Berry phase

The Berry phase (actually first discovered by Pancharatnam) is the equivalent in Quantum mechanics of Hannay angles, so it is natural to introduce the concept of adaiabaticity. Consider a time-dependent Hamiltonian such that,

\[ H[\mathbf{x}(t)]\ket{n,\mathbf{x}(t)} = E_n[\mathbf{x}(t)]\ket{n,\mathbf{x}(t)} \]

Gauge invariance

The BP is gauge-invariant modulo \(2\pi\), \[ \ket{n,x}\rightarrow e^{i\beta(x)}\ket{n,x} \]