Berry phase and adiabatic evolution
A slowly transported eigenstate acquires a geometric phase in addition to its dynamical phase.
$$\gamma_n(C)=i\oint_C\langle n(R)|\nabla_R n(R)\rangle\cdot dR$$
What this means
A slowly transported eigenstate acquires a geometric phase in addition to its dynamical phase.
Deep dive: mathematical and physical mechanism
Write $|\psi(t)\rangle=e^{-i\int E_n dt/\hbar}e^{i\gamma_n(t)}|n(R(t))\rangle$. Inserting into the Schrödinger equation in the adiabatic limit gives $\dot\gamma_n=i\langle n|\partial_t n\rangle$. A closed cycle yields holonomy.
How it appears in quantum phenomena
Adiabatically steering a spin in a magnetic field produces a Berry phase.
Assumptions, caveats and limitations
Requires sufficiently slow evolution and suitable spectral separation; phase is defined modulo $2\pi$.
Connect the ideas
For context, see the Geometric Quantum Mechanics topic guide and the reading references.