Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond

Whenever a quantum system undergoes a cyclic evolution governed by a slow change of parameters, it acquires a phase factor: the geometric phase. Its most common formulations are known as the Aharonov–Bohm phase and the Pancharatnam and Berry phase, but both earlier and later manifestations exist. Although traditionally attributed to the foundations of quantum mechanics, the geometric phase has been generalized and become increasingly influential in many areas from condensed-matter physics and optics to high-energy and particle physics and from fluid mechanics to gravity and cosmology. Interestingly, the geometric phase also offers unique opportunities for quantum information and computation. In this Review, we first introduce the Aharonov–Bohm effect as an important realization of the geometric phase. Then, we discuss in detail the broader meaning, consequences and realizations of the geometric phase, emphasizing the most important mathematical methods and experimental techniques used in the study of the geometric phase, in particular those related to recent works in optics and condensed-matter physics. The geometric phase is a deep and influential concept in modern physics and related sciences. This Review briefly discusses its origin, mathematical formulation and various forms, some of which are topological; then elaborates on contemporary optical and condensed-matter applications. The Aharonov–Bohm phase, acquired by charged particles encircling a confined magnetic flux, is topological, gauge invariant and realistic, highlighting the unique role of electromagnetic potentials in quantum mechanics. The Aharonov–Bohm phase can be seen as a manifestation of Berry’s geometric phase accumulated whenever a quantum system is adiabatically transported around a cyclic circuit on an abstract surface in the parameter space (with additional generalizations to degenerate and open systems, and to non-adiabatic, non-cyclic, non-unitary evolutions). The geometric phase is an example of a holonomy (failure of parallel transport around closed cycles to preserve the geometrical information being transported) and its profound role in physics. The two main types of geometric phase in optics originate from ‘spin redirection’ (when light with a fixed state of polarization is changing direction) and from a slow change in polarization (of light propagating through an anisotropic medium in a fixed direction), giving rise to the Pancharatnam–Berry phase. In condensed-matter physics, the geometric phase manifests itself in the electronic Bloch states, quantum Hall effect, electric polarization, exchange statistics and many other phenomena. The Aharonov–Bohm phase, acquired by charged particles encircling a confined magnetic flux, is topological, gauge invariant and realistic, highlighting the unique role of electromagnetic potentials in quantum mechanics. The Aharonov–Bohm phase can be seen as a manifestation of Berry’s geometric phase accumulated whenever a quantum system is adiabatically transported around a cyclic circuit on an abstract surface in the parameter space (with additional generalizations to degenerate and open systems, and to non-adiabatic, non-cyclic, non-unitary evolutions). The geometric phase is an example of a holonomy (failure of parallel transport around closed cycles to preserve the geometrical information being transported) and its profound role in physics. The two main types of geometric phase in optics originate from ‘spin redirection’ (when light with a fixed state of polarization is changing direction) and from a slow change in polarization (of light propagating through an anisotropic medium in a fixed direction), giving rise to the Pancharatnam–Berry phase. In condensed-matter physics, the geometric phase manifests itself in the electronic Bloch states, quantum Hall effect, electric polarization, exchange statistics and many other phenomena.