How to Measure the Quantum Geometry of Bloch Bands

Understand

Single-particle states in electronic Bloch bands form a Riemannian manifold whose geometric properties are described by two gauge invariant tensors, one being symmetric the other being antisymmetric, that can be combined into the so-called Fubini-Study metric tensor of the projective Hilbert space.

• The latter directly controls the Hall conductivity.
• Here we show that the symmetric part of the Fubini-Study metric tensor also has measurable consequences by demonstrating that it enters the current noise spectrum.
• In particular, we show that a non-vanishing equilibrium current noise spectrum at zero temperature is unavoidable whenever Wannier states have non-zero minimum spread, the latter being quantifiable by the symmetric part of the Fubini-Study metric tensor.