Pancharatnam Phase and Berry Phase

The Indian physicist S. Pancharatnam, workingPhasePancharatnam on quantum Optics in 1956, introduced the novel concept of a geometrical phase. Let \(H(\xi )\) denote a Hamiltonian that depends on some parameters \(\xi \) , with ground state \(|\psi (\xi )\rangle \) . One can define a phase difference \(\Delta \varphi _{12}\) between two ground states \(|\psi (\xi _{1})\rangle \)  and \(|\psi (\xi _{2})\rangle \) : \( \langle \psi (\xi _{1})|\psi (\xi _{2})\rangle =|\langle \psi (\xi _{1})|\psi (\xi _{2})\rangle | e^{-i \Delta \varphi _{12} }. \) .