The Bott Metric: A Real-Space Bridge Between Topology and Quantum Metric

Abstract

The Bott index has become an indispensable tool to probe the topology of quantum matter, particularly in systems lacking translational symmetry. Constructed from a plaquette operator, it retains the phase information while discarding the amplitude. Here we introduce and develop the Bott metric, which captures this complementary amplitude information and provides a measure of the underlying quantum metric of the system. We show that, in the thermodynamic limit, the Bott metric converges to the trace of the integrated quantum metric. Our framework provides a new route to reveal the quantum metric structure in non-periodic systems, which we illustrate using representative examples ranging from disordered to amorphous models. More broadly, our definition of the Bott metric unifies the notion of topological invariants and quantum metric under the same overarching plaquette operator construction.