Twisted Magnetism Can Both Create and Destroy Key Components for Quantum Computing

Researchers are increasingly exploring heterostructures combining superconductors and magnetic textures as a promising pathway towards realising topological quantum computation. Bastien Fajardo, T. Pereg-Barnea, Arun Paramekanti et al. from McGill University, the University of Toronto, and Argonne National Laboratory demonstrate that Majorana zero modes (MZMs) can emerge in superconductor-magnet heterostructures featuring d-wave superconductivity. Their work extends previous theoretical predictions for s-wave superconductors and reveals a surprising sensitivity of these MZMs to the strength of d-wave pairing and skyrmion-induced spin twisting, potentially leading to topological phase transitions. This finding is significant because it highlights the crucial role of pairing symmetry in designing robust MZMs and informs material selection for future topological quantum devices.

Suppression of Majorana zero modes in d-wave superconductor-magnetic skyrmion heterostructures

Researchers have uncovered a surprising phenomenon concerning Majorana zero modes (MZMs) within hybrid superconductor-magnet heterostructures. These MZMs, considered crucial for building fault-tolerant quantum computers, are typically sought in systems combining superconductors with magnetic textures like magnetic skyrmions.

Previous theoretical work established a pathway to induce these modes in skyrmion-vortex pairs using conventional s-wave superconductors. This new study extends that investigation to fully gapped d+is and d+id superconductors, revealing a counterintuitive result: enhanced d-wave pairing or stronger skyrmion-induced spin twisting can actually eliminate the topological conditions necessary for stable MZMs.

This unexpected behaviour stems from the unique spatial structure of d-wave pairing and the mixing of pairing channels with differing angular momentum when the skyrmion texture is effectively untwisted through a coordinate transformation. The work employs exact diagonalisation of Bogoliubov, de Gennes tight-binding models to analyse how MZMs emerge in these heterostructures.

Analysis demonstrates that while stable MZMs are present across much of the system’s parameter space, increasing the d-wave pairing strength or the rate of radial winding of the skyrmion spin texture can disrupt the topological state. This contrasts with s-wave superconductors, where stronger coupling and larger gaps generally promote MZM stability.

By performing a local spin rotation, researchers generated a spatially uniform spin, orbit coupling, similar to the s-wave case, but observed a highly nonlocal structure in the superconducting gap. The transformed d-wave pairing decomposed into a superposition of both even and odd angular-momentum pairing channels, altering the criteria for topological protection.

When the d-wave amplitude or skyrmion-induced spin, orbit coupling becomes sufficiently large, the odd-angular-momentum components dominate, driving the system into a trivial superconducting phase despite the presence of both vortices and skyrmion winding. These findings highlight a critical constraint for realising MZMs in unconventional superconductors, suggesting that a delicate balance between order parameter amplitudes and magnetic texture is essential for maintaining a robust Majorana bound state.

The research informs the feasibility of designing topological quantum devices based on layered heterostructures incorporating unconventional superconductors and skyrmions, particularly those utilising cuprates exhibiting time-reversal-symmetry-breaking superconductivity at elevated temperatures. Experiments on twisted Bi2Sr2CaCu2O8+δ (Bi-2212) junctions have already reported signatures of this symmetry breaking, further motivating the exploration of these hybrid systems.

Delineating Topological Phase Boundaries using Zero Mode Wavefunction Overlap

Researchers employed an overlap criterion to delineate phase boundaries for topological superconductivity induced by magnetic skyrmions in proximity to d+is and d+id superconductors. This method involved scanning parameter space and evaluating the integral of the overlap between the wavefunctions of zero modes, denoted as Iθ⋆, to precisely determine transitions between topological and trivial phases.

A comparison metric, the ratio of the second lowest positive energy eigenvalue (E2) to the lowest positive energy eigenvalue (E1), was also calculated; however, this proved less reliable in identifying clear phase boundaries than the overlap method. Radial probability distributions of edge Majorana modes were then computed for both the d+is and d+id models to visually confirm the localization of modes in topological and trivial phases.

The study utilised a numerical procedure to generate phase diagrams for both superconducting models, with the overlap criterion consistently producing sharper distinctions between phases compared to the energy ratio method. Representative radial probability distributions were analysed to demonstrate instances where the energy ratio incorrectly indicated a topological phase due to the detection of a trivial Andreev bound state, while the overlap criterion accurately identified the trivial state.

