arxiv: 2602.11278 · v3 · submitted 2026-02-11 · 🪐 quant-ph · cond-mat.stat-mech· cond-mat.str-el

Recognition: 2 theorem links

· Lean Theorem

Long-Range Pairing in the Kitaev Model: Krylov Subspace Signatures

Rishabh Jha , Heiko Georg Menzler

Authors on Pith no claims yet

Pith reviewed 2026-05-16 02:13 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechcond-mat.str-el

keywords Kitaev modelKrylov subspaceLanczos coefficientstopological phasesMajorana edge modeslong-range pairingboundary modesoperator growth

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The pith

In the Kitaev chain the Krylov staggering parameter is exactly constant in the balanced short-range limit and its sign distinguishes the topological Majorana phase from the trivial phase.

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

The paper shows that Krylov subspace diagnostics built from Lanczos coefficients of local boundary operators can identify whether the lowest energy excitations in the Kitaev model are localized at the boundaries or spread in the bulk. In the short-range version with balanced hopping and pairing this staggering parameter stays exactly constant no matter the chain length and its sign flips between the topological phase containing Majorana edge modes and the trivial phase. The same sign pattern continues to track the boundary-versus-bulk distinction even when long-range couplings are added and the simple constancy is lost. An exact recursion algorithm reduces the problem to finite linear algebra allowing precise calculations on chains of hundreds of sites. These tools matter for characterizing topological superconductors whose interactions decay algebraically rather than exponentially.

Core claim

Lanczos coefficients generated from local boundary operators sharply diagnose whether the lowest excitation gap is controlled by boundary-localized or bulk-extended modes. In the short-range Kitaev chain with balanced hopping and pairing the introduced Krylov staggering parameter is exactly constant for arbitrary system size in the thermodynamic limit and its sign cleanly distinguishes the topological phase with Majorana edge modes from the trivial phase.

What carries the argument

The Krylov staggering parameter extracted from the sequence of Lanczos coefficients produced by an exact single-particle operator Lanczos algorithm applied to local boundary operators.

If this is right

The sign pattern of the staggering parameter tracks boundary-mode control of the gap even in the long-range Kitaev chain.
The exact single-particle Lanczos algorithm achieves machine precision for systems of hundreds of sites by reducing the recursion to a finite-dimensional linear problem.
These diagnostics work for topological superconductors with broken U(1) symmetry and algebraically decaying couplings.
Analytical constancy is lost away from the balanced short-range limit but the diagnostic utility of the sign remains.

Where Pith is reading between the lines

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

This method could be tested on other quadratic fermionic models to see if the staggering sign remains a reliable phase indicator.
Applying the approach to time-dependent or driven systems might reveal how boundary modes evolve under perturbations.
Extensions to interacting models would require checking if the single-particle reduction still holds or if many-body effects alter the staggering behavior.

Load-bearing premise

Lanczos coefficients generated from local boundary operators can sharply diagnose whether the lowest excitation gap is controlled by boundary-localized or bulk-extended modes and the analytical constancy result holds in the thermodynamic limit.

What would settle it

Numerical computation of the staggering parameter on a large finite balanced short-range Kitaev chain that shows deviation from exact constancy.

Figures

Figures reproduced from arXiv: 2602.11278 by Heiko Georg Menzler, Rishabh Jha.

**Figure 1.** Figure 1: BdG spectrum Eν/ p Tr(H2) versus θ/π at ϵ = −0.2 for three long-range exponents α (N = 1000, open boundaries). (a) Short-range-like behavior (α = 3) displays a gapless region centered at θ/π ≈ 0.5 with sparse spectral density near E = 0 elsewhere. (b) Intermediate regime (α = 1) retains a similar gapless range with modified spectral density. (c) Strong long-range limit (α = 1/3) lifts the degeneracy throug… view at source ↗

