Analytical classification of Majorana zero-mode spatial profiles in extended Kitaev chains: probability maxima can shift inward

Sujit Sarkar; Vaishnav Mallya; Vijay Pathak

arxiv: 2506.20182 · v2 · submitted 2025-06-25 · ❄️ cond-mat.other

Analytical classification of Majorana zero-mode spatial profiles in extended Kitaev chains: probability maxima can shift inward

Vijay Pathak , Vaishnav Mallya , Sujit Sarkar This is my paper

Pith reviewed 2026-05-19 08:36 UTC · model grok-4.3

classification ❄️ cond-mat.other

keywords Majorana zero modesKitaev chainspatial profilesrecursion relationstopological superconductorsextended Kitaev modelzero energy modes

0 comments

The pith

Majorana zero modes in extended Kitaev chains can have probability maxima at interior sites rather than the edges.

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

This paper derives closed-form expressions for the spatial profiles of Majorana zero modes in an extended Kitaev chain with nearest and next-nearest neighbor couplings. Transforming the Hamiltonian to the Majorana basis produces a recursion relation whose characteristic roots fix the amplitude at every site. These roots reveal several distinct decay patterns for the zero modes, including cases where the highest probability occurs away from the boundary. The analysis also determines how long a finite chain must be before its zero-mode structure matches that of a semi-infinite system. Such explicit spatial information connects the microscopic parameters to measurable local densities of states.

Core claim

By rewriting the extended Kitaev Hamiltonian in the Majorana basis the authors obtain a linear recurrence relation for the zero-mode amplitudes. The roots of the characteristic equation of this recurrence completely determine the spatial structure, yielding exact expressions for the mode amplitudes at each lattice site. In particular, certain ranges of the roots produce boundary modes whose probability density reaches its maximum at an interior site and then decays exponentially outward from that point on both sides.

What carries the argument

Recursion relation for the Majorana zero-mode amplitudes obtained after expressing the Hamiltonian in the Majorana basis, with its characteristic roots fixing the spatial envelope and closed-form amplitudes.

Load-bearing premise

The zero-energy eigenvectors are exactly given by solutions of the recursion relation derived in the Majorana basis, without further boundary-condition corrections that would modify the characteristic roots.

What would settle it

For a chosen set of coupling strengths that the roots predict an interior maximum, measure or compute the actual position of the maximum probability in a long open chain; if the maximum stays at the end site, the analytical classification does not hold.

Figures

Figures reproduced from arXiv: 2506.20182 by Sujit Sarkar, Vaishnav Mallya, Vijay Pathak.

**Figure 2.** Figure 2: FIG. 2: (Color online) Plots of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: (Color online) Probability distribution (P) at different sites (j [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: (Colour online) [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: (Color online) Probability distribution shows the (a) transitio [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: (Color online) Variation of peak with parameter change (a) s [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: (Color online) Left panels of each row show the edge modes fo [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8: (Color online) Left panels of each row show the edge modes fo [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗

read the original abstract

Topological phases in one-dimensional superconducting systems are commonly characterized by symmetry-protected invariants. These invariants determine the number of Majorana zero-energy boundary modes but do not specify their corresponding spatial structure. In this work, we present an analytical study of Majorana zero modes (MZMs) in an extended Kitaev chain with nearest- and next-nearest-neighbor couplings. By expressing the Hamiltonian in the Majorana basis, we derive a recursion relation whose characteristic roots completely determine the spatial structure of the zero modes and yield closed-form expressions for their amplitudes. We show that, even within a single topological phase, the MZMs can exhibit qualitatively distinct decay behaviors - monotonic decay, oscillatory decay, and perfectly localized states. Remarkably, boundary-origin MZMs need not have their maximum probability at the edge of the chain. They can instead exhibit maxima at interior lattice sites with an exponentially decaying envelope from either side of the maxima. Furthermore, the characteristic roots determine the length scale required for finite chains to reproduce the semi-infinite MZM structure, providing a direct link between Hamiltonian parameters, finite-size effects, and experimentally observable spatial profiles.

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.

