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

TL;DR

This paper introduces Krylov subspace diagnostics, especially the Krylov staggering parameter, to distinguish boundary-localized from bulk-extended modes in long-range Kitaev chains, providing a practical tool for analyzing topological superconductors.

Contribution

The authors develop an exact single-particle operator Lanczos algorithm and demonstrate how Krylov diagnostics can identify boundary versus bulk excitations in long-range Kitaev models.

Findings

Krylov staggering parameter is constant in the short-range Kitaev chain and distinguishes topological phases.

Sign patterns of the diagnostic track whether boundary or bulk modes control the lowest excitation gap.

The Lanczos algorithm achieves machine precision for chains of hundreds of sites, enabling practical analysis.

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.