Berry’s geometric phase is a correction to the wave function arising in the study of adiabatic quantum systems; it has been discovered by M. V. Berry. There are analogous effects for other wave phenomena; there has been also much earlier work of Pancharatnam on the related phenomenon in optics. The origin of the Berry’s phase is in nonflatness of a parallel transport which appears in the corresponding phase factors.
M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. London Ser. A 392 (1984), no. 1802, 45–57, doi
M. V. Berry, The quantum phase, five years after, in: Geometric phases in physics, 7–28, Adv. Ser. Math. Phys., 5, World Sci. Publ., Teaneck, NJ, 1989.
Barry Simon, Holonomy, the quantum adiabatic theorem, and Berry’s phase, Phys. Rev. Lett. 51 (1983), no. 24, 2167–2170, MR85e:81024
Dariusz Chruściński, Andrzej Jamioƚkowski, Geometric phases in classical and quantum mechanics, Progress in Math. Physics 36, Birkhäuser 2004. xiv+333 pp. ISBN: 0-8176-4282-X
D. Rohrlich, Berry’s phase, entry in Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy, ed. F. Weinert, K. Hentschel, D. Greenberger and B. Falkenburg (Springer), to appear; arxiv/0708.3749
wikipedia: geometric phase
Frank Wilczek, Alfred Shapere, Geometric phases in physics, World Scientific, 1989
M. O. Katanaev, On geometric interpretation of the Berry phase, Rus. Phys. J. 54(2012)1082–1092, Izv. VUZov. Fizika 10(2011) 26–35 arxiv/1212.1782