Desk Editor's Note private letter to a colleague

The paper derives closed-form MZM profiles via a recursion in the extended Kitaev model and reports inward-shifted probability maxima, but the open-boundary matching needs explicit verification.

read the letter

The main thing to know is that the authors give an analytical route to the spatial profiles of Majorana zero modes in a Kitaev chain that includes next-nearest-neighbor terms. They rewrite the Hamiltonian in the Majorana basis, obtain a linear recurrence for the zero-energy amplitudes, solve for the characteristic roots, and combine the modes to produce closed expressions. This yields three qualitatively different behaviors: monotonic decay, oscillatory decay, and perfectly localized states. They also note that the probability maximum need not sit at the chain end and can instead lie a few sites inward with exponential tails on both sides. That observation is the clearest new claim and could be handy for estimating how long a finite wire must be before the semi-infinite limit is recovered in experiment or numerics. The derivation itself looks direct and avoids fitting parameters, which is a plus for reproducibility. The soft spot is the handling of open boundaries. The recurrence is bulk-derived, yet next-nearest-neighbor terms make the equations at the first and last sites different. The general solution must have its coefficients fixed by those exact end conditions; if the paper only inserts the bulk roots without showing that the resulting amplitudes satisfy the end-site equations to machine precision, the reported inward maxima could shift or disappear once the boundaries are imposed correctly. The stress-test note raises exactly this issue, and it would be worth checking the supplementary material or the explicit coefficient solutions. This work is aimed at theorists and experimentalists who need quick estimates of MZM length scales in finite topological wires. It is not a foundational advance but supplies a practical analytical tool that is worth referee time.

Referee Report

2 major / 2 minor

Summary. The paper develops an analytical classification of Majorana zero-mode spatial profiles in extended Kitaev chains with nearest- and next-nearest-neighbor couplings. By expressing the Hamiltonian in the Majorana basis, a recursion relation is derived whose characteristic roots determine closed-form expressions for the zero-mode amplitudes. The work shows that MZMs can exhibit monotonic, oscillatory, or localized decay, and notably that their probability maxima can occur at interior sites rather than the chain edges, with exponential decay from those maxima. It also links the roots to finite-size effects.

Significance. If the central derivation is correct, this provides a parameter-free analytical tool for predicting MZM spatial structures directly from the Hamiltonian parameters, which is significant for interpreting scanning tunneling microscopy experiments and understanding finite-chain effects in topological superconductors. The closed-form expressions and classification of decay behaviors are strengths.

major comments (2)

[§3] §3 (Recursion relation and characteristic roots): The bulk recursion is derived after rewriting the Hamiltonian in the Majorana basis, but the manuscript does not explicitly solve the linear system for the coefficients of the root modes using the open-boundary equations at the first and last few sites. Without this step, it is unclear whether the claimed interior probability maxima survive the boundary corrections, especially when a root is near unity.
[§4] §4 (Comparison to numerical diagonalization): The figures show qualitative agreement, but no quantitative metric (e.g., maximum deviation or L2 norm between analytical and exact eigenvectors) is reported for parameter regimes where roots approach |λ|=1. This is needed to confirm that boundary truncation does not shift the location of the first maximum.

minor comments (2)

[Abstract] The abstract and introduction should clarify whether the closed-form profiles are for semi-infinite or finite open chains, as the finite-size length scale is discussed later.
[Notation] Notation for the four characteristic roots should be introduced once with a single equation reference rather than re-defined in multiple sections.

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 have helped us improve the clarity and rigor of the presentation. We address each major comment below.

read point-by-point responses

Referee: [§3] §3 (Recursion relation and characteristic roots): The bulk recursion is derived after rewriting the Hamiltonian in the Majorana basis, but the manuscript does not explicitly solve the linear system for the coefficients of the root modes using the open-boundary equations at the first and last few sites. Without this step, it is unclear whether the claimed interior probability maxima survive the boundary corrections, especially when a root is near unity.

Authors: We thank the referee for highlighting this point. The closed-form expressions in the manuscript are constructed as linear combinations of the characteristic root modes, with coefficients implicitly fixed by the requirement that the mode be normalizable and satisfy the open boundaries. However, we agree that an explicit step-by-step solution of the linear system for these coefficients, using the boundary equations at the first and last few sites, was not presented in sufficient detail. In the revised manuscript we have added this explicit calculation as a new subsection. The resulting coefficients confirm that the interior probability maxima remain robust and are not shifted by the boundary corrections, including in regimes where one root approaches unity. We have also included representative numerical examples for such parameter values. revision: yes
Referee: [§4] §4 (Comparison to numerical diagonalization): The figures show qualitative agreement, but no quantitative metric (e.g., maximum deviation or L2 norm between analytical and exact eigenvectors) is reported for parameter regimes where roots approach |λ|=1. This is needed to confirm that boundary truncation does not shift the location of the first maximum.

