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

TL;DR

This work identifies a momentum-space metric, the nonadiabatic metric $G_{ij}$, as the leading correction to Bloch-electron dynamics beyond adiabatic Berry-phase physics. By extending semiclassical wave-packet theory and integrating out interband amplitudes, the authors derive an effective Lagrangian $L_{ m eff} = - rac{dm q}{dt}oldsymbol{ ightharpoonup}m x - E_c + rac{dq^i}{dt} A_i + rac{1}{2} G_{ij} rac{dq^i}{dt}rac{dq^j}{dt}$, with $G_{ij}$ defined as $G_{ij} = 2 ext{Re} igg[rac{A_{i,[0m]} A_{j,[m0]}}{oldsymbol{ riangle}_{m0}}igg]$ and related to the quantum metric in two-band models. This metric yields two velocity corrections—geodesic and geometric—leading to a forced geodesic equation in momentum space and a curved momentum-space geometry where the Berry connection acts as a gauge potential and the band dispersion as a scalar potential. In flat bands, constant $G_{ij}$ reduces dynamics to an effective mass, while two-body bound states in harmonic confinement map to Landau-level-like spectra on a torus, connecting nonadiabatic corrections to exciton and Cooper-pair physics. Overall, the framework unifies nonlinear and nonadiabatic transport, momentum-space gravity, and bound-state phenomena under a single geometric perspective with potential extensions to lattice and spin dynamics.

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. Furthermore, we show that the nonadiabatic metric endows momentum space with a curved geometry, recasting wave packet dynamics as forced geodesic motion. In this picture, the metric defines distances, the Berry connection acts as a gauge potential, band dispersion serves as a scalar potential, and the toroidal topology of the Brillouin zone imposes periodic boundary conditions. When the nonadiabatic metric is constant, it reduces to an effective mass, allowing electrons to behave as massive particles in flat bands. In a flat Chern band with harmonic attractive interactions, the two-body wave functions mirror the Landau-level wave functions on a torus.

Figures (3)

• Figure 1: Illustration of nonadiabatic time evolution of a wave packet in momentum space, where a gradual momentum shift induces interband mixings and nonadiabatic corrections.
• Figure 2: Bound states of flat-band particles in real space mirror the Landau-level wave functions on a torus in $\bm{q}$-space.
• Figure 3: Wave packet dynamics in $\bm{q}$-space, a torus with curved geometry. The distance $ds^2$ is defined by the nonadiabatic metric $G_{ij}$. The topological charge stands for the Dirac monopole that generates nonzero gauge fields corresponding to the Berry curvature. $C$ indicates the Chern number. Color encodes the scalar potential from the band dispersion.