Universal Wilson-Loop Bound of Quantum Geometry: Z₂ Bound and Physcial Consequences

Topological bounds of quantum geometry have various physical consequences, such as bounding the superfluid weight. In general, nontrivial topology must provide a lower bound of the quantum geometric tensor, but the key question is finding the expression of the bound. In this work, we will provide a universal Wilson-loop bound of the quantum geometry. The Wilson-loop bound naturally reproduces the known Chern-number and Euler-number bounds of the quantum geometric tensor, and remarkably provides an explicit expression for the time-reversal-invariant Z₂ bound, which has been an open question. Physical consequences of the new Z₂ bound will also be discussed.