Contents

# Contents

## Idea

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).

## References

### General

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 torusZak phases:

Review and further discussion:

With focus on topological phases of matter (topological insulators, semimetals, etc.):

### For anyon statistics

On anyon phases (specifically in the quantum Hall effect) as Berry phases of a adiabatic transport of anyon positions:

### Experimental observation

Experimental observation of Zak phases:

• Marcos Atala, Monika Aidelsburger, Julio T. Barreiro, Dmitry Abanin, Takuya Kitagawa, Eugene Demler, Immanuel Bloch: Direct measurement of the Zak phase in topological Bloch bands, Nature Physics 9 (2013) 795–800 (doi:10.1038/nphys2790)

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 21:50:05. See the history of this page for a list of all contributions to it.