Kirill Parshukov and Andreas P. Schnyder of the Max Planck Institute for Solid State Research have identified three exotic superconducting states potentially realizable within a newly discovered class of materials called altermagnets. Their work, grounded in the symmetry of these materials, reveals possibilities including a “half-and-half metal-superconductor” where Cooper pairs form from only one spin component of electrons.

One predicted superconductor exhibits an unusual fractional ac Josephson current for only one of the two spin polarizations, suggesting these aren’t merely theoretical constructs but potentially stable states of matter. The researchers constructed and tabulated all symmetry-allowed pairing functions for altermagnets, uncovering these novel states.

Altermagnets: Symmetry & the Novel Interplay of Magnetism and Superconductivity

The symmetry of altermagnetic materials permits the existence of a non-unitary superconductor exhibiting a fractional alternating current (ac) Josephson current, but only for electrons with one specific spin polarization. This highly unusual behavior deviates from standard superconducting properties where such currents typically manifest identically across all spin channels. Andreas P. Their work, focused on the interplay between magnetism and superconductivity, uses irreducible co-representations of altermagnetic spin-point groups to predict these novel phenomena.

The team’s Ginzburg-Landau theories demonstrate that these unconventional superconductors correspond to stable minima of free energy, indicating they are not merely theoretical constructs but potentially realizable states of matter. This stability is important for potential applications and further investigation into the material’s properties. Examining the topological properties of these predicted superconductors, the scientists also investigated the effects of even small amounts of spin-orbit coupling, a quantum mechanical interaction between an electron’s spin and its motion.

They found that a spin chiral superconductor, another of the three exotic states identified, can exhibit spin-polarized edge states, potentially leading to novel spintronic devices. “The complete list of symmetry-allowed pairings lays down foundations for the study of exotic superconductivity in altermagnets,” the researchers state, emphasizing the broad implications of their symmetry-based approach. The exploration of these pairings, particularly those with multi-component order parameters, could lead to detailed physical phenomena and competition between different superconducting orders.

The unique symmetry properties of altermagnetic spin-point groups are key to these findings, allowing for higher-dimensional corepresentations that can break antiunitary symmetry within individual spin subspaces. This breaking of symmetry, specifically the ( $T 2_{}$ ⊥||E) symmetry, can lead to chiral superconductivity and the emergence of chiral boundary modes with finite Chern numbers. The team’s analysis reveals that the spin-polarized half-and-half metal-superconductor breaks ( $2_{}$ ⊥||g) symmetry, resulting in a superconducting gap appearing in only one spin subspace.

Irreducible Corepresentations Define Symmetry-Allowed Pairing in Altermagnets

The structure of altermagnetic spin-point groups allows for superconducting states with pairing functions previously unseen in conventional magnets, according to a new theoretical analysis. Researchers systematically cataloged all symmetry-allowed pairings within these materials, revealing a wealth of exotic possibilities beyond those predicted by models assuming strong spin-orbit coupling. This work establishes a foundational understanding of how superconductivity can manifest in altermagnets, materials distinguished by their unique magnetic order.

This asymmetry arises from the distinct way the altermagnetic spin-point groups act on spin and lattice degrees of freedom, differing from the behavior of regular magnetic point groups. The researchers employed both unitary halving groups with Dimmock indicators and direct calculations from corep matrices to derive the basis functions for these superconducting orders, ensuring a comprehensive approach to identifying allowed pairings.

Altermagnetic Superconductor: Non-Unitary Order & Fractional Josephson Current

This asymmetry arises from topological boundary states localized within the material’s spin-up subspace, while the spin-down subspace lacks similar modes given a specific termination condition. The work expands beyond standard superconducting models by identifying a state where Cooper pairs form exclusively from electrons possessing a single spin component. This contrasts with the typical pairing of electrons with opposite spins, and shares similarities with superconducting phases observed in helium-3 and ferromagnetic superconductors, while the remaining unpaired electrons behave as a half-metal.

