Quaternion-Kahler geometry of time reversal symmetric crystals

Quantum geometry reveals how the shape of Bloch wave functions governs correlated quantum phenomena. Its standard formulation describes isolated complex bands, where Berry curvature is Abelian and ideal geometry is Kahler. However, time reversal symmetric crystals with spin require a different language since Kramers degeneracy pairs Bloch states and turns Berry curvature into a non-Abelian SU(2) field. Here we show that Kramers pair band geometry is quaternionic. A minimal Kramers pair defines a map into quaternion projective space, and its quaternionic quantum geometric tensor unifies the quantum metric with the three SU(2) Berry curvature components. The non-negativity of this tensor imposes local metric-curvature inequalities, whose saturation defines the non-Abelian counterpart of ideal Chern bands. In four dimensions, the ideal limit further yields an algebraic structure related to the four-dimensional quantum Hall effect. Our results promote ideal quantum geometry from the Abelian geometry of Chern bands to the quaternionic, non-Abelian geometry of time reversal symmetric quantum matter.