Quaternionic Hermitian Band Geometry in Four Dimensions: Realization on $S^4$ and Obstruction on $T^4$

Abstract: We establish a realization-obstruction dichotomy for quaternionic Hermitian band geometry in four-dimensional parameter spaces: the minimal-charge lowest Landau level on $S<sup>{4}$ provides a global realization, whereas an everywhere nondegenerate saturated realization induced by a single occupied quaternionic band is obstructed on $T<sup>{4}$. An antiunitary symmetry $\mathcal{J}$ satisfying $\mathcal{J}<sup>{2}=-1$ makes the occupied doublet a quaternionic band. Within this setting, we formulate the quaternionic Wirtinger inequality as a band-geometric bound involving the quantum metric and the second Chern density. At every nondegenerate saturation point, the canonical geometry of the quaternionic projective space pulls back to a compatible quaternionic structure on the parameter space; if these conditions hold everywhere, the parameter space acquires quaternionic Hermitian band geometry. On $S<sup>{4}$, we express the minimal-charge states as quaternionic Perelomov coherent states and establish everywhere nondegenerate saturation, thereby realizing quaternionic Hermitian band geometry. On $T<sup>{4}$, by contrast, a minimal four-band system saturates the inequality everywhere, but topology forces the quantum metric to become degenerate somewhere, obstructing a globally induced quaternionic structure. An explicit lattice Dirac Hamiltonian exhibits this obstruction. Adding unoccupied bands cannot remove this obstruction when the inequality is saturated everywhere, since the image of any nondegenerate saturated projector remains confined to a fixed $\mathbb{H}P<sup>{1}$. These results provide a symmetry-aware framework for non-Abelian band geometry and show how parameter-space topology constrains the global realization of quaternionic Hermitian band geometry.