Interplay between relativistic spin-momentum locking and breaking of inversion symmetry: conditions for effective p-wave magnetism

Abstract

We investigate the interplay between relativistic spin-momentum locking arising from altermagnetism and various forms of inversion symmetry breaking. Depending on the specific symmetry breaking, this can give rise to Rashba-type spin–orbit coupling (SOC), Weyl-type SOC, or the coexistence of two distinct spin-momentum lockings. We focus on the altermagnetic Ca₂RuO₄ as a testbed material. Our results reproduce the experimentally observed ground state, which is an A-centered magnetic order with the Néel vector aligned along the b-axis, hosting spin cantings along the a- and c-axes but without weak ferromagnetism. Ca₂RuO₄ exhibits relativistic spin-momentum locking, with different even-parity wave orders for each of the three spin components. We interpret the experimental results on doped samples as evidence for a transition from a pure altermagnetic phase to a weak ferromagnetic phase. We investigate which nodal planes persist when inversion symmetry is broken in the relativistic case, and we find that, in general, the spin-momentum locking of the spin component parallel to the electric field survives in the presence of Rashba coupling. The spin-momentum lockings of the other components adopt effective p-wave character in the case of Rashba; in contrast, Weyl-type SOC disrupts all nodal planes, leaving only nodal lines. The presence of the p-wave depends on the interplay between relativistic spin-momentum locking and Rashba–Weyl SOC, which should have a common nodal plane to generate an effective p-wave. Finally, to simulate a stripe phase with structural distortions along the z-axis, we investigated a modulated electric field that induces atomic displacements within a single layer of the Ca₂RuO₄ unit cell. This produces a magnetic phase transition to an exotic altermagnetic state with two non-relativistic spin-momentum lockings with a common nodal plane, thereby producing an effective p-wave magnet. Our research presents a comprehensive analysis of various possible scenarios in altermagnets with breaking of inversion symmetries under relativistic effects.