Berry phaseeffect
In 1984, British physicist Michael Berry generalized an old idea about a phase accumulated during adiabatic (slow) changes of parameters. It turned out this phase is geometric: its magnitude is determined not by time or speed but by the curvature of parameter space. Earlier partial manifestations had been noticed before Berry, but he showed their universality and connection to gauge fields.
How it works
The Berry phase appears in the Aharonov-Bohm effect, molecular spectroscopy, and condensed matter physics. For example, it explains why some crystals conduct electricity only on the surface while remaining insulators inside.
💡 The famous Foucault pendulum, which rotates its swing plane due to Earth's rotation, is a classical analogue of the Berry phase: the pendulum 'senses' geometry, not just forces.
\gamma = i \oint_C \langle n(\mathbf{R}) | \nabla_{\mathbf{R}} n(\mathbf{R}) \rangle \cdot d\mathbf{R}
γ — geometric Berry phase; i — imaginary unit; ∮_C — integral over closed contour C in parameter space; ⟨n(R)| — bra-vector of an eigenstate of the Hamiltonian depending on parameters R; ∇_R — gradient with respect to parameters; |n(R)⟩ — ket-vector of the same state; dR — contour element. The quantity A = i⟨n|∇_R n⟩ is called the Berry connection.