pith. sign in

arxiv: 1906.11784 · v1 · pith:RAZVHVKQnew · submitted 2019-06-19 · ❄️ cond-mat.supr-con · cond-mat.mes-hall

Unveiling Signatures of Topological Phases in Open Kitaev Chains and Ladders

Alfonso Maiellaro , Francesco Romeo , Carmine Antonio Perroni , Vittorio Cataudella , Roberta Citro This is my paper

Pith reviewed 2026-05-25 20:00 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.mes-hall
keywords Kitaev chaintopological superconductordifferential conductanceopen quantum systemBogoliubov-de Gennesdisorder effectsnanowire deviceedge modes
0
0 comments X

The pith

Differential conductance in open Kitaev chains and ladders carries signatures of topological phases, some of which survive moderate disorder but vanish upon hybridization with external reservoirs.

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

The work addresses how to characterize topological phases when Kitaev chains and ladders are no longer isolated but coupled to external normal and superconducting leads that inject and extract current. A scattering method formulated in the Bogoliubov-de Gennes framework is used to compute differential conductance as a function of model parameters and the position of a local probe electrode. The calculated response shows clear marks of the topological order known from the closed systems. A subset of these conductance features remains stable even when moderate disorder is added. The same calculation reveals that states originally protected in the isolated chain or ladder lose their topological character once they mix with reservoir states.

Core claim

Applying a scattering technique within the Bogoliubov-de Gennes formulation to Kitaev chains and ladders connected to normal and superconducting leads shows that the differential conductance encodes the topological phase diagram of the isolated systems. Some of these conductance signatures remain robust against moderate disorder. Local spectroscopic measurements obtained by varying the position and distance of the normal electrode further map the spatial structure of the edge modes. Examination of the internal modes demonstrates that topological protection disappears when the original edge states hybridize with states belonging to the external reservoirs.

What carries the argument

Scattering technique within the Bogoliubov-de Gennes formulation for open systems, used to obtain differential conductance and local response as functions of lead position and system parameters.

If this is right

  • Differential conductance can serve as an electrical probe of topological order in nanowire devices.
  • A fraction of the conductance signatures of topology remain detectable even in the presence of moderate disorder.
  • The position of a local normal electrode controls whether topological edge modes are visible in the measured response.
  • Topological protection present in an isolated chain or ladder is lost once the system states hybridize with reservoir states.

Where Pith is reading between the lines

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

  • Device designs must limit the strength of coupling to external leads if topological protection is required for operation.
  • The same scattering approach could be applied to test whether other one-dimensional topological superconductors exhibit similar loss of protection in open geometries.
  • Experimental searches for topological signatures in mesoscopic wires should include controlled variation of lead coupling to distinguish protected from hybridized regimes.

Load-bearing premise

The scattering technique within the Bogoliubov-de Gennes formulation applied to the open system with external leads faithfully reproduces the topological invariants and edge-mode properties without artifacts from lead modeling or coupling strength.

What would settle it

Compute or measure the differential conductance spectrum for a Kitaev chain with increasing normal-lead coupling strength and check whether the zero-bias peak or other predicted topological features disappear at the same coupling values where the calculated mode hybridization begins.

Figures

Figures reproduced from arXiv: 1906.11784 by Alfonso Maiellaro, Carmine Antonio Perroni, Francesco Romeo, Roberta Citro, Vittorio Cataudella.

Figure 1
Figure 1. Figure 1: FIG. 1. A schematic representation of the ideal Kitaev chain [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Energy bands of the Kitaev chain in the non-topologic [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Topological phase diagram of the ladder in the ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Topological phase diagram of the ladder in the (∆ [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Low energy part of the ladder energy spectra as a funct [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Topological phase diagram of the ladder in the ( [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Panel (a): Schematic of a tunnel conductance measure [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. N-KC-SC device: Zero-temperature differential cond [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. N-KC-SC device: Differential conductance (in units o [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. N-KL-SC device. A Kitaev ladder (central region) co [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. N-KL-SC device: zero-temperature differential con [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. N-KL-SC device in the presence of disorder. The pane [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. In-line configuration of the N-KC-SC device. [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. N-KC-SC device. Zero-temperature differential con [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Transparent limit of the N-KL-SC device. Zero-temp [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. site-dependent charge density [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
read the original abstract

In this work, the general problem of the characterization of the topological phase of an open quantum system is addressed. In particular, we study the topological properties of Kitaev chains and ladders under the perturbing effect of a current flux injected into the system using an external normal lead and derived from it via a superconducting electrode. After discussing the topological phase diagram of the isolated systems, using a scattering technique within the Bogoliubov de Gennes formulation, we analyze the differential conductance properties of these topological devices as a function of all relevant model parameters. The relevant problem of implementing local spectroscopic measurements to characterize topological systems is also addressed by studying the system electrical response as a function of the position and the distance of the normal electrode (tip). The results show how the signatures of topological order affect the electrical response of the analyzed systems, a subset of which being robust also against the effects of a moderate amount of disorder. The analysis of the internal modes of the nanodevices demonstrates that topological protection can be lost when quantum states of an initially isolated topological system are hybridized with those of the external reservoirs. The conclusions of this work could be useful in understanding the topological phases of nanowire-based mesoscopic devices.