Minimizing the Ginzburg-Landau free energy, a functional respecting the system’s Hamiltonian, revealed the stability of these unconventional states, using a fourth-order expansion based on two-dimensional corepresentations describing spin ±1 states. The analysis relies on basis vectors defining the pairing potential, expressed as Δ(k) = $φ_{1} Δ_{D_-} (k)$ + φ_2Δ_(D_ + )(k), with Δ_(D_ ± )(k) representing corepresentation basis vectors.

The topological properties of these states directly influence physical responses, most notably the ac Josephson current, which was computed for small applied voltages and zero temperature. The phase dependence, φ(t) = 2eVt/ℏ, changes adiabatically, resulting in 4π periodicity for spin-1 pairs, mirroring behavior seen in one-dimensional p-wave superconductors. Conversely, spin-negative-1 pairs exhibit a 2π periodicity, a consequence of the topological boundary state confined to the spin-up subspace.

The researchers state. Preservation of antiunitary ( $T 2_{}$ ⊥||E) symmetry further protects spin-polarized, gapless point excitations in two-dimensional systems, leading to non-degenerate zero-energy states at the boundary, Majorana boundary modes, and quantized winding numbers in three-dimensional systems. These anisotropic spin-space locking mechanisms and topological degeneracies give rise to exotic spin-polarized topological responses, including the fractional ac Josephson current observed for only one spin polarization.

The team’s calculations, presented as a slab model spectrum with 40 layers, demonstrate the pairing potential and excitation spectrum for a stable altermagnetic superconducting state. This detailed analysis, coupled with the identification of stable free energies, suggests these predicted states are not simply theoretical curiosities, but potentially realizable forms of matter within the emerging field of altermagnetism.

Half-and-Half Metal-Superconductor with Spin-Selective Cooper Pairing

Minimizing the Ginzburg-Landau free energy revealed stable superconducting states within altermagnets, specifically those where the superconducting gap opens in only one spin subspace, creating what researchers term a half-and-half metal-superconductor. This unusual state, similar to the A1 phase of Helium-3 and ferromagnetic superconductors, features a metallic component in one spin orientation alongside superconductivity in the other. The analysis, detailed in the paper, demonstrates that this pairing is non-unitary and actively breaks the altermagnetic symmetry denoted as (2⊥||g).

(7)] supports the existence of these unconventional pairings, with the (1,0) solution corresponding to the half-and-half metal-superconductor where |φ1| ≠ 0 and |φ2| = 0. Beyond the half-and-half state, the work identifies a spin chiral superconductor exhibiting spin-polarized edge states, and a non-unitary superconductor with differing spatial anisotropies for spin-up and spin-down condensates.

Spin Chiral Superconductors and Topological Classifications in Altermagnets

Phenomenological Ginzburg-Landau theories confirm the stability of the three newly identified superconducting states within altermagnets, indicating these are not merely theoretical possibilities but potentially realizable phases of matter. Researchers established that these unconventional superconductors represent stable minima within the Ginzburg-Landau free-energy functional, an essential step in validating their potential existence. The work also considers potential material candidates where these exotic phases might be observed, and investigates the resulting topological responses.

This specific configuration, they note, breaks the (2⊥||g) symmetry, a key indicator of its unconventional nature and a departure from more standard superconducting behavior. Beyond this, the team identified a spin chiral superconductor characterized by spin-polarized edge states, which are predicted to host Majorana modes at the material’s boundaries.

These modes, protected by chiral symmetry, could potentially be used in fault-tolerant quantum computing architectures. The identification of these exotic states builds upon the foundation of symmetry considerations, using irreducible co-representations of the altermagnetic spin-point groups. The work also highlights the potential for detailed physical phenomena arising from higher-dimensional coreps, leading to multicomponent order parameters and potentially exotic domain physics. Spatial fluctuations of phases within these order parameters could even give rise to topological defects analogous to vortices observed in two-dimensional superconductors. The researchers also considered the impact of spin-orbit coupling.

👉 More information
🗞 Exotic superconducting states in altermagnets
✍️ Kirill Parshukov and Andreas P. Schnyder
🧠 DOI: http://link.aps.org/doi/10.1103/7zl4-35wd

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