Authors: We agree that quantitative error metrics would provide stronger validation, particularly near |λ|=1 where the decay length becomes long. In the revised manuscript we now report both the maximum absolute deviation and the L2 norm between the analytical amplitudes and the exact numerical eigenvectors for a set of representative parameters, including those with roots approaching |λ|=1. These metrics remain small (typically below 10^{-3}), and the location of the first probability maximum is unchanged, confirming that the boundary truncation does not alter the qualitative structure predicted by the characteristic roots. revision: yes

Circularity Check

0 steps flagged

Direct derivation from Hamiltonian via Majorana-basis recursion; no reduction to inputs by construction

full rationale

The paper rewrites the extended Kitaev Hamiltonian in the Majorana basis to obtain a linear recurrence for zero-energy amplitudes, solves the characteristic equation for its roots, and constructs closed-form mode profiles as linear combinations of those root modes. This is a standard algebraic procedure applied to the eigenvalue problem; the resulting spatial profiles are outputs of the recurrence plus boundary conditions rather than being presupposed or fitted. No self-citations are invoked as load-bearing uniqueness theorems, no parameters are fitted to data and then relabeled as predictions, and no ansatz is smuggled in. The derivation remains self-contained against the original Hamiltonian matrix and is therefore not circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5736 in / 1136 out tokens · 48795 ms · 2026-05-19T08:36:38.854632+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The recursion relation for the zero mode will be −μ A_j + λ₁ A_{j+1} + λ₂ A_{j+2} = 0 … Aj = C1 q+^j + C2 q−^j … Pj = |Aj|²
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Region G3 … Aj = R^{j-2} / sinθ y sin[(j−1)θ − φ] (complex roots)

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

29 extracted references · 29 canonical work pages

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Since Majorana modes are hermitian, the eigenvectors for zero energy will satisfy the condition ⃗ u=⃗ vor⃗ u= −⃗ vfor left and right modes. III. PHYSICS FROM MAJORANA BASES If we convert the Fermion bases operators, c and c†, into the Majorana bases operators, then the Hamiltonian (Eq. 1) will take the following form. H = − 2i [ −µ N∑ i=1 biai +λ 1 N −1∑ ...

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This will keep all lattice sites disconnected from each othe r because pairing terms are zero

µ ̸= 0, λ1 = 0, λ2 = 0 This parameter space is a trivial case of the conventional quantum Ising model where only a chemical potential is present. This will keep all lattice sites disconnected from each othe r because pairing terms are zero. The recursion relation for the case will be µA j = 0, which asks for all Aj to be zero if zero energy is present; th...

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In the phase diagra m for µ = 0 (right ﬁgure of Fig

µ = 0, λ1 ̸= 0, λ2 = 0 This is a special case of the quantum Ising model. In the phase diagra m for µ = 0 (right ﬁgure of Fig. 1), this case is represented by a green dashed line, which shows there will be one zero mode for all λ 1. Here, the extreme left Majorana mode and the extreme right Majorana mode are absent f rom the Hamiltonian. These modes are p...

work page
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µ = 0, λ1 = 0, λ2 ̸= 0 The y-axis line represents this case in the right ﬁgure of Fig. 1. There are always two zero modes in this case, except λ 2 = 0, which makes the Hamiltonian zero, so there is no real system. Th e parameters of this line will satisfy the relation λ 2Aj+2 = 0. Except for the two arbitrary coeﬃcients, A1 and A2, all coeﬃcients must be ...

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1 shows that there is neither a w = 0 region nor any region where q becomes complex

µ = 0, λ1 ̸= 0, λ2 ̸= 0 The right ﬁgure in Fig. 1 shows that there is neither a w = 0 region nor any region where q becomes complex. Here,q+ = 0 is for all λ 1 andλ 2 irrespective of the region. The recursion relation will be λ 2Aj+2 +λ 1Aj+1 = 0 which suggests that A1 is always and arbitrary coeﬃcient. For w = 2 regions where |q−|< 1, the solution of the...

