Engineering tunable p-wave magnetism in antiferromagnetic bilayers

The gist: An in-plane relative shift between two AB-stacked antiferromagnetic square lattices selectively breaks symmetries protecting spin degeneracy while preserving time-reversal-related constraints, thereby generating odd-parity spin splitting in a fully compensated magnetic state without spin-orbit coupling

Symmetry Analysis and Mechanism

The study proposes a symmetry-guided route to engineer tunable p-wave magnetism in a bilayer system composed of two AB-stacked antiferromagnetic square lattices with their collinear moments nonparallel to each other. The in-plane relative shift between the two layers is shown to selectively break the symmetries protecting spin degeneracy while preserving time-reversal-related constraints, which generates odd-parity spin splitting in a fully compensated magnetic state without spin-orbit coupling. This mechanism establishes geometric displacement as a generic symmetry-control parameter capable of selectively breaking spin-space symmetries and converting compensated AFM into p-wave magnetic states.

The analysis of the spin space group symmetry reveals that for spins along the z-axis, the even-parity relation E(k, sz) = E(-k, sz) is protected by spatial inversion P and the combined symmetry C2zP. Conversely, an arbitrary in-plane relative shift η illustrated in Fig. 1(b) breaks both the spatial inversion P and the combined symmetry C2zP, while preserving the timereversal T and the combined symmetry C2zT. This selective symmetry breaking naturally allows for p-wave spin splitting exclusively along the z-direction, whereas the spin degeneracy for all in-plane directions z⊥ remains intact.

Phenomenological Modeling and Tuning

The bilayer system is mapped onto an effective bond-modulated square lattice where the nearest-neighbor (NN) hoppings are anisotropically modulated while the next-NN (NNN) hoppings remain uniform. The corresponding phenomenological Hamiltonian is given by Hp = −X i j,σ ti jc†iσ cjσ − h X i ei · mˆ i.

The vector order parameter for the p-wave spin splitting is defined as Sp = X k s z(k)ϕk, s z(k) = X n s z nk f(Enk), where ϕk = (sin kx,sin ky) ensures that Eq. (2) projects out the odd-parity (l = 1) component of the momentum-resolved spin splitting. The direction of the p-wave order follows roughly θp = −θ, consistent with the Landau free energy analysis. Furthermore, the strength of p-wave spin splitting S p is nearly a constant of θ, with a weak π/2-periodic oscillation due to the C4v symmetry of the system.

Spin Response and Detection

The odd parity nature of the spin splitting in p-wave magnetism forbids any spin Hall response while allowing a finite Edelstein effect. An external electric field can induce a nonzero spin-polarization density, δS a = P b χ ab Eb, where the charge-to-spin conversion tensor is given by χ ab = eħ / 4NΓA X n,k ⟨unkτaunk⟩⟨unk∂H(k)/∂kbunkδ(Enk). For the p-wave magnetic state with spin splitting along the z-axis, the nonzero elements of the response tensor are χ zx and χ zy, from which we define a response vector χ = (χ zx, χzy) ≡ χ(cos θχ,sin θχ). The estimated magnitude of Edelstein response is estimated to be ≃ 1.58ħ/VÅ.

Microscopic Model Realization

The system is investigated using the half-filled Hubbard model on the bond-modulated square lattice, with the mean-field Hamiltonian reading HMF = −X i j,σ ti jc†iσ cjσ + U 4 X i h 2ninˆi − n 2i − 2mi · mˆ i + m 2i i.

Mean-field calculations indicate that p-wave magnetism can emerge spontaneously from sufficiently strong on-site Coulomb interactions at large t' /t ratio, with either coplanar or chiral spin configurations. The resulting mean-field ground state phase diagram shows that the AFM state in a narrow regime near t' ≃ 0.8t is a triple-Q chiral AFM, sandwiched by the conventional Neel AFM phase with collinear moments at small t' and the double-Q coplanar AFM at large t'. The finite spin chirality breaks both parity and time-reversal symmetry, leading to topologically nontrivial electronic bands.

