Defines the absolute Wilson loop winding of a band system and proves it lower-bounds the integrated quantum (Fubini-Study) metric. This single bound reproduces the known Chern and Euler lower bounds and, crucially, supplies an explicit lower bound tied to the time-reversal-protected Z2 index — resolving a previously open question — and extends to any topological invariant expressible through Wilson loop winding (e.g. the particle-hole Z2 index). Physical consequences: the time-reversal Z2 index bounds superfluid weight and optical conductivity from below and the direct band gap from above.
We define the absolute Wilson loop winding and prove that it bounds the (integrated) quantum metric from below. This Wilson loop lower bound naturally reproduces the known Chern and Euler bounds of t…