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 addresses the characterization of topological phases in open Kitaev chains and ladders coupled to normal and superconducting leads. After reviewing the phase diagram of the isolated systems, it employs a scattering technique in the Bogoliubov-de Gennes formalism to compute differential conductance as a function of model parameters, normal-electrode position, and moderate disorder. The central claims are that topological signatures appear in the electrical response (including zero-bias features), a subset of these signatures remain robust to moderate disorder, and hybridization with external reservoirs can destroy topological protection.

Significance. If the conductance signatures are shown to be independent of lead modeling details, the work would provide a concrete route to local spectroscopic characterization of topological nanowires via transport, directly relevant to mesoscopic experiments on Majorana modes.

major comments (2)
  1. [scattering technique and differential conductance analysis] The central claim that differential-conductance signatures directly reflect the topological invariants of the isolated chain/ladder requires explicit verification that these features survive in the weak-coupling limit and remain invariant under changes in lead bandwidth or coupling strength. The abstract itself notes that hybridization destroys protection, yet no such limit or invariance test is described.
  2. [disorder analysis] The statement that 'a subset' of signatures is robust to moderate disorder is load-bearing for the experimental relevance claim, but the manuscript supplies no quantitative definition of 'moderate' (e.g., relative to gap or hopping) nor statistics over disorder realizations; without these, the robustness cannot be assessed.
minor comments (2)
  1. [methods] Notation for the normal-lead coupling and the position-dependent tip conductance should be defined explicitly before the numerical results are presented.
  2. [introduction] Standard references to the closed Kitaev chain phase diagram and to prior BdG scattering studies on Majorana nanowires should be added for context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below and will revise the manuscript accordingly to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [scattering technique and differential conductance analysis] The central claim that differential-conductance signatures directly reflect the topological invariants of the isolated chain/ladder requires explicit verification that these features survive in the weak-coupling limit and remain invariant under changes in lead bandwidth or coupling strength. The abstract itself notes that hybridization destroys protection, yet no such limit or invariance test is described.

    Authors: We agree that an explicit check in the weak-coupling limit and invariance tests under lead-parameter changes would strengthen the central claim. Our scattering calculations employ finite but tunable coupling; the reported conductance features are obtained in regimes where the lead-system hybridization is weak enough for topological signatures to remain visible. In the revised manuscript we will add a dedicated subsection (or appendix) that (i) systematically reduces the coupling strength toward the weak-coupling limit while tracking the zero-bias and finite-bias features, and (ii) varies lead bandwidth and coupling strength over a representative range, confirming that the identified signatures are stable within the parameter window used in the main text. These additions will also make explicit the crossover to the hybridization-dominated regime already alluded to in the abstract. revision: yes

  2. Referee: [disorder analysis] The statement that 'a subset' of signatures is robust to moderate disorder is load-bearing for the experimental relevance claim, but the manuscript supplies no quantitative definition of 'moderate' (e.g., relative to gap or hopping) nor statistics over disorder realizations; without these, the robustness cannot be assessed.

    Authors: We acknowledge that a quantitative definition of 'moderate' disorder and ensemble statistics are necessary for a rigorous assessment. In the original calculations, disorder amplitudes were chosen to lie below the bulk gap scale while still affecting some conductance features. In the revision we will (i) state the disorder strength explicitly in units of the hopping parameter t (e.g., W ≤ 0.3t–0.5t, with the precise range tied to the gap size), and (ii) present averages and standard deviations over at least 50–100 independent disorder realizations for the key conductance traces, thereby quantifying the fraction of signatures that remain robust. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper first discusses the topological phase diagram of the isolated Kitaev systems, then applies the standard scattering technique in the Bogoliubov-de Gennes formulation to the open system with external leads. No load-bearing steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the conductance signatures are computed from the model Hamiltonian and lead couplings without renaming known results or smuggling ansatzes. The derivation remains self-contained against external benchmarks of BdG scattering theory, consistent with the absence of any quoted reduction in the abstract or claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard Bogoliubov-de Gennes formalism and scattering theory for superconducting systems; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Bogoliubov-de Gennes formulation accurately describes the superconducting nanowire systems under study
    Invoked as the basis for the scattering technique used to compute conductance.

