arXiv: 2608.10556 by the authors.

Figure 1
Figure 1. FIG. 1. 1D cold atom chain in an optical lattice. The phys [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Dissipative (damping) gap of the Kitaev model [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Dissipative (damping) gap and (b) the topolog [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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