phase picked up by an eigenstate of a parameter-dependent Hamiltonian when transported adiabatically around a closed trajectory in parameter space. It turns out that in addition to the well-known dynamical phase there is also a geometric component to the phase, which is observable, at least in principle. While the dynamical phase provides a measure of the duration of the Hamiltonian’s evolution and is independent of the geometry of the trajectory followed, conversely, the geometric phase is independent of the rate at which the state is transported around the loop (as long as this is slow enough for the adiabatic theorem to rule out transitions to neighbouring, orthogonal states) and depends solely upon the geometry of the trajectory. The geometric phase had hitherto been widely overlooked as just another unphysical phase factor, and certainly had not been granted the level of recognition it enjoys nowadays.

Before going on to explain how Berry’s phase may be used to probe for QPTs, it is useful here to first restate a simple example given in Berry’s original paper, which serves to illustrate the concept of geometric phase. To that end, consider a single spin-$1/2$ particle which is coupled to an external magnetic field, and 