**Figure 2.** Figure 2: Lanczos coefficients {bn} for the long-range Kitaev chain at ϵ = −0.2 with open boundaries and Hermitian boundary seed γ1 (N = 1000). Each panel shows the two interleaved subsequences (odd and even steps of the recursion), whose relative ordering determines the sign of the staggering parameter ηn = ln(b2n−1/b2n) and hence the crossing count Ncross (Eq. (39)). Panels (a), (b) show parameter points in the bu… view at source ↗

**Figure 3.** Figure 3: Phase diagram for the long-range Kitaev chain at ϵ = −0.2 with open boundaries (N = 1000) and boundary seed γ1, generating 2000 Lanczos coefficients. As discussed in Sec. IV B, the black region indicates parameters where the Krylov staggering parameter ηn = ln(b2n−1/b2n) exhibits nonzero robust sign changes (Ncross ≥ 1), while the white region corresponds to Ncross = 0. As discussed in Sec. IV A, solid c… view at source ↗

**Figure 4.** Figure 4: Finite-size robustness of the edge-versus-bulk [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: Numerical verification of Eqs. (C3)–(C4) for the balanced short-range Kitaev chain at |t| = |∆| = 1 (ϵ = 0), with seed γ1 and N = 1000 sites. (a) Offdiagonal Lanczos coefficients bn at µ/t = 4 (trivial regime). (b),(c) Off-diagonal Lanczos coefficients bn at µ/t = 1.7 and µ/t = 0.3 (topological regime). In all three cases, the numerical coefficients match the exact alternating values set by |µ| and 2|t| w… view at source ↗

**Figure 6.** Figure 6: BdG spectrum Eν/ p Tr(H2) versus θ/π at ϵ = 1 for three long-range exponents α (N = 1000, open boundaries). The α-dependence mirrors [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗

**Figure 7.** Figure 7: BdG spectrum Eν/ p Tr(H2) versus θ/π at extreme pairing-dominated ϵ = 10 for three long-range exponents α (N = 1000, open boundaries). The systematic α-dependence observed in Figs. 1 and 6 persists: gapless region for α = 3, similar gapless range for α = 1 with gap opening, and degeneracy lifted except for a small near-degenerate regime for α = 1/3. Spectral density near E = 0 is further increased relative… view at source ↗

**Figure 8.** Figure 8: Lanczos coefficients {bn} for the long-range Kitaev chain at ϵ = −0.2 with open boundaries and boundary seed γ1 + γ2 (N = 1000). Each panel shows the two interleaved subsequences (odd and even recursion steps), whose relative ordering determines the staggering parameter ηn = ln(b2n−1/b2n) and crossing count Ncross. Panels (a) (α = 2, θ/π = 0.1) and (c) (α = 2/3, θ/π = 0.1) lie in the bulk-gap regime and ex… view at source ↗

**Figure 9.** Figure 9: Lanczos coefficients {bn} for the long-range Kitaev chain at ϵ = −0.2 with open boundaries and bulk seed γN (N = 1000). Each panel shows the two interleaved subsequences (odd and even recursion steps), whose relative ordering determines the staggering parameter ηn = ln(b2n−1/b2n) and crossing count Ncross. Panels (a) (α = 2, θ/π = 0.1) and (c) (α = 2/3, θ/π = 0.1) lie in the bulk-gap regime (Ncross = 0), w… view at source ↗

**Figure 10.** Figure 10: Lanczos coefficients {bn} for the long-range Kitaev chain at ϵ = −0.2 with open boundaries and bulk seed γN + γN+1 (N = 1000). Each panel shows the two interleaved subsequences whose relative ordering determines the staggering parameter ηn = ln(b2n−1/b2n) and crossing count Ncross. Panels (a), (c) (α = 2, 2/3; θ/π = 0.1) show bulkgap regime (Ncross = 0), while panels (b), (d) (α = 2, 2/3; θ/π = 0.4) show… view at source ↗

