Direction-selective triplet pairing and spin-edge locking in altermagnetic metals

The gist: Momentum-dependent altermagnetic spin splitting suppresses opposite-spin singlet pairing and stabilizes highly anisotropic equal-spin triplet order, leading to nearly dispersionless Majorana boundary states associated with effective one-dimensional topological channels

Model and Framework

The study develops a minimal self-consistent framework for superconductivity in a two-dimensional d-wave altermagnetic metal with RSOC and short-range attractive interactions. The normal-state Hamiltonian is given by H0 = Pk c†k h0(k)ck, where h0(k) is defined as Hn = ϵ(k)σ0 + 2J0(cos kx − cos ky)σz + 2λ(sin kyσx − sin kxσy>. The interaction Hamiltonian describes pairing arising from magnetic exchange interactions on nearest-neighbor bonds, leading to bond order parameters ∆s i,j and ∆t,σ i,j>.

Pairing Selection Mechanism

The self-consistent solutions reveal several robust features [65]. For the spin-singlet channel, the result is always ∆s x = −∆s y, indicating a dominant dx2−y2-wave pairing (hereafter referred to as d-wave). In the absence of RSOC (λ = 0), the triplet order becomes quasi-one-dimensional within each spin sector: for spin up, only the y-directed bond component is finite (∆t,↑ y ≠ 0, ∆t,↑ x ≈ 0); for spin down, the opposite holds (∆t,↓ x ≠ 0, ∆t,↓ y ≈ 0). The preserved C4zT symmetry further relates the two spin sectors by enforcing ∆t,↑ y = ∆t,↓ x>.

Role of Rashba Spin-Orbit Coupling (RSOC)

RSOC qualitatively modifies this pairing structure. As indicated by the Fermi surface in the inset of Fig. 1(b), the electronic states acquire mixed spin character, relaxing the spin-selective pairing pattern at λ = 0 and activating otherwise suppressed pairing components, allowing singlet and triplet orders to coexist in a mixed-parity superconducting state. Most notably, in each spin sector the originally quasi-one-dimensional triplet pairing acquires a finite px component, evolving into a px + ipy-like state, as evidenced by the finite ∆↑,px seen in Fig. 1(b) at finite λ.

Boundary Properties and Spin-Edge Locking

The momentum-space phase winding of the triplet pairing motivates an analysis of the boundary spectrum. In the λ = 0 limit, each spin sector realizes p-wave pairing predominantly along a single crystalline direction, rather than a two-dimensional chiral px ± ipy state, which is directly reflected in the cylinder spectrum shown in Fig. 2(a). With finite RSOC, the boundary spectrum changes qualitatively as shown in Fig. 2(c), where the nearly flat zero-energy modes become dispersive away from the highsymmetry momenta. The spin-resolved boundary spectra reveal a characteristic spin-edge locking, in which the dominant spin polarization of the boundary states is determined by the crystallographic orientation of the edge. This behavior follows from the C4zT symmetry and establishes a clear correlation between boundary orientation and spin polarization.

Conclusion on Mechanism

Altermagnetic spin splitting acts as a symmetry-selective mechanism that suppresses singlet pairing, generates a quadratic anisotropy between the px and py triplet channels, and selects direction-dependent equal-spin triplet components. The resulting superconducting state supports Majorana boundary states whose spin-resolved spectral weights reveal a characteristic spin-edge locking dictated by the altermagnetic symmetry. These results establish altermagnetic metals as a platform for symmetry-controlled pairing and spin-selective Majorana boundary phenomena without external magnetic fields.

Supplementary material for “Direction-selective triplet pairing and spin-edge locking in altermagnetic metals”

In this Supplementary Material, we provide additional theoretical analysis and supporting results for the main text. Section S-1 presents a Ginzburg–Landau analysis of the two-component p-wave order parameter and shows how altermagnetic anisotropy leads to direction-selective triplet pairing. Section S-2 gives a complete account of the selfconsistent superconducting state, including the pairing amplitudes in both spin sectors, their symmetry relations, the momentum-space phase winding of the triplet pairing, and the spin-resolved boundary spectral functions underlying the spin–edge correspondence.

S-1. GINZBURG–LANDAU ANALYSIS OF ALTERMAGNETIC ANISOTROPY

We analyze the effect of altermagnetic anisotropy on the equal-spin triplet order parameter within a Ginzburg–Landau (GL) theory [8]. The quadratic GL free energy for one spin sector can therefore contain an anisotropic term, F(2)AM = αxηx2 + αyηy2, with αx ≠ αy in general. This anisotropy changes the mechanism of order-parameter selection. The resulting leading pairing structure is ηy↑ ≠ 0, ηx↓ ≠ 0, ηx↑ ≃ 0, ηy↓ ≃ 0.

