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

Abstract: Krylov subspace methods quantify operator growth in quantum many-body systems through Lanczos coefficients that encode how operators spread under time evolution. While these diagnostics have been proposed to distinguish quantum chaos from integrability, quadratic fermionic Hamiltonians are widely expected to exhibit trivial Lanczos structure. Here we demonstrate that Lanczos coefficients generated from local boundary operators provide a quantitative diagnostic of whether the lowest excitation gap is controlled by boundary-localized or bulk-extended modes in the long-range Kitaev chain, the model for topological superconductivity with algebraically decaying couplings. We introduce \emph{Krylov staggering parameter}, defined as the logarithmic ratio of consecutive odd and even Lanczos coefficients, whose sign structure correlates robustly with the edge versus bulk character of the gap across the full phase diagram. This correlation arises from a bipartite Krylov structure induced by pairing, power-law couplings, and open boundaries. We derive an exact single-particle operator Lanczos algorithm that reduces the recursion from exponentially large operator space to a finite-dimensional linear problem, achieving machine precision for chains of hundreds of sites. These results establish Krylov diagnostics as operational probes of how low-energy excitations are localized along the chain and how strongly they are tied to the boundaries with broken U(1) symmetry, with potential applications to trapped-ion and cold-atom quantum simulators.

• The paper develops an exact single-particle Lanczos method for quadratic Kitaev chains, reducing operator dynamics to a 2N-dimensional Majorana problem and enabling stable calculations at N=1000.
• The Krylov staggering parameter, η_n = ln(b_{2n−1}/b_{2n}), shows sign changes when edge-localized modes control the gap and maintains a fixed sign when extended bulk modes set the lowest energy scale.
• The results closely reproduce BdG edge–bulk phase boundaries, demonstrating that local Krylov data can reveal excitation localization in integrable systems, while remaining sensitive to seed locality, finite size, and long-range threshold effects.

Overview and motivation

This paper addresses a specific question within the operator-growth program: can Lanczos coefficients, computed from local boundary operators, distinguish whether the lowest excitation gap of a quadratic fermionic chain is controlled by boundary-localized or bulk-extended modes? The setting is the long-range Kitaev chain with algebraically decaying hopping and pairing, a quadratic model that remains exactly solvable for all decay exponents $\alpha$ yet hosts a rich interplay between edge physics, $U(1)$-breaking pairing, and power-law hybridization (2602.11278).

The question is motivated by two strands of prior work. First, the "universal operator growth hypothesis" (UOGH) proposed linear-in-$n$ growth of Lanczos coefficients $\{b_n\}$ as a chaos diagnostic, but subsequent results showed that integrable systems can exhibit UOGH-like growth and that quadratic models can be tuned to display "universal-looking" Lanczos behavior. In particular, the widely repeated expectation that Lanczos structure in free fermionic systems is trivial (constant $b_n$) is identified by the authors as potentially misleading: free systems can host nontrivial boundary localization and gap mechanisms invisible to any single growth law. Second, long-range Kitaev chains are known to exhibit massless versus massive edge modes depending on $\alpha$ and hopping–pairing imbalance, with gapped phases lacking conventional bulk gap closures. The paper's contribution is to show that these static distinctions leave sharp, robust signatures in operator Krylov data—despite the absence of chaos or thermalization.

Model and exact single-particle Liouvillian

The Hamiltonian studied is a spinless-fermion chain of $N$ sites with open boundaries,

$$H_{\mathrm{LRK}}=\sin\theta \sum_{i<j}\frac{c_i^\dagger c_j+(1+\epsilon)c_i c_j+\mathrm{h.c.}}{|i-j|^\alpha}+2\cos\theta\sum_i n_i,$$

where $\alpha$ controls algebraic decay, $\theta$ interpolates between kinetic/pairing and chemical-potential terms, and $U(1)$0 biases pairing relative to hopping ($U(1)$1 recovers particle-number conservation). In the Majorana basis the model takes the standard quadratic form $U(1)$2 with real antisymmetric $U(1)$3. A key structural fact, proved in an appendix, is that the commutator algebra closes on operators linear in Majoranas:

$U(1)$4

Consequently the Liouvillian acts on this subspace as multiplication by the Hermitian single-particle generator $U(1)$5, reducing the operator Krylov problem from an exponentially large many-body space to a $U(1)$6-dimensional linear problem. This reduction is the technical backbone of the paper: it enables machine-precision Lanczos recursions for chains of hundreds of sites ($U(1)$7 throughout), where standard many-body implementations would be limited both in size and recursion depth.

For Hermitian seeds such as the boundary Majorana $U(1)$8, all diagonal Lanczos coefficients vanish identically ($U(1)$9), proved generally for seeds satisfying $n$0 via trace cyclicity. The tridiagonal Krylov representation is therefore purely off-diagonal, and recovering $n$1 serves as a stringent internal benchmark.