**Figure 11.** Figure 11: Phase diagrams for the long-range Kitaev chain at [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗

read the original abstract

Krylov subspace methods quantify operator growth in quantum many-body systems through Lanczos coefficients that encode how operators spread under time evolution. Although these diagnostics were originally motivated by questions of chaos and integrability, quadratic fermionic Hamiltonians are often expected to exhibit trivial Lanczos structure. Here we show that, in the long-range Kitaev chain, Lanczos coefficients generated from local boundary operators sharply diagnose whether the lowest excitation gap is controlled by boundary-localized or bulk-extended modes. We introduce the $Krylov$ $staggering$ $parameter$ for the Lanczos coefficients. In the short-range Kitaev chain with balanced hopping and pairing, we derive analytically for arbitrary system size (valid in the thermodynamic limit) and show that this quantity is exactly constant and its sign cleanly distinguishes the topological phase with Majorana edge modes from the trivial phase. Away from that limit, long-range couplings and pairing-hopping imbalance deform the simple flat structure and analytical control is lost, nevertheless, we show that the sign pattern of the diagnostic still tracks whether the lowest excitation gap is controlled by boundary modes or by bulk excitations. These results are enabled by an exact single-particle operator Lanczos algorithm, as derived in this work, which reduces the recursion from exponentially large operator space to a finite-dimensional linear problem and achieves machine precision for chains of hundreds of sites. Krylov diagnostics thus emerge as practical probes of boundary-versus-bulk low-energy physics in topological superconductors with broken U(1) symmetry and algebraically decaying couplings.

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 Krylov staggering parameter is exactly constant in the balanced short-range Kitaev chain and its sign distinguishes the topological phase analytically.

read the letter

The main takeaway is that this paper derives an exact constancy for the Krylov staggering parameter in the short-range Kitaev model with balanced hopping and pairing, valid for arbitrary system size and thus the thermodynamic limit, with the sign cleanly separating the phase that has Majorana edge modes from the trivial one. They also introduce an exact single-particle Lanczos algorithm that reduces the operator recursion to a finite linear problem and delivers machine precision up to hundreds of sites. For long-range couplings the flat structure deforms but the sign pattern still tracks whether the lowest gap is set by boundary or bulk modes. This is genuinely new as a diagnostic; earlier Lanczos applications to quadratic models did not isolate this boundary-versus-bulk distinction with an analytical constant. The algorithm itself is a concrete practical step that makes the numerics feasible without exponential cost. The analytical claim rests on a direct derivation from the Hamiltonian rather than a fit, which is a strength if the algebra holds. The long-range numerics are presented as tracking the expected behavior, though they lose closed-form control. A minor soft spot is that the abstract does not spell out the derivation steps or error bounds for the large-system numerics, so a referee would want to see those explicitly to confirm no hidden assumptions creep in. Overall the work is for people working on Krylov methods in integrable or topological fermionic systems who want operator-growth tools that speak to edge-mode protection. It has enough new analytical content and a reproducible method to deserve serious peer review rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes Krylov-subspace diagnostics in the Kitaev chain with long-range pairing. It introduces a Krylov staggering parameter constructed from Lanczos coefficients generated by local boundary operators. For the short-range balanced hopping-pairing case the authors derive that this parameter is exactly constant for arbitrary finite size (hence in the thermodynamic limit) and that its sign distinguishes the topological phase hosting Majorana edge modes from the trivial phase. For long-range couplings and pairing-hopping imbalance the flat structure is lost, but the sign pattern of the diagnostic is shown numerically to continue to track whether the lowest gap is controlled by boundary-localized or bulk-extended modes. The results rest on a new exact single-particle operator Lanczos algorithm that reduces the recursion to a finite-dimensional linear problem and achieves machine precision up to hundreds of sites.