S-2. COMPLETE SELF-CONSISTENT PAIRING STRUCTURE

A. Pairing amplitudes and C4zT symmetry

In the spin-decoupled limit, C4zT relates the dominant components ∆↑,py and ∆↓,px. With finite Rashba spin–orbit coupling, additional components appear, but the hierarchy of the triplet order parameters continues to reflect the underlying altermagnetic anisotropy.

B. Momentum-space phase structure of the triplet pairing

The resulting ϕσ(θ) is shown in Figs. S2(b) and S2(d). As θ evolves from 0 to 2π, the phase winds by approximately ±2π, giving a nonzero winding number.

C. Spin-resolved boundary spectral functions

The complete spin-resolved boundary spectra, together with the schematic in Fig. S4, provide a comprehensive confirmation of the spin–edge correspondence discussed in the main text. The boundaries normal to the y direction are dominated by spin-up spectral weight, whereas those normal to the x direction are dominated by spin-down spectral weight. This edge-dependent spin polarization follows from the C4zT symmetry of the altermagnetic state. The boundaries normal to the y direction are dominated by spin-up spectral weight, while those normal to the x direction are dominated by spin-down spectral weight. This edge-dependent spin polarization reflects the underlying C4zT symmetry of the altermagnetic state. The boundaries normal to the y direction are dominated by spin-up spectral weight, whereas those normal to the x direction are dominated by spin-down spectral weight. This edge-dependent spin polarization follows from the C4zT symmetry of the altermagnetic state. The boundaries normal to the y direction are dominated by spin-up spectral weight, whereas those normal to the x direction are dominated by spin-down spectral weight. This edge-dependent spin polarization follows from the C4zT symmetry of the altermagnetic state. The boundaries normal to the y direction are dominated by spin-up spectral weight, whereas those normal to the x direction are dominated by spin-down spectral weight. This edge-dependent spin polarization follows from the C4zT symmetry of the altermagnetic state.

1. Incorporation of Altermagnetic Symmetry into Neural Network Architectures The model's finding that altermagnetic spin splitting acts as a symmetry-selective mechanism suggests designing neural network layers whose connectivity or activation functions are explicitly constrained by crystalline symmetries like C4zT. This would allow the AI to inherently learn the directional selection of pairing, potentially leading to more physically motivated and robust representations of quantum states.

2. Development of Spin-Resolved Topological Feature Detectors The paper demonstrates that the boundary orientation determines the dominant spin character through spin-resolved spectral functions. An improved AI system could be trained to analyze spectroscopic data (like those from scanning tunneling microscopy) and predict the boundary orientation based on the observed spin polarization, effectively turning it into a symmetry-sensing diagnostic tool for emergent quantum phases.

3. Creation of Self-Consistent Pairing Prediction Modules Since the pairing amplitudes are determined dynamically, an AI system could be developed to solve the self-consistent BdG equations (Eq. 5) or the Ginzburg-Landau free energy minimization (S1) in real-time for novel material parameters, allowing it to predict the superconducting order parameters from a set of normal-state Hamiltonian inputs without needing pre-defined pairing symmetries.

4. Modeling Mixed-Parity State Evolution The paper shows that Rashba spin–orbit coupling relaxes this spin-selective structure by mixing spin sectors, thereby promoting a mixed-parity superconducting state. An AI system could be used to map the phase diagram (Figures 1(e) and S1(c)) to predict how adding external perturbations like RSOC drives the system from a direction-selective triplet state into a mixed-parity phase with coexisting pairing components.

5. Generating Momentum-Space Phase Winding Metrics The quantification of the phase winding number, Wσ = 1/2π ∫0 2π dθ ∂θϕσ(k), provides a metric for the internal structure of triplet pairing. An AI system could be trained to calculate this winding number from simulated or experimental momentum-space data to characterize the topological nature of the superconducting order parameter in complex materials.

We investigate self-consistent unconventional superconductivity in a two-dimensional d-wave altermagnetic metal. We find that momentum-dependent altermagnetic spin splitting suppresses opposite-spin singlet pairing and stabilizes highly anisotropic equal-spin triplet order. In the spin-conserving limit, this directional triplet pairing gives rise to nearly dispersionless Majorana boundary states associated with effective one-dimensional topological channels. Rashba spin-orbit coupling mixes spin sectors, activates additional pairing components, and drives the system into a mixed-parity superconducting state with dispersive Majorana boundary states. The spin-resolved boundary spectra further reveal a characteristic locking between boundary orientation and spin polarization, reflecting the underlying altermagnetic symmetry. These results identify altermagnetic spin splitting as an intrinsic mechanism for selecting unconventional pairing and generating spin-resolved Majorana boundary states without external magnetic fields.