pith-pipeline@v0.9.0 · 5759 in / 1319 out tokens · 30521 ms · 2026-05-25T20:00:48.164422+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    Topological Quantum Computation

    Pachos, J.K. Topological Quantum Computation. In Formal Methods for Dynamical Systems; Bernardo, M., de Vink, E., Di Pierro, A., Wiklicky, H., Eds.; Springer Berlin Heidelberg: Berlin, Heidelberg, 2013; Vol. 7938, pp. 150-179 (ISBN 978-3-642-38873-6)

    work page 2013

  2. [2]

    Unpaired Majorana fermions in quantum wires

    Kitaev, A.Y. Unpaired Majorana fermions in quantum wires. Physics-Uspekhi 2001, 44, 131-136

    work page 2001

  3. [3]

    Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator

    Fu, L.; Kane, C.L. Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator. Physical Review Letters 2008, 100, 096407

    work page 2008

  4. [4]

    Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices

    Mourik, V.; Zuo, K.; Frolov, S.M.; Plissard, S.R.; Bakkers, E.P.A.M.; Kouwenhoven, L.P. Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices. Science 2012, 336, 1003-1007

    work page 2012

  5. [5]

    Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor

    Nadj-Perge, S.; Drozdov, I.K.; Bernevig, B.A.; Yazdani, A. Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor. Physical Review B 2013, 88, 020407(R)

    work page 2013

  6. [6]

    Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor

    Nadj-Perge, S.; Drozdov, I.K.; Li, J.; Chen, H.; Jeon, S.; Seo, J.; MacDonald, A.H.; Bernevig, B.A.; Yazdani, A. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science 2014, 346, 602-607

    work page 2014

  7. [7]

    Search for Majorana Fermions in Superconductors

    Beenakker, C.W.J. Search for Majorana Fermions in Superconductors. Annual Review of Condensed Matter Physics 2013, 4, 113–136

    work page 2013

  8. [8]

    Majorana quasiparticles in condensed matter

    Aguado, R. Majorana quasiparticles in condensed matter. La Rivista del Nuovo Cimento 2017, 40, 523–593

    work page 2017

  9. [9]

    Signatures of topological phase transitions in Josephson current-phase discontinuities

    Marra, P.; Citro, R.; Braggio, A. Signatures of topological phase transitions in Josephson current-phase discontinuities. Physical Review B 2016, 93, 220507(R)

    work page 2016

  10. [10]

    Majorana Fermions in One-Dimensional Structures at LaAlO3/SrTiO3 Oxide Interfaces

    Mazziotti, M.; Scopigno, N.; Grilli, M.; Caprara, S. Majorana Fermions in One-Dimensional Structures at LaAlO3/SrTiO3 Oxide Interfaces. Condensed Matter 2018, 3, 37

    work page 2018

  11. [11]

    Atomtronics: Ultracold-atom analogs of electronic devices

    Seaman, B.T.; Kr\" a mer, M.; Anderson, D.Z.; Holland, M.J. Atomtronics: Ultracold-atom analogs of electronic devices. Physical Review A 2007, 75, 023615

    work page 2007

  12. [12]

    Effects of gapless bosonic fluctuations on Majorana fermions in an atomic wire coupled to a molecular reservoir

    Hu, Y.; Baranov, M.A. Effects of gapless bosonic fluctuations on Majorana fermions in an atomic wire coupled to a molecular reservoir. Physical Review A 2015, 92, 053615

    work page 2015

  13. [13]

    Multichannel Generalization of Kitaev's Majorana End States and a Practical Route to Realize Them in Thin Films

    Potter, A.C.; Lee, P.A. Multichannel Generalization of Kitaev's Majorana End States and a Practical Route to Realize Them in Thin Films. Physical Review Letters 2010, 105, 227003

    work page 2010

  14. [14]

    Majorana Bound States without Vortices in Topological Superconductors with Electrostatic Defects

    Wimmer, M.; Akhmerov, A.R.; Medvedyeva, M.V.; Tworzydo, J.; Beenakker, C.W.J. Majorana Bound States without Vortices in Topological Superconductors with Electrostatic Defects. Physical Review Letters 2010, 105, 046803

    work page 2010

  15. [15]

    Crossover from Majorana edge- to end-states in quasi-one-dimensional p-wave superconductors