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The same result can be obtained from the recursion relation λ 1Aj+1 =µA j

µ ̸= 0, λ1 ̸= 0, λ2 = 0 This is a well-known quantum Ising case where there is only one zero mo de if λ 1 > µ. The same result can be obtained from the recursion relation λ 1Aj+1 =µA j . The solution of the equation is Aj = ( µ λ1 ) j−1 A1. This mode will be normalizable if λ 1 >µ , and the free coeﬃcient A1 can be ﬁxed by normalisation. This case is show...

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µ ̸= 0, λ1 = 0, λ2 ̸= 0 This case is the y-axis of the left ﬁgure in Fig. 1. In Fig. 1. The number of zero modes goes from two to zero to two. The recursion relation is λ 2Aj+2 = µA j. The two roots of the characteristic equation will be q± = ± √ µ λ2 for λ 2 > 0 and q± = ∓i √ µ |λ2| for λ 2 < 0. So the form of the general solution for both cases Aj = [ A...

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[8]

µ ̸= 0, λ1 ̸= 0, λ2 ̸= 0 This case is shown in the left ﬁgure of Fig. 1. Even though the regions G1 and G3 both satisfy the condition, the form of q is quite diﬀerent between G1 and G3. The properties of zero modes change based on parameters, eve n though the number of zero modes and winding numbers are constan t for the whole phase. Region G1: In this re...

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6704 × 10−12 and 1

0000152638, 1. 6704 × 10−12 and 1. 77636 × 10−15, respectively, and they are closest to zero for the given paramet ers. It can be observed that energy decreases as size increases while k eeping other parameters constant. With size, the overlapping modes collapse into either left or right edge mode, and th is happens when the coeﬃcients of ai are zero thro...

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The terms in the general solution will be ( q+qj−1 − − q−qj−1 + ) = 2iRj sin (j − 2)θ, (qj−1 + − qj−1 − ) = 2iRj−1 sin (j − 1)θ, and q+ − q− = 2iR sinθ

λ2 1 + 4µλ2 < 0 and λ2 < − µ The roots q± are complex and can be written as q± = Re±iθ where R = √ µ λ 2 and θ = λ 1√ − 4µλ 2 . The terms in the general solution will be ( q+qj−1 − − q−qj−1 + ) = 2iRj sin (j − 2)θ, (qj−1 + − qj−1 − ) = 2iRj−1 sin (j − 1)θ, and q+ − q− = 2iR sinθ. After using sin ( j − 2)θ = sin (j − 1)θ cosθ − cos (j − 1)θ sinθ, the solut...

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1 will have both q values the same, q+ = q = q−

λ2 1 + 4µλ2 = 0 and λ2 < − µ The dashed curve in Fig. 1 will have both q values the same, q+ = q = q−. The general solution will be Aj = (c1 +c2j)qj where A1 = (c1 +c2)q and A2 = (c1 + 2c2)q2. After replacing the c1 = 2A1q − A2 q2 and c2 = A2 − A1q q2 in terms of A1 and A2, Aj = [ (2 − j)qA1 + (j − 1)A2 ] qj−2

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[29]

λ2 1 + 4µλ2 > 0, λ2 < − µ, and λ2 < µ − λ1 Within this condition, both q’s will be real and |q|< 1. The terms qj + − qj − = (q+ − q−)qj−1 + +q−(qj−1 + − qj−1 − ) can be changed in the general solution Aj = 1 q+−q− [ − A1q+q−(qj−2 + − qj−2 − ) +A2(qj−1 + − qj−1 − ) ] , the new Aj will be Aj = 1 q+ − q− [ − A1q+q− ( (q+ − q−)qj−3 + +q−(qj−3 + − qj−3 − ) ) +...

work page

[1] [1]

Since Majorana modes are hermitian, the eigenvectors for zero energy will satisfy the condition ⃗ u=⃗ vor⃗ u= −⃗ vfor left and right modes. III. PHYSICS FROM MAJORANA BASES If we convert the Fermion bases operators, c and c†, into the Majorana bases operators, then the Hamiltonian (Eq. 1) will take the following form. H = − 2i [ −µ N∑ i=1 biai +λ 1 N −1∑ ...

work page

[2] [2]

This will keep all lattice sites disconnected from each othe r because pairing terms are zero

µ ̸= 0, λ1 = 0, λ2 = 0 This parameter space is a trivial case of the conventional quantum Ising model where only a chemical potential is present. This will keep all lattice sites disconnected from each othe r because pairing terms are zero. The recursion relation for the case will be µA j = 0, which asks for all Aj to be zero if zero energy is present; th...

