Nonadiabatic Origin of Quantum-Metric Effects via Momentum-Space Metric Tensor

TL;DR

This paper uncovers a fundamental geometric structure in momentum space caused by nonadiabatic effects in Bloch electrons, introducing a nonadiabatic metric tensor that influences electron dynamics and unifies various quantum metric effects.

Contribution

It introduces the nonadiabatic metric tensor in momentum space, extending semiclassical theory to include nonadiabatic effects and unifying quantum metric phenomena in electronic responses.

Findings

Defines the nonadiabatic metric tensor from nonadiabatic evolution.

Shows the nonadiabatic metric influences electron velocities and dynamics.

Reveals momentum space as a curved geometry affecting wave packet motion.

Abstract

We reveal a fundamental geometric structure of momentum space arising from the nonadiabatic evolution of Bloch electrons. By extending semiclassical wave packet theory to incorporate nonadiabatic effects, we introduce a momentum-space metric tensor -- the nonadiabatic metric. This metric gives rise to two velocity corrections, dubbed geometric and geodesic velocities, providing a unified and intuitive framework for understanding nonlinear and nonadiabatic transport phenomena beyond Berry phase effects. The geometric velocity is related to the nonadiabatic metric itself, whereas the geodesic velocity is a Christoffel symbol of the nonadiabatic metric. As the nonadiabatic metric is related to the energy-gap renormalized quantum metric, it unifies the broad quantum metric effects in electronic responses. When the nonadiabatic metric is constant, it reduces to an effective mass, modifying flat-band electron dynamics in confining potentials. In a flat Chern band with harmonic attractive interactions, the two-body wave functions mirror the Landau-level wave functions on a torus. Furthermore, we show that the nonadiabatic metric endows momentum space with a curved geometry, recasting wave packet dynamics as forced geodesic motion.