    Zhou, B.; Shen, S.-Q. Crossover from Majorana edge- to end-states in quasi-one-dimensional p-wave superconductors. Physical Review B 2011, 84, 054532

    work page 2011

  16. [16]

    Majorana fermions and multiple topological phase transition in Kitaev ladder topological superconductors

    Wakatsuki, R.; Ezawa, M.; Nagaosa, N. Majorana fermions and multiple topological phase transition in Kitaev ladder topological superconductors. Physical Review B 2014, 89, 174514

    work page 2014

  17. [17]

    Low-field topological threshold in Majorana double nanowires

    Schrade, C.; Thakurathi, M.; Reeg, C.; Hoffman, S.; Klinovaja, J.; Loss, D. Low-field topological threshold in Majorana double nanowires. Physical Review B 2017, 96, 035306

    work page 2017

  18. [18]

    Majorana zero modes in a ladder of density-modulated Kitaev superconductor chains

    Zhou, B.-Z.; Xu, D.-H.; Zhou, B. Majorana zero modes in a ladder of density-modulated Kitaev superconductor chains. Physics Letters A 2017, 381, 2426-2431

    work page 2017

  19. [19]

    Topological phase diagram of a Kitaev ladder

    Maiellaro, A.; Romeo, F.; Citro, R. Topological phase diagram of a Kitaev ladder. The European Physical Journal Special Topics 2018, 227, 1397-1404

    work page 2018

  20. [20]

    Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures

    Altland, A.; Zirnbauer, M.R. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Physical Review B 1997,55, 1142-1161

    work page 1997

  21. [21]

    Large Chern-number topological superfluids in a coupled-layer system

    Huang, B.; Chan, C.F.; Gong, M. Large Chern-number topological superfluids in a coupled-layer system. Physical Review B 2015, 91, 134512

    work page 2015

  22. [22]

    Transport in nanostructures; Cambridge studies in semiconductor physics and microelectronic engineering; 1st paperback ed.; Cambridge Univ

    Ferry, D.K.; Goodnick, S.M. Transport in nanostructures; Cambridge studies in semiconductor physics and microelectronic engineering; 1st paperback ed.; Cambridge Univ. Press: Cambridge, 1999; ISBN 978-0-521-66365-6

    work page 1999

  23. [23]

    Transition from metallic to tunneling regimes in superconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion

    Blonder, G.E.; Tinkham, M.; Klapwijk, T.M. Transition from metallic to tunneling regimes in superconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion. Physical Review B 1982, 25, 4515-4532

    work page 1982

  24. [24]

    Scattering formula for the topological quantum number of a disordered multimode wire

    Fulga, I.C.; Hassler, F.; Akhmerov, A.R.; Beenakker, C.W.J. Scattering formula for the topological quantum number of a disordered multimode wire. Physical Review B 2011, 83, 155429

    work page 2011

  25. [25]

    Tunneling characteristics of a chain of Majorana bound states

    Flensberg, K. Tunneling characteristics of a chain of Majorana bound states. Physical Review B 2010, 82, 180516(R)

    work page 2010

  26. [26]

    Two-dimensional chiral topological superconductivity in Shiba lattices

    Li, J.; Neupert, T.; Wang, Z.; MacDonald, A.H.; Yazdani, A.; Bernevig, B.A. Two-dimensional chiral topological superconductivity in Shiba lattices. Nature Communications 2016, 7, 12297

    work page 2016

  27. [27]

    High-resolution studies of the Majorana atomic chain platform

    Feldman, B.E.; Randeria, M.T.; Li, J.; Jeon, S.; Xie, Y.; Wang, Z.; Drozdov, I.K.; Andrei Bernevig, B.; Yazdani, A. High-resolution studies of the Majorana atomic chain platform. Nature Physics 2017, 13, 286–291

    work page 2017

  28. [28]

    Andreev Spectroscopy of Molecular States in Resonant and Charge Accumulation Regime

    Romeo, F.; Nappi, C.; Parlato, L.; Sarnelli, E. Andreev Spectroscopy of Molecular States in Resonant and Charge Accumulation Regime. IEEE Transactions on Applied Superconductivity 2018, 28, 1–7

    work page 2018

  29. [29]

    Quantum Interference in Single-Molecule Superconducting Field-Effect Transistors

    Nappi, C.; Romeo, F.; Parlato, L.; Di Capua, F.; Aloisio, A.; Sarnelli, E. Quantum Interference in Single-Molecule Superconducting Field-Effect Transistors. The Journal of Physical Chemistry C 2018, 122, 11498–11504

    work page 2018