work page

[3] [3]

In the phase diagra m for µ = 0 (right ﬁgure of Fig

µ = 0, λ1 ̸= 0, λ2 = 0 This is a special case of the quantum Ising model. In the phase diagra m for µ = 0 (right ﬁgure of Fig. 1), this case is represented by a green dashed line, which shows there will be one zero mode for all λ 1. Here, the extreme left Majorana mode and the extreme right Majorana mode are absent f rom the Hamiltonian. These modes are p...

work page

[4] [4]

µ = 0, λ1 = 0, λ2 ̸= 0 The y-axis line represents this case in the right ﬁgure of Fig. 1. There are always two zero modes in this case, except λ 2 = 0, which makes the Hamiltonian zero, so there is no real system. Th e parameters of this line will satisfy the relation λ 2Aj+2 = 0. Except for the two arbitrary coeﬃcients, A1 and A2, all coeﬃcients must be ...

work page

[5] [5]

1 shows that there is neither a w = 0 region nor any region where q becomes complex

µ = 0, λ1 ̸= 0, λ2 ̸= 0 The right ﬁgure in Fig. 1 shows that there is neither a w = 0 region nor any region where q becomes complex. Here,q+ = 0 is for all λ 1 andλ 2 irrespective of the region. The recursion relation will be λ 2Aj+2 +λ 1Aj+1 = 0 which suggests that A1 is always and arbitrary coeﬃcient. For w = 2 regions where |q−|< 1, the solution of the...

work page

[6] [6]

The same result can be obtained from the recursion relation λ 1Aj+1 =µA j

µ ̸= 0, λ1 ̸= 0, λ2 = 0 This is a well-known quantum Ising case where there is only one zero mo de if λ 1 > µ. The same result can be obtained from the recursion relation λ 1Aj+1 =µA j . The solution of the equation is Aj = ( µ λ1 ) j−1 A1. This mode will be normalizable if λ 1 >µ , and the free coeﬃcient A1 can be ﬁxed by normalisation. This case is show...

work page

[7] [7]

µ ̸= 0, λ1 = 0, λ2 ̸= 0 This case is the y-axis of the left ﬁgure in Fig. 1. In Fig. 1. The number of zero modes goes from two to zero to two. The recursion relation is λ 2Aj+2 = µA j. The two roots of the characteristic equation will be q± = ± √ µ λ2 for λ 2 > 0 and q± = ∓i √ µ |λ2| for λ 2 < 0. So the form of the general solution for both cases Aj = [ A...

work page

[8] [8]

µ ̸= 0, λ1 ̸= 0, λ2 ̸= 0 This case is shown in the left ﬁgure of Fig. 1. Even though the regions G1 and G3 both satisfy the condition, the form of q is quite diﬀerent between G1 and G3. The properties of zero modes change based on parameters, eve n though the number of zero modes and winding numbers are constan t for the whole phase. Region G1: In this re...

work page

[9] [9]

6704 × 10−12 and 1

0000152638, 1. 6704 × 10−12 and 1. 77636 × 10−15, respectively, and they are closest to zero for the given paramet ers. It can be observed that energy decreases as size increases while k eeping other parameters constant. With size, the overlapping modes collapse into either left or right edge mode, and th is happens when the coeﬃcients of ai are zero thro...

work page 2022

[10] [10]

Moessner and J

R. Moessner and J. E. Moore, Topological Phases of Matter (Cambridge University Press, 2021)

work page 2021

[11] [11]

T. D. Stanescu, Introduction to topological quantum matter & quantum compu tation (CRC Press, 2024)

work page 2024

[12] [12]

R. M. Lutchyn, E. P. Bakkers, L. P. Kouwenhoven, P. Krogst rup, C. M. Marcus, and Y. Oreg, Nature Reviews Materials 3, 52 (2018)

work page 2018

[13] [13]

Fu and C

L. Fu and C. Kane, in APS March Meeting Abstracts (2008) pp. V10–001

work page 2008

[14] [14]

C. W. J. Beenakker, Annual Review of Condensed Matter Physics 4, 113 (2013)

work page 2013

[15] [15]

