Beyond Local Berry Geometry: A First-Principles Finite-Momentum Theory of Electronic Position
Electronic position controls how a crystal polarizes and responds to an external field. In crystals, it is usually described through local changes of electronic states in momentum space. This Berry framework has reshaped modern solid-state physics, but strong fields drive electrons across a finite momentum range, where coherence between different momenta becomes part of the response. Here we establish a first-principles theory of electronic position at finite momentum that retains this missing information. We show that unequal-momentum coherence can cancel under spatial averaging and still produce polarization, forming a coherence dipole. We obtain the matrix directly from material wave functions, without model bands or fitted transition elements. In Si, the finite-momentum geometry sets a material momentum scale. Comparing this scale with the momentum change driven by the field predicts when finite-momentum physics becomes active. Crossing the scale strongly reorganizes the fifth and higher harmonics, showing that momentum-space geometry, rather than emitted photon energy alone, controls the nonlinear response. HHG is the first demonstration, but the theory applies whenever driven electrons explore a finite momentum range. It therefore extends quantum geometry beyond the local Berry limit and provides a general basis for predicting field-driven phenomena in real materials.