Significance. If the central claims hold, the work supplies a practical, numerically robust probe of boundary-versus-bulk low-energy physics in topological superconductors with algebraically decaying interactions. The exact analytical constancy result for the balanced short-range case and the machine-precision Lanczos algorithm constitute clear technical strengths that could be adopted by the community working on long-range Kitaev and related models.

major comments (2)

[single-particle Lanczos algorithm section] The analytical derivation that the Krylov staggering parameter is exactly constant in the short-range balanced case is load-bearing for the central claim. The manuscript must exhibit the explicit recursion relations or closed-form expression (presumably in the section presenting the single-particle Lanczos algorithm) that demonstrates constancy for arbitrary system size without additional assumptions.
[numerical results for long-range cases] The assertion that the sign of the staggering parameter cleanly diagnoses whether the lowest excitation gap is controlled by boundary-localized versus bulk-extended modes rests on the choice of local boundary operators. A direct comparison between the Lanczos-generated diagnostic and the actual spatial profile of the lowest eigenmode (e.g., via the Majorana wave-function weight at the edges) should be provided for at least one long-range parameter set to confirm the diagnostic is not merely correlative.

minor comments (2)

[introduction of staggering parameter] The precise definition of the Krylov staggering parameter (how it is extracted from the sequence of Lanczos coefficients) should be stated as an explicit formula immediately after its introduction.
[figures] Figure captions for the long-range numerical scans should explicitly state the system sizes used and the convergence criterion with respect to the Lanczos depth.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comments. We address each major comment below.

read point-by-point responses

Referee: [single-particle Lanczos algorithm section] The analytical derivation that the Krylov staggering parameter is exactly constant in the short-range balanced case is load-bearing for the central claim. The manuscript must exhibit the explicit recursion relations or closed-form expression (presumably in the section presenting the single-particle Lanczos algorithm) that demonstrates constancy for arbitrary system size without additional assumptions.

Authors: We agree that explicit recursion relations and the closed-form expression are needed for full transparency. In the revised manuscript we will expand the single-particle Lanczos algorithm section to display the recursion relations obtained from the exact mapping to a finite-dimensional linear problem and the resulting closed-form Lanczos coefficients for the balanced short-range case, from which constancy of the staggering parameter follows directly for arbitrary finite N (and hence in the thermodynamic limit) with no further assumptions. revision: yes
Referee: [numerical results for long-range cases] The assertion that the sign of the staggering parameter cleanly diagnoses whether the lowest excitation gap is controlled by boundary-localized versus bulk-extended modes rests on the choice of local boundary operators. A direct comparison between the Lanczos-generated diagnostic and the actual spatial profile of the lowest eigenmode (e.g., via the Majorana wave-function weight at the edges) should be provided for at least one long-range parameter set to confirm the diagnostic is not merely correlative.

Authors: We thank the referee for this suggestion. In the revised manuscript we will add, for at least one representative long-range parameter set, a direct side-by-side comparison of the sign of the staggering parameter with the edge weight of the lowest eigenmode (computed from the Majorana wave-function amplitudes at the boundaries). This will explicitly confirm that the diagnostic tracks boundary versus bulk control of the gap rather than being merely correlative. revision: yes

Circularity Check

0 steps flagged

No circularity: analytical constancy derived directly from Kitaev Hamiltonian

full rationale

The Krylov staggering parameter is defined from Lanczos coefficients of local boundary operators. The paper then derives its exact constancy analytically for arbitrary system size in the balanced short-range Kitaev chain, valid in the thermodynamic limit, directly from the model's Hamiltonian. This is presented as an exact result rather than a fit, renaming, or self-citation reduction. No load-bearing self-citations, ansatz smuggling, or input-output equivalence by construction appear in the derivation chain. The diagnostic's sign distinguishing topological vs. trivial phases follows from this independent analytical step.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the definition of the staggering parameter from Lanczos coefficients of boundary operators and the analytical evaluation of that parameter in the balanced short-range Kitaev Hamiltonian. No free parameters are introduced. The main background assumptions are standard properties of the Lanczos recursion for quadratic fermionic operators and the standard form of the Kitaev chain.

axioms (2)

standard math Lanczos recursion reduces operator growth in quadratic fermionic Hamiltonians to a finite-dimensional linear problem
Stated as enabling machine-precision computation for chains of hundreds of sites.
domain assumption The Kitaev chain Hamiltonian with possible long-range pairing terms
Standard model for topological superconductors with broken U(1) symmetry.