Experimental Realization Potential

A possible experimental realization of the proposal may be pursued in oxide heterostructures composed of layered AFM such as Sr2IrO4 or La2CuO4. Nonparallel spin orientations between the two layers may be engineered through layer-selective magnetic anisotropy induced by strain or octahedral rotations. An in-plane relative shift between the two layers can be introduced through substrate-induced lattice mismatch, controlled shear strain, or deliberately designed stacking offsets during growth.

The shifted bilayer system considered here maps onto a bond-modulated square lattice. Mean-field calculations indicate that p-wave magnetism can emerge spontaneously from sufficiently strong on-site Coulomb interactions at large t' /t ratio, with either coplanar or chiral spin configurations. The in-plane shift provides a continuous control of the direction and the magnitude of the p-wave spin splitting. The results validate that the in-plane shift provides an efficient knob to realize and manipulate p-wave magnetism and the associated spin response in AB-stacked, AFM-ordered square lattices without relying on SOC.

Conclusion

In this work, we have theoretically proposed a plausible route to engineer unconventional p-wave magnetism by introducing an in-plane relative shift in a bilayer system composed of two AB-stacked, AFM-ordered square lattices with nonparallel collinear moments. Symmetry analysis shows the fully compensated zero net magnetization is protected by spin-space symmetries that map spins onto opposite partners. The p-wave magnetism emerges from selectively breaking the symmetries protecting spin degeneracy, while preserving the time-reversal-related constraints responsible for odd-parity spin splitting. Supported by phenomenological modeling, we demonstrate that the direction of the pwave order, the magnitude of the spin splitting, and the associated spin responses can all be continuously tuned through the in-plane shift, providing an efficient control parameter for spintronic applications. The work establishes a minimal equilibrium platform for realizing and manipulating p-wave magnetism in two-dimensional systems.

Acknowledgments

This work is supported by the National Key Research and Development Program of China (Grant No. 2022YFA1403800), the National Natural Science Foundation of China (Grant Nos. 12374153 and 12447101), and the Postdoctoral Fellowship Program of CPSF (Grant No. GZC20241749). Numerical calculations in this work were performed on the HPC Cluster of ITP-CAS. The corresponding author is wangzi@bc.edu and the corresponding author is zhousen@itp.ac.cn. This work was supported by the U.S. Department of Energy, Basic Energy Sciences (Grant No. DE-FG02-99ER45747). Numerical calculations in this work were performed on the HPC Cluster of ITP-CAS.

1. No-SOC p-wave magnetism realization: The system can generate odd-parity spin splitting in a fully compensated magnetic state without spin-orbit coupling, which allows for fundamental study of unconventional magnetism in materials where SOC is negligible, such as certain antiferromagnets.

2. Tunable control knob: The direction and the magnitude of the resulting p-wave spin-splitting, together with the associated spin responses, can be continuously tuned by the in-plane displacement, providing a precise knob for control over magnetic properties via simple geometric displacement rather than complex external driving.

3. Spin response prediction: The system allows for purely electrical detection of the Neel vector via nonlinear conductivities and enables measurement of the transverse spin-current generation at heterojunctions, which can be utilized by AI to design sensors capable of detecting magnetic order through these specific transport signatures.

4. Spontaneous phase mapping: The results establish a minimal equilibrium platform for realizing and manipulating p-wave magnetism in two-dimensional systems, enabling AI models to rapidly map the magnetic ground-state phase diagram over a broad parameter regime from microscopic Hubbard model calculations without needing complex, time-consuming experimental synthesis.

5. Spontaneous state prediction: Mean-field calculations demonstrate that p-wave magnetism with either coplanar or chiral spin configurations is stabilized in an extended region of the phase space, allowing AI to predict the emergence of specific coplanar or chiral AFM phases based on controllable parameters like interlayer shift and interaction strength.