Static classification: edge versus bulk gaps

The reference diagnostic is purely spectral. Each positive BdG eigenvalue $n$2 is assigned an edge weight—the fraction of its Nambu amplitude within boundary windows of width $n$3 at each end—and classified as edge-localized if this weight exceeds a threshold $n$4. Defining $n$5 and $n$6 as the minimum energies among edge-localized and bulk-extended modes respectively, parameter points $n$7 are labeled edge-gap phase when $n$8 and bulk-gap phase otherwise. This construction follows the physical logic established for long-range topological phases in earlier work on the same model.

The authors are explicit that $n$9 and $\{b_n\}$0 are operational rather than universal parameters. They argue the classification is asymptotically well-defined because bulk-mode edge weights scale as $\{b_n\}$1 while localized modes retain finite weight, and they verify stability by varying $\{b_n\}$2 from $\{b_n\}$3 to $\{b_n\}$4. Residual sensitivity persists in the strong long-range regime ($\{b_n\}$5) at large $\{b_n\}$6, where algebraically decaying eigenmodes extend over large fractions of the chain—a finite-size limitation conceded plainly in the text.

The BdG spectra themselves exhibit clear $\{b_n\}$7-driven reorganization at fixed $\{b_n\}$8: for $\{b_n\}$9 a gapless region centered near $b_n$0 with sparse low-energy density elsewhere; for $b_n$1 a crossover; and for $b_n$2 degeneracy lifting across nearly the whole interval with markedly denser spectral weight near $b_n$3, consistent with the known edge-mode mass acquisition mechanism for $b_n$4. Appendix spectra at $b_n$5 and $b_n$6 confirm these trends persist under strong pairing dominance, with increased low-energy density uniformly across $b_n$7.

Dynamical diagnostic: Krylov staggering

The central observable is the Krylov staggering parameter

$b_n$8

the logarithmic ratio of consecutive odd and even Lanczos coefficients. Its significance rests on a bipartite structure of the Krylov space: since $b_n$9 with $\alpha$0 real, Lanczos vectors generated from the real seed $\alpha$1 alternate between real and purely imaginary subspaces, which are separately invariant under $\alpha$2. The coefficients $\alpha$3 encode how these two sectors couple at each step, and $\alpha$4 measures their imbalance.

The empirical pattern, stated as a correlation rather than a theorem, is:

• Edge-gap phase ($\alpha$5): $\alpha$6 exhibits nonzero sign changes as recursion depth increases.
• Bulk-gap phase ($\alpha$7): $\alpha$8 retains a fixed sign over the stable window.

The physical interpretation is that a boundary seed couples preferentially to a low-energy edge-localized mode when one exists, imprinting an odd/even asymmetry on the recursion; when the lowest scale is set by extended modes, the two invariant subspaces remain evenly matched. Three mechanisms breaking the symmetry are identified: $\alpha$9-breaking pairing, competition between on-site and kinetic scales even in the particle-conserving limit, and long-range hybridization at $N$0, which delays staggering to larger Krylov depth.

Sign changes are counted via a tolerance-filtered crossing count $N$1 over the numerically stable portion of the recursion, restricted to $N$2 to avoid initialization effects.

Results: quantitative agreement of two independent phase diagrams

The main result is a joint phase diagram on a $N$3 grid in $N$4 for $N$5: the region $N$6 closely tracks the BdG-derived boundary $N$7 within grid resolution. Two features strengthen this claim:

Robustness to threshold choice. The correspondence sharpens systematically as $N$8 decreases from $N$9 to $$H_{\mathrm{LRK}}=\sin\theta \sum_{i<j}\frac{c_i^\dagger c_j+(1+\epsilon)c_i c_j+\mathrm{h.c.}}{|i-j|^\alpha}+2\cos\theta\sum_i n_i,$$0, with residual discrepancies confined to the strong long-range regime at large $$H_{\mathrm{LRK}}=\sin\theta \sum_{i<j}\frac{c_i^\dagger c_j+(1+\epsilon)c_i c_j+\mathrm{h.c.}}{|i-j|^\alpha}+2\cos\theta\sum_i n_i,$$1 and attributable to grid resolution and finite-size effects.

