Bulk-dissociated topological bands without spin-orbit coupling in hetero-dimensional superconducting metamaterials

The gist: The authors theoretically study a square superconducting network decorated with spin-polarized magnetic adatoms and find that localized Yu-Shiba-Rusinov bound states collectively form a weak topological superconducting phase despite the absence of spin-orbit interactions, and by tuning the Fermi energy, the system can transition to a bulk-dissociated TSC phase where edge state bands separate from the bulk, giving rise to unexpected features such as nodal lines and co-existing bulk-dissociated edge and corner modes.

Topological Phase Generation without Spin-Orbit Coupling

The study investigates how a square superconducting network decorated with spin-polarized magnetic adatoms can generate a weak topological superconducting phase even when assuming no spin-orbit interactions or intrinsic SOC <ref:2604.08675#pg5>. This is achieved by modeling magnetic adatoms as impurities with spin S using an exchange interaction with coupling J at the sites of the impurities, defining a real space Kondo lattice forming a square network with a lattice constant a and plaquette constant (superlattice constant) l <ref:2604.08675#pg6>. The system is described using the Bogoliubov de-Gennes (BdG) method with a mean-field superconducting gap ∆ <ref:2604.08675#pg7>. Despite assuming collinear spin structure and no spin-orbit interactions in the model, the network geometry allows a weak topological superconducting phase to develop, hosting spinpolarized weak topological boundary modes <ref:2604.08675#pg5>. This demonstrates that a ferromagnetic spin configuration of the decorated superconducting network can generate a TSC <ref:2604.08675#pg5>.

Transition to Bulk-Dissociated States

The research demonstrates that by tuning the Fermi energy of the network, the system can transition from a weak TSC phase to a bulk-dissociated TSC phase where the edge state bands separate from the bulk, giving rise to unexpected features such as nodal lines and co-existing bulk-dissociated edge and corner modes <ref:2604.08675#pg3>. Such bulk-dissociated topological phases can be rigorously defined by their susceptibility to Anderson disorder, specifically when boundary modes are localizable by disorder and local boundary perturbations <ref:2604.08675#pg5>. The emergence of nodal topological superconductivity is identified when the bulk superconducting gap can be suppressed and, yet, topological edge bands remain intact by modulating the Fermi energy <ref:2604.08675#pg5>.

Characterization of Topological Phases

The bulk topology is characterized by calculating the strong topological invariant; according to the Altland-Zirnbauer classification, YSR superlattices belong to class D and the Chern number is the corresponding topological invariant <ref:2604.08675#pg6>. For two-dimensional superconductors in class D, Z2 weak topological symmetry indicators (TIs) vx,π and vy,π are constructed from Pfaffians evaluated at high-symmetry points in the Brillouin zone <ref:2604.08675#pg6>. The classification of phases is summarized in Table I, which distinguishes between first-order bulk-coupled (I), first-order bulk-dissociated (II), hybrid bulk-coupled (III), and hybrid bulk-dissociated (IV) weak TSC phases <ref:2604.08675#pg9>. Phase II, for instance, is characterized by vx/y,π = +1, ∆b ≥ 0 and δSEB > 0 <ref:2604.08675#pg8>.

Bulk-Dissociated Edge and Corner Modes

The study presents the excitation band structure showing a flat band emerging near E = 4∆ in a YSR network geometry <ref:2604.08675#pg10>. In phase II, gapless edge bands are dissociated from the bulk bands that cross the Fermi energy in phase II where ∆b = 0 <ref:2604.08675#pg10>. This bulk dissociation of the edge bands presents with the lack of edge-bulk hybridization at momenta where edge and bulk bands cross <ref:2604.08675#pg10>. The local density of states (LDOS) analysis distinguishes between phases by showing that the bulk-dissociated edge mode has a large spectral weight at the boundary with bulk penetration ∼ l before a complete suppression of the LDOS further into the bulk <ref:2604.08675#pg10>.

Corner Mode Properties and Robustness

The corner modes in phase IV are found to be highly localized corner modes in the network <ref:2604.08675#pg13>. The positive energy corner modes are four-fold degenerate and spinpolarized, indicating they are not spinless Majorana modes <ref:2604.08675#pg14>. The scaling of the energy of a corner mode E0 with L for JS = 0.6t and EF = 1.924t shows a rapid convergence leading to an anomalous L-independent energy for L > 200a, which is used to characterize bulk-dissociated corner modes <ref:2604.08675#pg14>. The localization of the corner state is characterized by an exponential fit ∼ e −λcx/a, and smaller λc leads to larger Lc <ref:2604.08675#pg14>.

Hetero-dimensional Metamaterial Application

The YSR network is viewed as a hetero-dimensional nanoscale metamaterial (HNM), where 1D components are geometrically assembled to form a 2D structure which is then decorated with 0D magnetic adatoms <ref:2604.08675#pg17>. This heterodimensional system serves as a useful template for controlling the coupling between modes of different dimensionalities, specifically coupling of 2D bulk, 1D edge, and 0D corner modes <ref:2604.08675#pg17>. Manipulating boundary morphology can help distinguish topological and trivial modes while also being a useful tool to control the hybridization of bulk and boundary modes <ref:2604.08675#pg16>.

Conclusion on Topological Phases

The work concludes that neither spin-orbit interactions nor spin textures are necessary to generate a wide variety of topological superconducting phases in magnetic adatom decorated superconducting networks <ref:2604.08675#pg18>. The findings highlight the nuanced role of bulk and boundary mode coupling in determining spectral properties of topological phases that may affect topological phases in other symmetry classes and dimensions <ref:2604.08675#pg18>. The network geometry is shown to be a useful system to study bulk-boundary coupling directly by manipulating boundary morphology <ref:2604.08675#pg18>.

