= Berry curvature
{c}
{title2=$\mathcal F=d\mathcal A$}
The Berry curvature measures the local twisting of the <quantum phase> convention of a smoothly varying, nondegenerate <energy eigenstate>. For the <Berry connection> $\mathcal A=\sum_j A_j\,dR_j$, it is the <differential form>
$$
\mathcal F=d\mathcal A,\qquad F_{ij}=\partial_iA_j-\partial_jA_i.
$$
Changing 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.<Quantal phase factors accompanying adiabatic changes by Michael Victor Berry>[§2, surface-integral formula; §4, spin in a magnetic field]{ref}
Back to article page
