This formula can be regarded as the quantum adiabatic limit of the Kubo formula. The eigenstates of the system are labeled by the index n . These are in general many body states, and the energies are in general many body energies. At finite temperatures a thermal average over n 0 is implicit. The field B can be regarded as the rotor of "vector potential" A (mathematically known as the "one-form"). Namely, B → → = ∇ ∇ ∧ ∧ A → → . The `` Berry phase " which is acquired by a wavefunction at the end of a closed cycle is