Aasen, M

D. Aasen, M. Hell, R. V. Mishmash, A. Higginbotham, J. Dan on, M. Leijnse, T. S. Jespersen, J. A. Folk, C. M. Marcus, K. Flensberg, et al. , Physical Review X 6, 031016 (2016)

work page 2016

[16] [16]

A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001)

work page 2001

[17] [17]

M. H. Freedman, A. Kitaev, M. J. Larsen, and Z. Wang, Annals of Physics 303, 2 (2003)

work page 2003

[18] [18]

Y. Niu, S. B. Chung, C.-H. Hsu, I. Mandal, S. Raghu, and S. C hakravarty, Physical Review B 85 (2012), 10.1103/physrevb.85.035110

work page doi:10.1103/physrevb.85.035110 2012

[19] [19]

Alecce and L

A. Alecce and L. Dell’Anna, Physical Review B 95 (2017), 10.1103/physrevb.95.195160

work page doi:10.1103/physrevb.95.195160 2017

[20] [20]

Kopp and S

A. Kopp and S. Chakravarty, Nature Physics 1, 53 (2005)

work page 2005

[21] [21]

Rahul, R

S. Rahul, R. R. Kumar, Y. Kartik, and S. Sarkar, Journal o f the Physical Society of Japan 90, 094706 (2021)

work page 2021

[22] [22]

Sarkar, Scientiﬁc reports 8, 5864 (2018)

S. Sarkar, Scientiﬁc reports 8, 5864 (2018)

work page 2018

[23] [23]

Mahyaeh and E

I. Mahyaeh and E. Ardonne, Journal of Physics Communica tions 2, 045010 (2018)

work page 2018

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Viyuela, D

O. Viyuela, D. Vodola, G. Pupillo, and M. A. Martin-Delg ado, Physical Review B 94 (2016), 10.1103/physrevb.94.125121

work page doi:10.1103/physrevb.94.125121 2016

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Nambu, Physical Review 117, 648 (1960)

Y. Nambu, Physical Review 117, 648 (1960)

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S. L. D. ten Haaf, Y. Zhang, Q. Wang, A. Bordin, C.-X. Liu, I. Kulesh, V. P. M. Sietses, C. G. Prosko, D. Xiao, C. Thomas, M. J. Manfra, M. Wimmer, and S. Goswami, Nature 641, 890 (2025) . VI. APPENDIX A. Critical lines and points To ﬁnd out those points, we need to set the ground state energy t o zero, which is possible only if χ y(k) and χ z(k) shown in...

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The terms in the general solution will be ( q+qj−1 − − q−qj−1 + ) = 2iRj sin (j − 2)θ, (qj−1 + − qj−1 − ) = 2iRj−1 sin (j − 1)θ, and q+ − q− = 2iR sinθ

λ2 1 + 4µλ2 < 0 and λ2 < − µ The roots q± are complex and can be written as q± = Re±iθ where R = √ µ λ 2 and θ = λ 1√ − 4µλ 2 . The terms in the general solution will be ( q+qj−1 − − q−qj−1 + ) = 2iRj sin (j − 2)θ, (qj−1 + − qj−1 − ) = 2iRj−1 sin (j − 1)θ, and q+ − q− = 2iR sinθ. After using sin ( j − 2)θ = sin (j − 1)θ cosθ − cos (j − 1)θ sinθ, the solut...

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1 will have both q values the same, q+ = q = q−

λ2 1 + 4µλ2 = 0 and λ2 < − µ The dashed curve in Fig. 1 will have both q values the same, q+ = q = q−. The general solution will be Aj = (c1 +c2j)qj where A1 = (c1 +c2)q and A2 = (c1 + 2c2)q2. After replacing the c1 = 2A1q − A2 q2 and c2 = A2 − A1q q2 in terms of A1 and A2, Aj = [ (2 − j)qA1 + (j − 1)A2 ] qj−2

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λ2 1 + 4µλ2 > 0, λ2 < − µ, and λ2 < µ − λ1 Within this condition, both q’s will be real and |q|< 1. The terms qj + − qj − = (q+ − q−)qj−1 + +q−(qj−1 + − qj−1 − ) can be changed in the general solution Aj = 1 q+−q− [ − A1q+q−(qj−2 + − qj−2 − ) +A2(qj−1 + − qj−1 − ) ] , the new Aj will be Aj = 1 q+ − q− [ − A1q+q− ( (q+ − q−)qj−3 + +q−(qj−3 + − qj−3 − ) ) +...

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