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

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 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.