平凡系统中的隐藏拓扑与强量子度量界
Hidden topology and strong quantum metric bounds in trivial systems
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中文总结 AI 辅助
研究在二维系统中，传统陈数对量子度量积分（QMI）的界定在某些平凡系统中失效的问题。通过降维框架将二维QMI分解为低维分量，为其建立非零下界，并在多个模型中验证，还研究了高阶拓扑相，推广了量子几何与拓扑的关系。
中文摘要 AI 辅助
二维系统中的量子度量积分（QMI）传统上由陈数从下方界定。然而，对于陈数为零或贝里曲率恒为零的系统，此界变得平凡，无法提供有用的几何约束。本文开发了一种降维框架，以嵌套循环方式将二维QMI分解为低维分量。通过该方法，即使传统二维拓扑平凡，我们也能基于一维拓扑障碍为QMI建立非零下界。我们在倾斜的二维Su-Schrieffer-Heeger模型和具有手征对称性的各向异性Wilson-Dirac模型中明确展示了这一机制。QMI的所得下界由沿两个不同方向的量子化万尼尔能带决定。我们还采用相同策略研究了高阶拓扑相中的量子几何。通过引入从嵌套威尔逊回路获得的万尼尔能带基，我们证明万尼尔能带QMI由高阶拓扑不变量从下方界定，如在Benalcazar-Bernevig-Hughes模型中的四极矩。我们的结果从降维框架为QMI建立了非零下界，从而推广了量子几何与拓扑之间的基本关系。
英文摘要
The quantum metric integral (QMI) in two-dimensional (2D) systems is conventionally bounded from below by the Chern number. For systems with zero Chern number or identically vanishing Berry curvature, however, this bound becomes trivial and provides no useful geometric constraints. Here, we develop a dimension-reduction framework that decomposes the 2D QMI into lower-dimensional components in a nested-loop way. With this method, we establish a nonzero lower bound on the QMI arising from one-dimensional topological obstructions even when the conventional 2D topology is trivial. We explicitly demonstrate this mechanism in a tilted 2D Su-Schrieffer-Heeger model and an anisotropic Wilson-Dirac model with chiral symmetry. The resulting lower bounds of QMI are determined by the quantized Wannier bands along two different directions. We further investigate the quantum geometry in higher-order topological phases following the same strategy. By introducing Wannier-band basis obtained from the nested Wilson loop, we demonstrate that the Wannier-band QMI is bounded from below by the higher-order topological invariant, e.g. the quadrupole moment in Benalcazar-Bernevig-Hughes model. Our results establish nonzero lower bounds on QMI from a dimension-reduction framework, thereby generalizing the fundamental relation between quantum geometry and topology.
