Quasisymmetry protected Ising spin-orbit coupling

Quasisymmetry is an approximated symmetry in the sense that it is not the crystalline symmetry but commutes with the Hamiltonian at a lower order. Unlike exact symmetry, quasisymmetry emerges in the context of low-energy effective theory. One of the most desirable properties added by quasisymmetry is the large Berry curvatures due to the perturbatively small band gaps, which are robust against chemical and physical perturbations. Ising spin-orbit coupling (SOC) is a special type of SOC with electron spins polarizing to out-of-plane direction and can protect the spin coherence against the relaxation. For Ising-type superconductors, Ising SOC significantly affects the superconducting properties such as Pauli limit. It is commonly believed that the out-of-plane mirror symmetry Mz is necessary for Ising SOC, which puts a constraint on the allowed materials. Here we demonstrate that the strong Ising SOC exists beyond the system with Mz symmetry and is protected by the quasisymmetry. Based on the group-theoretical analysis and density-functional theory calculations, we show that the lower-order SOC Hamiltonian for certain representation is Ising type, which is protected by the quasisymmetry. Several promising systems with strong Ising SOC of several hundreds of meV near the Fermi energy are identified. In addition, we propose the general design principles to search for the realistic materials. Our results advance the fundamental understanding of Ising spin-orbit physics and greatly broaden the range of the materials hosting the Ising SOC.