Quaternion-Kähler geometry of time reversal symmetric crystals

Abstract

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 Kähler. 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.

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BibTeX RIS

BibTeX

@misc{lim2026quaternionkähler,
  title = {{Quaternion-Kähler geometry of time reversal symmetric crystals}},
  author = {Hyeongmuk Lim and Junseo Jung and Yuting Qian and Bohm-Jung Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.03178},
  eprint = {2608.03178},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.03178},
}

RIS

TY  - UNPB
TI  - Quaternion-Kähler geometry of time reversal symmetric crystals
AU  - Hyeongmuk Lim
AU  - Junseo Jung
AU  - Yuting Qian
AU  - Bohm-Jung Yang
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2608.03178
UR  - https://arxiv.org/abs/2608.03178
AB  - 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 Kähler. 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.
ER  -

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