invented entities (1)

Krylov staggering parameter no independent evidence
purpose: Diagnostic extracted from the pattern of Lanczos coefficients that distinguishes phases and mode localization
Newly defined in this work; no independent external evidence supplied.

pith-pipeline@v0.9.0 · 5585 in / 1328 out tokens · 106043 ms · 2026-05-16T02:13:06.798986+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 washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

for every recursion depth n and every system size N ... b_{2n-1} = |μ|, b_{2n} = 2|t|, a_n = 0 ... η_n = ln(|μ|/2|t|) = const. ... sgn(η_n) = −1 iff |μ|<2|t| (topological)
IndisputableMonolith/Foundation/ArithmeticFromLogic embed_add / LogicNat recovery echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the balanced short-range Kitaev chain in Majorana form is exactly the SSH chain ... Krylov recursion ... preserves this alternating-bond matrix structure

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

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Eckseler, M

J. Eckseler, M. Pieper, and J. Schnack, Escaping the Krylov space during the finite-precision Lanczos algo- rithm, Phys. Rev. E112, 025306 (2025). Appendix A: Majorana commutator closure This appendix provides a self contained derivation of the commutator identity [HLRK, γℓ] =i 2NX m=1 HM,mℓγm,(A1) used in Sec. II E to obtain a closed single-particle equa...

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With the normalization in Eq

Trace identities The Majorana operators satisfyγ † µ =γ µ and {γµ, γν}=δ µν. With the normalization in Eq. (8), one hasγ 2 µ = 1

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Hence, on a 2 N dimensional fermionic Hilbert space, Tr(γµ) = 0,Tr(γ µγν) = 2N−1 δµν.(A2) The second identity follows from Tr(γ 2 µ) = Tr(1/2) = 2N−1 and from Tr(γµγν) =−Tr(γ νγµ) forµ̸=ν

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Elementary commutator identity We use the operator identity [γjγk, γℓ] =γ j{γk, γℓ} − {γj, γℓ}γk.(A3) 18 To verify it, expand and reorder using anticommutators: [γjγk, γℓ] =γ jγkγℓ −γ ℓγjγk =γ j(γkγℓ)−(γ ℓγj)γk =γ j({γk, γℓ} −γ ℓγk)−({γ ℓ, γj} −γ jγℓ)γk =γ j{γk, γℓ} − {γℓ, γj}γk,(A4) which is Eq. (A3). With{γ µ, γν}=δ µν, this immediately yields [γiγj, γℓ...

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Derivation of[H LRK, γℓ] Starting from the quadratic Majorana form HLRK = i 2 2NX i,j=1 HM,ij γiγj,(A6) whereH M is real and antisymmetric,H M,ji =−H M,ij, we compute [HLRK, γℓ] = i 2 2NX i,j=1 HM,ij[γiγj, γℓ] = i 2 2NX i,j=1 HM,ij (γiδjℓ −δ iℓγj) = i 2 2NX i=1 HM,iℓγi − i 2 2NX j=1 HM,ℓj γj.(A7) Relabelj→iin the second term and use antisymmetry HM,ℓi =−H...

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Using Eq

Closure for linear operators Consider any generic operatorO= P2N k=1 ukγk. Using Eq. (A1), [HLRK,O] = 2NX k=1 uk[HLRK, γk] =i 2NX k=1 uk 2NX m=1 HM,mk γm =i 2NX m=1 (HM u)m γm, (A9) showing that the commutator maps the linear Majorana subspace to itself. Appendix B: Algorithmic details

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Using Tr(γ µγν) = 2 N−1 δµν for the Ma- jorana normalization adopted in Eq

Inner product on the linear Majorana subspace Consider operatorsO v = P µ vµγµ andO w =P µ wµγµ. Using Tr(γ µγν) = 2 N−1 δµν for the Ma- jorana normalization adopted in Eq. (8), the infinite- temperature Hilbert-Schmidt inner product becomes ⟨Ov,O w⟩HS = 1 2N Tr O† vOw = 1 2 2NX µ=1 v∗ µwµ = 1 2 v†w. (B1) Thus, within the linear Majorana sector, the Eucli...

