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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]

Michael Berry (physicist) showed that slowly carrying a nondegenerate energy eigenstate around a closed loop of external parameters can produce a geometric phase in addition to its dynamical phase. The geometric phase depends on the path through parameter space and survives after the ordinary energy-dependent contribution is removed. For a spin following a slowly varying magnetic field, it is proportional to the solid angle enclosed by the field direction, making the geometry measurable through interference.^[ref]

A theoretical physicist whose work includes geometric phases in quantum mechanics.

Take $\omega >0$ and $T=2\pi /\omega$. A full spin one-half rotation gives $R(T)=-I$, an essential sign. The cyclic phase of a spin in a rotating field has factors and total quantum phasesTo separate the adiabatic dynamical phase, set $B=\sqrt{{\Omega }_0^2+{\Omega }_R^2}>0$ and $\cos \theta ={\Omega }_0/B$. Then $\Omega =B-\omega \cos \theta +O({\omega }^2/B)$. The instantaneous laboratory energies are $\pm \hslash B/2$, so the adiabatic dynamical phases are $\mp BT/2$. Subtracting them from ${\Phi }_{\pm }$ givesThe oriented solid angle $\mathcal{A}$ is swept out by the field direction. These are the Berry phases for the eigenstates parallel and antiparallel to that direction. Retaining the spinor sign makes the quantum phase vanish modulo $2\pi$ when the field does not trace a nontrivial loop. The assumption $\omega \ll B$ suffices for adiabatic following; near degeneracy it must not be silently dropped.^[ref]