basics
Examples
Generally, a Berry phase is a non-trivial phase picked up by a quantum state of definite energy under adiabatic changes of the quantum system‘s Hamiltonian around a loop in its parameter space.
Specifically, in condensed matter theory and for a crystalline material with a set of isolated (gapped) electronic bands, the partial derivatives along momentum-vectors of the corresponding Bloch states, projected back onto these states, turn out to canonically define a connection 1-form on the vector bundles (over the Brillouin torus) which is spanned by these states. This is called a Berry connection, due to (Berry 84, Simon 83). The parallel transport along this connection describes the change of Bloch states under the adiabatic change of their momentum/wave vector. The curvature 2-form of a Berry connection is accordingly called the Berry curvature.
The holonomy of such a Berry connection is called a Berry phase, in general, and a Zak phase (Zak 89) when evaluated along one of the non-trivial 1-cycles of the Brillouin torus.
By default, this is understood to apply to the valence bundle of a crystalline material, but the construction works more generally.
If in the case of a gapped valence bundle, hence a topological insulator-phase, the holonomy of the Berry connection is non-abelian (which may happen, as originally highlighted in Wilczek & Zee 84) then one also says that the topological phase exhibits topological order.
For semimetals the Berry phases of the valence bundle around their nodal loci of codimension 2 are a measure for the obstruction to adiabatically deforming the semimetal such as to open its gap closures, hence to become a (topological) insulator (eg. Vanderbilt 18, 5.5.2).
Berry's phase (needs to be merged)
The original article:
The formulation in terms of connections on fiber bundles and their holonomy:
Generalization to (connections with) non-abelian holonomies:
The special case of Berry phases around the 1-cycles of a Brillouin torus – Zak phases:
Review and further discussion:
Michael Berry, The quantum phase, five years after, in: Geometric phases in physics, Adv. Ser. Math. Phys. 5, World Scientific (1989) 7–28 (pdf, doi:10.1142/0613)
Ming-Che Chang, Qian Niu, Berry curvature, orbital moment, and effective quantum theory of electrons in electromagnetic fields, J. Phys.: Condens. Matter 20 (2008) 193202 (doi:10.1088/0953-8984/20/19/193202)
Di Xiao, Ming-Che Chang, Qian Niu, Berry Phase Effects on Electronic Properties, Rev. Mod. Phys. 82 (2010) 1959-2007 (arXiv:0907.2021, doi:10.1103/RevModPhys.82.1959)
With focus on topological phases of matter (topological insulators, semimetals, etc.):
David Vanderbilt, Berry Phases in Electronic Structure Theory – Electric Polarization, Orbital Magnetization and Topological Insulators, Cambridge University Press (2018) (doi:10.1017/9781316662205)
Jérôme Cayssol, Jean-Noël Fuchs, Section IV.C of: Topological and geometrical aspects of band theory, J. Phys. Mater. 4 (2021) 034007 (arXiv:2012.11941, doi:10.1088/2515-7639/abf0b5)
Tudor D. Stanescu, Chapter 2 of: Introduction to Topological Quantum Matter & Quantum Computation, CRC Press 2020 (ISBN:9780367574116)
See also:
On anyon phases (specifically in the quantum Hall effect) as Berry phases of a adiabatic transport of anyon positions:
Experimental observation of Zak phases:
Proposal for experimental realization of Berry phases around codimension=2 nodal loci of (quantum simulations of time+space inversion symmetric) semi-metals:
Dan-Wei Zhang, Y. X. Zhao, Rui-Bin Liu, Zheng-Yuan Xue, Shi-Liang Zhu, Z. D. Wang, Quantum simulation of exotic PT-invariant topological nodal loop bands with ultracold atoms in an optical lattice, Phys. Rev. A 93 (2016) 043617 (arXiv:1601.00371, doi:10.1103/PhysRevA.93.043617)
(see sec II.A, these authors stand out as mentioning the relevant KO-theory)
Last revised on June 13, 2022 at 17:50:05. See the history of this page for a list of all contributions to it.