What is a proper definition of spin current? -- Lessons from the Kane-Mele Model

Spin current, a key concept in spintronics that carries spin angular momentum, has a non-unique definition due to the non-conservation of spins in solids. While two primary definitions exist -- conventional spin current and conserved spin current -- their validity has not been quantitatively examined. Here, we examine the validity of these definitions of spin current by comparing their spin Hall conductivities to the spin accumulation on edges of materials calculated in a real-time evolution simulation. Employing the Kane-Mele model with the Rashba term, which explicitly violates spin conservation, we reveal that the spin Hall conductivities calculated under both definitions fail to reproduce the simulated results of spin accumulation when the Rashba term is large. Our results suggest that the standard definitions of spin current and the associated spin Hall conductivity do not give an accurate quantitative estimate of spin accumulation. This conclusion indicates that real-time simulations are necessary to accurately estimate spin accumulation on edges/surfaces of materials.