The generation of pure spin current, that is, spin angular momentum transport without charge flow, is crucial for developing energy-efficient spintronic devices with minimal Joule heating. Here, we introduce the intrinsic nonlinear pure spin Hall effect (NPSHE), where both linear and second-order charge Hall currents vanish. We show intrinsic second-order spin angular momentum transport in metals and insulators through a detailed analysis of the quantum geometric origin of different spin current contributions. Our comprehensive symmetry analysis identifies 39 magnetic point groups that support NPSHE, providing a foundation for material design and experimental realization. Our first principle's calculations demonstrate a significant NPSHE in a prototypical Kramers-Weyl metal RhGe, even at room temperature. This positions Kramers-Weyl metals as potential candidates for NPSHE-based spin-torque devices. Our work lays a practical pathway for realizing charge-free angular momentum transport for the development of next-generation, energy-efficient spintronic devices.

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• Spin Hall effect
• Spin current
• Topological materials
• First-principles calculations
• Methods in transport