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Proof thata n = 0forO(0) † =±O(0) LetHbe Hermitian and define the Hilbert-Schmidt inner product⟨A,B⟩ HS = Tr(A†B)/2N. For any operator AsatisfyingA † =±A, we have ⟨A,[H,A]⟩ HS = 0.(B2) Using cyclicity of the trace, Tr A†[H,A] = Tr A†HA − A †AH = Tr HAA † − A†AH .(B3) IfA † =A(Hermitian), thenAA † =A 2 =A †Aand the terms cancel. IfA † =−A(anti-Hermitian), ...

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(30) and has entriesT nn =a n andT n,n+1 =T n+1,n =b n+1

Derivation of the Krylov subspace evolution equation The tridiagonal Lanczos matrixTis defined in Eq. (30) and has entriesT nn =a n andT n,n+1 =T n+1,n =b n+1. For the Hermitian seed (as is considered in this work), an = 0 andTis a real symmetric matrix with a vanishing diagonal. The coefficient vector evolves according to (using Eq. (13) in Eq. (12)) du(...

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[75]

This unfortunate fact is almost impossible to get around; even when using sophisticated re-orthogonalization routines as they are used across the literature

Reorthogonalization and stability checks In finite precision arithmetic, the Lanczos basis is bound to loose orthogonality for large number of Lanc- zos stepsn, leading to strong numerical instability. This unfortunate fact is almost impossible to get around; even when using sophisticated re-orthogonalization routines as they are used across the literatur...

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[76]

Majorana basis and coefficient vectors.Following Eqs

Majorana representation and the matrixH M a. Majorana basis and coefficient vectors.Following Eqs. (8)–(9), each sitejcontributes two Majorana oper- ators, γ2j−1 = cj +c † j√ 2 , γ 2j = c† j −c j i √ 2 ,(C5) satisfyingγ † µ =γ µ and{γ µ, γν}=δ µν. We callγ 2j−1 theA-typeandγ 2j theB-typeMajorana on sitej. The Hamiltonian is written asH SRK = i 2 P µν HM,µ...

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[77]

,2N−1: vn = (−i)n eγn+1 ,(C23) witha n = 0 for alln,b 2n−1 =|µ|, andb 2n = 2|t|

Proof by induction Claim.ForH SRK at ∆ =t, with seedv 0 =e γ1, and without loss of generalityt, µ >0, the Lanczos recursion wn =Lsp vn−1 −a n−1vn−1 −b n−1vn−2, bn =∥wn∥, v n = wn bn , (C22) produces, for alln= 0,1,2, . . . ,2N−1: vn = (−i)n eγn+1 ,(C23) witha n = 0 for alln,b 2n−1 =|µ|, andb 2n = 2|t|. Note that (−i) n cycles through +1,−i,−1,+iwith pe- r...

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[78]

The Su–Schrieffer–Heeger (SSH) model.The SSH model [39, 40] is the simplest 1D tight-binding chain with a topological phase

SSH interpretation and topological criterion a. The Su–Schrieffer–Heeger (SSH) model.The SSH model [39, 40] is the simplest 1D tight-binding chain with a topological phase. Label the two sublattice sites within unit celljasA j (left) andB j (right), withc j,A andc j,B the corresponding fermionic annihilation opera- tors. The Hamiltonian on an open chain o...

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[79]

Be- cause these two values alternate with strict period two for everyµ,t, andN, the staggering parameter defined in Eq

Physical interpretation and topological criterion The Krylov basis vectors traverse the Majorana chain (C20) sequentially, andb n records the bond strength encountered at each step:|µ|for the intra-site (A–B) bond and 2|t|for the inter-site (B–A) bond. Be- cause these two values alternate with strict period two for everyµ,t, andN, the staggering parameter...

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