Experimental Realization Potential

A bottom-up nanoassembly approach can be pursued for realizing a YSR network, such as stacking a network of 1D metallic carbon nanotubes onto a 2D or 3D superconducting substrate and decorating the network with magnetic impurities <ref:2604.08675#pg18>. Alternatively, one can take advantage of superconducting networks emerging in twisted van der Waals materials where the moire potential naturally creates an emergent conductive honeycomb network in the graphene bilayer <ref:2604.08675#pg18>. Both approaches represent a more general construction of hybrid materials using a hetero-dimensional design <ref:2604.08675#pg18>.

Methods Overview

The tight binding model starts by considering a one-dimensional metallic chain decorated with magnetic impurities, defining the Hamiltonian on a Kondo lattice <ref:2604.08675#pg19>. The YSR chain model is extended to a square network using the BdG formalism with a mean-field approximation of the superconducting gap ∆ <ref:2604.08675#pg20>. The BdG Hamiltonian in real space for the YSR chain is defined by equation (6) <ref:2604.08675#pg21>. When extending to a square network, the Hamiltonian H(network)BdG includes terms describing hopping, exchange interaction with magnetic impurities, and superconducting pairing <ref:2604.08675#pg21>. Numerical results were found using the Python package Kwant62 <ref:2604.08675#pg21>.

References

The paper cites numerous foundational works in topological insulators and superconductors, including those by Hasan and Kane on topological insulators <ref:2604.08675#pg24>, Kitaev on majorana fermions in quantum wires <ref:2604.08675#pg24>, and various studies on Shiba lattices and hybrid structures <ref:2604.08675#pg25>. Key references include those concerning the proposal for realizing majorana fermions in chains of magnetic atoms on a superconductor <ref:2604.08675#pg26> and studies on topological superconductivity in 2D ferromagnetic shiba lattices <ref:2604.08675#pg27>. The paper also references works on surface spectroscopy and surface–bulk hybridization of Weyl semimetals <ref:2604.08675#pg28> and non-hermitian origin of detachable boundary states in topological insulators.

Acknowledgments

The authors acknowledge that the work at Sandia is supported by a LDRD project, and S.A.A.G. was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No DE-SC0024942 <ref:2604.08675#pg18>.

Competing Interests

The authors declare no competing interests <ref:2604.08675#pg19>.

Data Availability

All data are available from the corresponding author upon reasonable request <ref:2604.08675#pg19>.

Extended Data Visualization

Extended Data Fig. 1 illustrates corner mode manipulation with boundary morphology, showing LDOS of corner modes with a trivial superconductor attached to the bottom of the network and with a trivial superconductor attached to the bottom left side of the network <ref:2604.08675#pg20>. Extended Data Fig. 2 shows experimental realization of a YSR network using superconducting proximity effect in carbon nanotubes functionalized with single-molecule magnets and magic-angle twisted bilayer graphene decorated with single-molecule magnets <ref:2604.08675#pg20>.

Tight Binding Model Details

The tight binding model starts by considering a one-dimensional metallic chain decorated with magnetic impurities, defining the Hamiltonian on a Kondo lattice <ref:2604.08675#pg19>.

1. A new class of topological materials can be synthesized without relying on intrinsic spin-orbit coupling, as demonstrated by coupling magnetic adatoms on the surface of a superconductor, forming a so-called YuShiba-Rusinov (YSR) chain which can create a spinless p-wave TSC. This allows AI systems to design and simulate novel topological superconducting architectures where geometric configuration dictates topological properties rather than requiring complex intrinsic material properties.

2. AI can predict the transition between different topological phases by tuning parameters like Fermi energy, as shown by the observation that by tuning the Fermi energy of the network, the system can transition from a weak TSC phase to a bulk-dissociated TSC phase. This enables AI to optimize network designs for specific desired boundary behaviors, such as realizing nodal topological superconductivity where the bulk superconducting gap can be suppressed and, yet, topological edge bands remain intact.

3. AI systems can accurately characterize and manipulate the coupling between different dimensionalities of electronic states by analyzing how manipulating boundary morphology can control the coupling between corner, edge, and bulk modes. This capability allows AI to design nanoscale metamaterials that serve as a useful template for controlling the coupling between electronic modes of different dimensionalities, leading to tailored interaction strengths.

Topological superconductors (TSCs) in superconducting hybrid heterostructures, which integrate superconducting and non-superconducting materials, have been intensely investigated with the hope of discovering exotic non-Abelian anyons for fault-tolerant quantum computing. In this effort, a challenge for hybrid superconducting systems is controlling hybridization, which is often a balance between enhancing the superconducting proximity effect at the cost of suppressing desirable electronic properties such as strong spin-orbit interactions. Hence, discovering hybrid superconducting systems with topological properties controlled and enhanced by material geometry design without spin-orbit interactions would be intriguing to explore. In this work, we theoretically study a square superconducting network decorated with spin-polarized magnetic adatoms. We find that localized Yu-Shiba-Rusinov bound states at magnetic adatom sites collectively form a weak topological superconducting phase despite the absence of spin-orbit interactions. We then demonstrate that by tuning the Fermi energy of the network, the system can transition from a weak TSC phase to a bulk-dissociated TSC phase where the edge state bands separate from the bulk, giving rise to unexpected features such as nodal lines and co-existing bulk-dissociated edge and corner modes. Moreover, our findings highlight how hetero-dimensional superconducting metamaterials can serve as a useful template for controlling the coupling and dissociation between electronic degrees of freedom of different dimensionalities.