Seed dependence. Boundary seeds ($$H_{\mathrm{LRK}}=\sin\theta \sum_{i<j}\frac{c_i^\dagger c_j+(1+\epsilon)c_i c_j+\mathrm{h.c.}}{|i-j|^\alpha}+2\cos\theta\sum_i n_i,$$2, and slightly less sharply $$H_{\mathrm{LRK}}=\sin\theta \sum_{i<j}\frac{c_i^\dagger c_j+(1+\epsilon)c_i c_j+\mathrm{h.c.}}{|i-j|^\alpha}+2\cos\theta\sum_i n_i,$$3) yield quantitative agreement; bulk seeds ($$H_{\mathrm{LRK}}=\sin\theta \sum_{i<j}\frac{c_i^\dagger c_j+(1+\epsilon)c_i c_j+\mathrm{h.c.}}{|i-j|^\alpha}+2\cos\theta\sum_i n_i,$$4, $$H_{\mathrm{LRK}}=\sin\theta \sum_{i<j}\frac{c_i^\dagger c_j+(1+\epsilon)c_i c_j+\mathrm{h.c.}}{|i-j|^\alpha}+2\cos\theta\sum_i n_i,$$5) preserve the qualitative zero-versus-nonzero crossing distinction at representative points but degrade substantially when scanned over the full parameter plane. This establishes seed locality as essential, supporting the broader principle that Krylov diagnostics are informative only when the seed couples selectively to the degrees of freedom under investigation. Notably, the pattern holds in both short-range-like ($$H_{\mathrm{LRK}}=\sin\theta \sum_{i<j}\frac{c_i^\dagger c_j+(1+\epsilon)c_i c_j+\mathrm{h.c.}}{|i-j|^\alpha}+2\cos\theta\sum_i n_i,$$6) and genuinely long-range ($$H_{\mathrm{LRK}}=\sin\theta \sum_{i<j}\frac{c_i^\dagger c_j+(1+\epsilon)c_i c_j+\mathrm{h.c.}}{|i-j|^\alpha}+2\cos\theta\sum_i n_i,$$7) regimes, indicating the diagnostic captures edge-versus-bulk control rather than an artifact of long-range couplings per se.

An important corollary follows immediately: since the model is quadratic and non-ergodic for all parameters, the observed structure must originate from eigenmode spatial localization rather than from chaos—directly contradicting the expectation that Lanczos data in free fermionic systems are featureless.

Numerical methodology and its limits

The paper devotes substantial care to Lanczos stability, acknowledging that orthogonality of the basis does not guarantee coefficient accuracy and citing evidence that finite-precision Lanczos sequences can escape disconnected subspaces. The protocol combines partial reorthogonalization (threshold $$H_{\mathrm{LRK}}=\sin\theta \sum_{i<j}\frac{c_i^\dagger c_j+(1+\epsilon)c_i c_j+\mathrm{h.c.}}{|i-j|^\alpha}+2\cos\theta\sum_i n_i,$$8), termination when $$H_{\mathrm{LRK}}=\sin\theta \sum_{i<j}\frac{c_i^\dagger c_j+(1+\epsilon)c_i c_j+\mathrm{h.c.}}{|i-j|^\alpha}+2\cos\theta\sum_i n_i,$$9 or when basis orthogonality loss exceeds $\alpha$0, and a cross-validation in which formally equivalent but numerically distinct Majorana-basis and Nambu-basis recursions (real versus complex arithmetic) must agree to $\alpha$1. Coefficients failing any criterion are discarded along with all subsequent ones. The authors concede that proving exactness of any long Lanczos sequence is essentially impossible; the claims rest on convergent multi-check evidence rather than proof. At $\alpha$2 the staggering signal is well resolved and $\alpha$3 suffices; smaller systems would require a finite tolerance.

Limitations and open questions

Several limitations bear directly on the strength of the central claim. The edge/bulk classification depends on operational thresholds whose residual sensitivity is largest precisely in the strong long-range regime where the physics is most novel. The correlation between $\alpha$4 sign structure and gap character is empirical, not derived; no analytic argument connects the bipartite Krylov structure quantitatively to the ratio $\alpha$5. When edge and bulk gaps become comparable (e.g., strongly pairing-dominated regimes), qualitative correspondence survives but quantitative matching becomes numerically difficult because the odd and even sub-chain responses nearly coincide. Finally, the single-particle closure exploited here does not extend to interacting systems, so the diagnostic's fate beyond quadratic models is untested.

The paper itself flags the most immediate open question: what sets the number of crossings, and does $\alpha$6 carry information beyond the binary zero/nonzero distinction? It also leaves open whether the crossing count remains meaningful in other Altland-Zirnbauer classes and in disordered long-range chains, where Anderson localization competes with topological edge physics.

Conclusion

This work demonstrates that Lanczos coefficients from local boundary operators constitute a quantitative, dynamical probe of edge-versus-bulk control of the excitation gap in the long-range Kitaev chain, with the Krylov staggering parameter's sign structure reproducing the BdG-based phase diagram across the full $\alpha$7 plane at $\alpha$8. The enabling advance—an exact single-particle operator Lanczos algorithm achieving machine precision via closure of the Majorana-linear commutator algebra—both refutes the presumption of trivial Lanczos structure in free fermionic chains and provides a computationally scalable route to Krylov diagnostics in quadratic models. The result reframes Krylov data as probes of excitation localization and boundary coupling rather than solely of chaos, with the principal caveats being the empirical nature of the staggering–gap correlation, threshold sensitivity in the deep long-range regime, and untested extension beyond the quadratic setting.