标题： 量子几何的普适威尔逊环边界：$Z_2$边界和物理后果 显示英文标题

标题： Universal Wilson loop Bound of Quantum Geometry: $Z_2$ Bound and Physical Consequences

摘要： 我们定义了绝对威尔逊环绕数，并证明它从下方界定了量子度量。 这个威尔逊环绕数下界自然地再现了已知的量子度量的陈和欧拉下界，并由于时间反演保护的$Z_2$指数提供了量子度量的显式下界，回答了一个至今未解的问题。 一般来说，威尔逊环绕数下界可以应用于任何由威尔逊环绕数表征的其他拓扑不变量，例如粒子-空穴$Z_2$指数。 作为$Z_2$界的物理后果，我们表明时间反演$Z_2$指数从下方界定了超流体重量和光学电导率，并从上方界定了能带绝缘体的直接能隙。 显示英文摘要

摘要： We define the absolute Wilson loop winding and prove that it bounds the quantum metric from below. This Wilson loop lower bound naturally reproduces the known Chern and Euler bounds of the quantum metric, and provides an explicit lower bound of the quantum metric due to the time-reversal protected $Z_2$ index, answering a hitherto open question. In general, the Wilson loop lower bound can be applied to any other topological invariants characterized by Wilson loop winding, such as the particle-hole $Z_2$ index. As physical consequences of the $Z_2$ bound, we show that the time-reversal $Z_2$ index bounds superfluid weight and optical conductivity from below, and bounds the direct gap of a band insulator from above.