Quantum System Analysis Reveals Hidden Details of Superconductivity’s Building Blocks

Scientists are increasingly focused on understanding operator growth and its connection to quantum chaos in complex many-body systems. Rishabh Jha and Heiko Georg Menzler, both from the Institute for Theoretical Physics at Georg-August-Universität Göttingen, demonstrate that Krylov subspace methods, typically expected to yield trivial results for quadratic fermionic Hamiltonians, can, in fact, serve as a powerful diagnostic tool. Their work reveals how Lanczos coefficients, generated from local boundary operators, quantify the character of the lowest excitation gap in the long-range Kitaev chain, a prominent model for topological superconductivity. By introducing a ‘Krylov staggering parameter’, the researchers correlate the sign structure of Lanczos coefficients with whether the gap is controlled by boundary-localized or bulk-extended modes, offering a new way to probe low-energy excitation localisation and boundary effects in this and potentially other systems, including cold-atom quantum simulators.

Scientists have developed a new method to pinpoint the origin of energy gaps in the long-range Kitaev chain, a specific type of superconductor, using Krylov subspace diagnostics. This approach moves beyond simply identifying whether a material is chaotic or orderly, instead focusing on where the lowest energy excitations reside, at the boundaries or within the bulk of the material. The research establishes that subtle patterns within the Lanczos coefficients, generated from local measurements, can reliably indicate whether the energy gap is determined by edge-localized or bulk-extended modes across the entire phase diagram of the system. This correlation stems from a unique bipartite structure within the Krylov space, induced by the pairing of electrons, the power-law nature of the interactions, and the open boundaries of the chain. Researchers achieved this breakthrough by deriving an exact single-particle operator Lanczos algorithm, significantly reducing computational demands and enabling calculations with machine precision for chains containing hundreds of sites. This algorithm transforms a problem that typically scales exponentially with system size into a manageable linear problem, crucial for distinguishing genuine physical phenomena from numerical artifacts. The study introduces a novel “Krylov staggering parameter”, calculated as the logarithmic ratio of consecutive odd and even Lanczos coefficients, which serves as a robust indicator of the gap’s character. To probe the character of low-energy excitations, researchers focused on local boundary operators as the initial seed for the Lanczos recursion, as these are particularly sensitive to the localization properties of the system’s excitations. The resulting Lanczos coefficients were used to define this parameter, providing a quantitative measure of the bipartite structure induced by pairing, power-law couplings, and open boundaries. The research demonstrates that the sign structure of the Lanczos coefficients accurately identifies the dominant character of the excitation gap, providing a quantitative measure of localization. Specifically, the bipartite Krylov structure arises from the interplay of long-range pairing, U symmetry breaking, and the competition between boundary and bulk excitations. The exact single-particle formulation circumvents numerical instabilities that typically plague many-body Krylov implementations, enabling analysis of larger system sizes and more precise identification of physical phenomena. These findings establish Krylov diagnostics as a powerful tool for probing the localization of low-energy excitations and their connection to boundaries where symmetry is broken. The implications extend to the development of advanced quantum simulators, particularly those based on trapped ions and cold atoms, offering a new avenue for understanding and controlling complex quantum systems. By focusing on the structure of the eigenmodes, this work reveals hidden details within seemingly simple quadratic models. Unlike short-range interactions, power-law couplings can lead to unusual behaviour, including violations of area laws and the emergence of gapped phases without conventional gap closure. This research demonstrates that even within this exactly solvable framework, Krylov subspace methods can uncover subtle but significant distinctions in the system’s behaviour, offering a new perspective on the interplay between long-range pairing, symmetry breaking, and the localization of quantum excitations. Scientists seeking to understand complex quantum systems often face a fundamental challenge: how to characterise the behaviour of many interacting particles. This work offers a novel diagnostic, leveraging Krylov subspace methods, to pinpoint where the key energy-determining excitations reside within a topological superconductor. The difficulty lies in the fact that many theoretical tools struggle to distinguish between these scenarios, particularly in systems with long-range interactions. What makes this approach notable is its ability to move beyond simply identifying a gap in energy, to actively locate the origin of that gap. By analysing how quantum operators evolve over time, the researchers have devised a ‘staggering parameter’ that reliably indicates whether the lowest energy excitations are localised at the boundaries or spread throughout the system. While bulk-localised operators still reveal qualitative differences, the precision diminishes, and extending the method to more complex, disordered systems remains an open question. Looking ahead, this technique could become invaluable for validating the performance of quantum simulators, providing a means to verify that these devices are accurately reproducing the behaviour of topological materials. More broadly, the development of robust diagnostics for operator localisation represents a significant step towards understanding the emergence of complex behaviour in quantum many-body systems, potentially unlocking new avenues for materials discovery and quantum technologies.