Pancharatnam Phase and Berry Phase

The Indian physicist S. Pancharatnam, working in 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 \(\varDelta \varphi _{12}\) between two ground states \(|\psi (\xi _{1})\rangle \) and \(|\psi (\xi _{2})\rangle \): $$ \langle \psi (\xi _{1})|\psi (\xi _{2})\rangle =|\langle \psi (\xi _{1})|\psi (\xi _{2})\rangle | e^{-i \varDelta \varphi _{12} }. $$