 Berry curvature

The Berry curvature measures the local twisting of the quantum phase convention of a smoothly varying, nondegenerate energy eigenstate. For the Berry connection $\mathcal{A}={\sum }_j{A}_{j}d{R}_j$, it is the differential formChanging the state representative by $e^{i\chi (\mathbf{R})}$ changes $\mathcal{A}$ by $-d\chi$ but leaves $\mathcal{F}$ invariant. Thus the Berry connection depends on a phase convention, while the Berry curvature has convention-independent local content.

For a surface whose boundary is a closed parameter path, Stokes theorem expresses the Berry phase as the integral of the Berry curvature, provided the chosen patch is smooth on the surface. For a spin one-half aligned with a slowly turning magnetic field, the curvature yields a Berry phase equal to minus half the oriented solid angle for the positive-spin branch. The spectral gap must remain open; a quantum degeneracy can obstruct a single smooth choice over the whole surface.^[ref]