Spin-Orbit Coupling Response Theory within the Oriented Spin Space Group Framework

Spin-orbit coupling (SOC) plays a pivotal role in a broad range of phenomena in condensed matter physics. The recently developed spin space group theory offers a new perspective for characterizing the fundamental symmetry-breaking features of SOC-related phenomena, yet its full potential remains to be explored. Here, we provide a general framework that relates the SOC-induced symmetry breaking to modifications of physical properties and emergence of new physical effects. This approach incorporates the recently developed oriented spin space group (OSSG) formalism, which can explicitly describe the symmetry reduction induced by SOC. By parameterizing the SOC Hamiltonian using a SOC tensor that connects real and spin spaces, an expansion in terms of the SOC components can be constructed for the response tensor describing any physical quantity. This framework determines not only whether a SOC-induced response is symmetry allowed, but also the lowest order in SOC at which it emerges and its dependence on magnetic anisotropy. As representative applications, we analyze the magnetoelectric coupling in Type-II multiferroics and show that magnetically induced electric polarization in different systems can be the result of SOC effects of different orders, and can exhibit varying characteristics of magnetic anisotropy. We further apply this theory to the anomalous Hall effect and spin magnetization in antiferromagnets, demonstrating that their distinct transformation properties under the OSSG allow a large anomalous Hall response to coexist with weak spin magnetization. A high-throughput analysis of materials in the MAGNDATA database identifies several symmetry-based material classes favorable for this coexistence. Overall, the proposed framework provides a unified symmetry description of SOC-induced phenomena and offers a systematic way for the identification of SOC-driven functional materials.