Berry's phase in nLab

Contents

Contents

Idea

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.

• Berry connection

• quantum adiabatic theorem

• adiabatic quantum computation

References

• Michael Berry, Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. London Ser. A 392 (1984), no. 1802, 45–57, doi

• Michael 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

• J. M. Robbins, Michael Berry, The geometric phase for chaotic systems, Proc. Roy. Soc. London Ser. A 436 (1992), no. 1898, 631–661, doi, 94a:81036

• V. I. Arnold, Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect,

  Selecta Mathematica 1:1, 1–19 (1995) doi

• 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

• Mikio Nakahara, Chapter 10.6 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)

A relation to Chern-Bott connection is explained in 4.1 of lecture notes

• Mauro Spera, Geometric methods of quantum mechanics, J. Geometry and Symmetry in Physics 24 1-44 (2011) euclid

• 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

• <http://www.mi.infm.it/manini/berryphase.html>

• 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

• Maxim Braverman, The Berry phase and the phase of the determinant, arxiv/1310.6332

Review in a context of (adiabatic) quantum computation:

• Jiang Zhang, Thi Ha Kyaw, Stefan Filipp, Leong-Chuan Kwek, Erik Sjöqvist, Dianmin Tong, Geometric and holonomic quantum computation [arXiv:2110.03602]