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

TL;DR

This work demonstrates that Krylov/Lanczos diagnostics can distinguish whether edge-localized or bulk-extended modes set the lowest excitation gap in the long-range Kitaev chain. By exploiting an exact single-particle Lanczos formulation in the Majorana basis, the authors reduce operator growth to a $2N$-dimensional problem and define a Krylov staggering parameter $\eta_n = \ln\left(b_{2n-1}/b_{2n}\right)$ whose sign pattern correlates with edge versus bulk physics across the phase diagram. They verify a quantitative match between the Krylov-phase diagram (based on $\eta_n$) and the BdG edge-vs-bulk classification, with boundary seeds providing the sharpest agreement. The results highlight the practical utility of Krylov diagnostics for probing localization of low-energy excitations and have potential implementations in trapped-ion and cold-atom quantum simulators. Overall, the paper establishes a new operational tool to study boundary effects in quadratic, long-range systems and paves the way for extensions to driven, disordered, and interacting settings.

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.

Figures (9)

• Figure 1: BdG spectrum $E_\nu/\sqrt{\operatorname{Tr}(H^2)}$ versus $\theta/\pi$ at $\epsilon=-0.2$ for three long-range exponents $\alpha$ ($N=1000$, open boundaries). \ref{['fig:spectrum_alpha30_eps_minus02']} Short-range-like behavior ($\alpha=3$) displays a gapless region centered at $\theta/\pi \approx 0.5$ with sparse spectral density near $E=0$ elsewhere. \ref{['fig:spectrum_alpha10_eps_minus02']} Intermediate regime ($\alpha = 1$) retains a similar gapless range with modified spectral density. \ref{['fig:spectrum_alpha013_eps_minus02']} Strong long-range limit ($\alpha=1/3$) lifts the degeneracy throughout the $\theta/\pi$ interval except for a small near-degenerate regime, with markedly denser spectral density near $E=0$ throughout.
• Figure 2: Lanczos coefficients $\{b_n\}$ for the long-range Kitaev chain at $\epsilon=-0.2$ with open boundaries and Hermitian boundary seed $\gamma_1$ ($N=1000$). Each panel shows the two interleaved subsequences (odd and even steps of the recursion), whose relative ordering determines the sign of the staggering parameter $\eta_n=\ln(b_{2n-1}/b_{2n})$ and hence the crossing count $N_{\mathrm{cross}}$ (Eq. \ref{['eq:Ncross_def']}). Panels \ref{['fig:lanczos_coefficients_shortrange_trivial']}, \ref{['fig:lanczos_coefficients_longrange_trivial']} show parameter points in the bulk-gap regime (discussed later in the context of Fig. \ref{['fig:lrk_edge_bulk_phase_diagram']}) and exhibit no interchange of the two subsequences (consistent with $N_{\mathrm{cross}}=0$), while panels \ref{['fig:lanczos_coefficients_shortrange_edge']}, \ref{['fig:lanczos_coefficients_longrange_edge']} lie in the edge-gap regime and show a clear interchanges (consistent with $N_{\mathrm{cross}}\ge 1$). To maintain numerical rigor, the Lanczos recursion is terminated when $b_n \lesssim 10^{-7}$, and all subsequent coefficients are excluded from the analysis. Consequently, the total number of Lanczos coefficients varies across parameter points (see Appendix \ref{['app:alg_numerical_stability']} for stability criteria).
• Figure 3: Phase diagram for the long-range Kitaev chain at $\epsilon=-0.2$ with open boundaries ($N=1000$) and boundary seed $\gamma_1$, generating $2000$ Lanczos coefficients. As discussed in Sec. \ref{['subsec:krylov_majorana_physics']}, the black region indicates parameters where the Krylov staggering parameter $\eta_n=\ln(b_{2n-1}/b_{2n})$ exhibits nonzero robust sign changes ($N_{\mathrm{cross}}\ge 1$), while the white region corresponds to $N_{\mathrm{cross}}=0$. As discussed in Sec. \ref{['subsec:edge_vs_bulk_gap']}, solid curves show the edge-bulk gap boundary $\Delta_{\mathrm{edge}}=\Delta_{\mathrm{bulk}}$ extracted from the BdG spectrum using the edge-weight criterion for three choices of threshold, $\omega_{\mathrm{edge}}=0.05,0.1,0.5$ (with $\ell_{\mathrm{edge}}=\sqrt{N}$). The Krylov-based and gap-based boundaries coincide within the numerical resolution of the $99\times 99$ grid in $(\alpha,\theta)$ with $\alpha\in(0,3]$ and $\theta\in(0,\pi)$. The phase boundary is robust across different $\omega_{\mathrm{edge}}$ values; discrepancies arise from grid resolution and finite-size effects, the latter being most pronounced in the strong long-range regime ($\alpha\lesssim 1$) at larger $\theta$ (see main text for further discussion).
• Figure 4: BdG spectrum $E_\nu/\sqrt{\mathrm{Tr}(H^2)}$ versus $\theta/\pi$ at $\epsilon=1$ for three long-range exponents $\alpha$ ($N=1000$, open boundaries). The $\alpha$-dependence mirrors Fig. \ref{['fig:spectrum_eps_minus02']}: panel (a) displays a gapless region at $\theta/\pi \approx 0.5$ with sparse spectral density near $E=0$ elsewhere; panel (b) shows modified spectral density; panel (c) shows the degeneracy lifted throughout the $\theta/\pi$ interval except for a small near-degenerate regime, with dense spectral density near $E=0$. Compared to $\epsilon=-0.2$, the spectral density near $E=0$ is increased across all $\alpha$ regimes.
• Figure 5: BdG spectrum $E_\nu/\sqrt{\mathrm{Tr}(H^2)}$ versus $\theta/\pi$ at extreme pairing-dominated $\epsilon=10$ for three long-range exponents $\alpha$ ($N=1000$, open boundaries). The systematic $\alpha$-dependence observed in Figs. \ref{['fig:spectrum_eps_minus02']} and \ref{['fig:spectrum_eps_1']} persists: gapless region for $\alpha=3$, similar gapless range for $\alpha=1$ with gap opening, and degeneracy lifted except for a small near-degenerate regime for $\alpha=1/3$. Spectral density near $E=0$ is further increased relative to smaller $\epsilon$, occurring uniformly across all $\alpha$ including the short-range limit. The qualitative $\alpha$-dependence remains unchanged across various $\epsilon$ regimes.