四维中的四元数厄米带几何：在$S^4$上的实现与在$T^4$上的阻碍
Quaternionic Hermitian Band Geometry in Four Dimensions: Realization on $S^4$ and Obstruction on $T^4$
AI总结：
该研究针对四维参数空间的四元数厄米带几何建立实现-阻碍二分法，在$S^4$上实现了该几何，而在$T^4$上受拓扑阻碍无法全局实现，为非阿贝尔带几何提供对称感知框架。
AI中文摘要：
我们针对四维参数空间中的四元数厄米带几何建立了实现-阻碍二分法：$S^4$上的最小电荷最低朗道能级提供了一种全局实现，而由单个占据的四元数带诱导的处处非退化的饱和实现则在$T^4$上受到阻碍。满足$\boldsymbol{\bigl(\boldsymbol{\tmathcal{J}}^2=-1\bigr)}$的反幺正对称性$\boldsymbol{\tmathcal{J}}$使占据的二重态成为四元数带。在该框架内，我们将四元数Wirtinger不等式表述为涉及量子度量和第二陈密度的带几何界。在每个非退化饱和点，四元数射影空间的典范几何拉回至参数空间上的相容四元数结构；若这些条件处处成立，则该参数空间具有四元数厄米带几何。在$S^4$上，我们将最小电荷态表示为四元数Perelomov相干态并证明其处处非退化饱和，从而实现了四元数厄米带几何。相比之下，在$T^4$上，一个最小四带系统处处饱和该不等式，但拓扑结构迫使量子度量在某处退化，阻碍了全局诱导的四元数结构。一个显式晶格Dirac哈密顿量展示了该阻碍。当不等式处处饱和时，添加未占据的带无法消除该阻碍，因为任何非退化饱和投影子的像仍局限于固定的$\boldsymbol{\tmathbb{H}P^1}$中。这些结果为非阿贝尔带几何提供了一种对称感知框架，并表明参数空间拓扑如何约束四元数厄米带几何的全局实现。
英文摘要：
We establish a realization-obstruction dichotomy for quaternionic Hermitian band geometry in four-dimensional parameter spaces: the minimal-charge lowest Landau level on $S^{4}$ provides a global realization, whereas an everywhere nondegenerate saturated realization induced by a single occupied quaternionic band is obstructed on $T^{4}$. An antiunitary symmetry $\mathcal{J}$ satisfying $\mathcal{J}^{2}=-1$ 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 $S^{4}$, we express the minimal-charge states as quaternionic Perelomov coherent states and establish everywhere nondegenerate saturation, thereby realizing quaternionic Hermitian band geometry. On $T^{4}$, 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 $\mathbb{H}P^{1}$. 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.
