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

TL;DR

This paper introduces the Bott metric, a new measure capturing quantum metric information in non-periodic systems, unifying topological invariants and quantum metric through a plaquette operator framework.

Contribution

It develops the Bott metric, which complements the Bott index by capturing amplitude information, and demonstrates its convergence to the quantum metric in the thermodynamic limit.

Findings

Bott metric converges to the trace of the integrated quantum metric in large systems.

The framework applies to disordered and amorphous models, revealing quantum metric structures.

Unifies topological invariants and quantum metric via a plaquette operator approach.

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.