Finite size effects were observed for small radial oscillation numbers (p) and zero d-wave pairing strength (∆d), leading to a revival of the trivial phase due to the divergence of the size of the MZMs. Notably, increasing either the superconducting strength (∆d) or the effective spin-orbit strength (p) was found to destroy the topological phase in the d+is model, a counterintuitive result explained through analysis in a rotated basis.

This transformation, achieved using a spatially varying spin rotation U(r), aligned the spin quantization axis with the skyrmion field, revealing an emergent, spatially varying spin-orbit coupling resembling the Rashba structure. The transformation of the pairing terms differed depending on the symmetry, with the d+is and d+id pairings exhibiting new contributions with angular dependence, ultimately influencing the topological phase boundaries and the stability of Majorana zero modes.

D-wave superconductivity and skyrmion interactions suppress Majorana zero mode topology

Magnetic skyrmions in proximity to fully gapped d+is and d+id superconductors exhibit a complex interplay influencing the emergence of Majorana zero modes. Analysis using exact diagonalisation of Bogoliubov, de Gennes tight-binding models revealed that increasing the strength of the d-wave component or the effective spin, orbit coupling generated by the skyrmion can drive the system into a topologically trivial phase.

This behaviour represents a significant departure from observations in s-wave superconductors, where larger spin, orbit coupling and superconducting gaps generally enhance the stability of Majorana zero modes. The research demonstrates that a delicate balance is required to maintain an effectively even total winding and a robust Majorana bound state.

Specifically, the study found that stronger d-wave components and skyrmion-induced spin twisting do not necessarily favour topology in these heterostructures. A local spin rotation aligning the skyrmion exchange field along a uniform z axis generates a spatially uniform spin, orbit coupling, similar to the s-wave case, but the superconducting gap acquires a highly nonlocal structure.

This transformed d-wave pairing decomposes into a superposition of pairing channels possessing both even and odd orbital angular momentum. Consequently, the standard topological criterion for supporting a localised Majorana mode, requiring an even total winding from the superconducting vortex, skyrmion texture, and superconducting order parameter, becomes inapplicable.

When the d-wave pairing amplitude or skyrmion-induced spin, orbit coupling exceeds a certain threshold, the odd-angular-momentum components dominate, leading to a trivial superconducting phase despite the presence of vortices and skyrmion winding. These findings provide crucial insights into the feasibility of realising Majorana zero modes with unconventional superconductors in heterostructured devices.

D-wave superconductivity destabilises skyrmion-hosted Majorana zero modes

Researchers have demonstrated that magnetic skyrmions in proximity to d-wave superconductors can host zero modes, but unlike s-wave superconductors, enhanced pairing or stronger spin twisting within the skyrmion can disrupt the topological state necessary for these modes. This finding stems from an investigation into the behaviour of these skyrmions when placed near d+is and d+id superconductors, revealing a complex interplay between the skyrmion’s spin texture and the unconventional pairing symmetry of the superconductor.

The study details how the spatial structure of d-wave pairing and the mixing of angular momentum channels influence the stability of these zero modes. The research clarifies the conditions under which Majorana zero modes, quasiparticles with potential applications in topological quantum computing, can emerge in heterostructures combining magnetic skyrmions and unconventional superconductors.

Specifically, the team explored how factors such as the rate of radial winding of the skyrmion spin texture, the strength of exchange coupling, and the amplitude of d-wave pairing affect the existence of these modes. Analysis within a rotated spin basis highlighted how the transformed d-wave pairing generates mixed angular-momentum channels, ultimately altering the topological conditions required for Majorana zero modes.

The authors acknowledge that their model assumes smooth variations in the skyrmion winding compared to the superconducting layer’s relevant length scales, which may not always hold true in experimental systems. Future research could focus on exploring the effects of more complex skyrmion textures and stronger coupling between the skyrmion and the superconductor. These findings are relevant to ongoing experimental efforts aimed at realising Majorana zero modes in skyrmion-superconductor heterostructures and twisted cuprate systems, providing valuable insights into the feasibility of utilising unconventional superconductors for topological quantum computation.