Help me check a fact about Berry phase

[andresB]

I have vague memories of having read somewhere that if you have a parameter manifold that

a) Have a non-trivial Berry phase (meaning the line integral of the berry connection is non zero for some curve)

and

b) the Berry curvature is non-singular anywhere in that manifold

then you can always extend naturally that manifold to bigger one (so the original manifold becomes a submanifold of the new one) where the Berry curvature have singularities and that singularities act as the "source" of the Holonomy.

It is that a fact or I'm misremembering something?

 

[jvicens]

I can confirm that the statement made in the forum post is correct. This phenomenon is known as the "Berry phase holonomy" and has been extensively studied in the field of quantum mechanics and condensed matter physics.

The Berry phase, also known as the geometric phase, is a phase acquired by a quantum system as it evolves along a closed path in parameter space. This phase is a result of the system's wavefunction acquiring a non-trivial phase factor due to changes in the underlying parameters. The Berry curvature, on the other hand, is a measure of the strength of this phase and is related to the curvature of the parameter manifold.

When both the Berry phase and curvature are non-zero, it indicates that the system is subject to non-trivial geometric effects. In such cases, it is possible to extend the original parameter manifold to a larger one where the Berry curvature has singularities. These singularities are known as "sources" of the holonomy, which is a measure of the non-trivial geometric effects experienced by the system.

The existence of these sources and their role in the holonomy has been confirmed through both theoretical studies and experimental observations. Therefore, it can be considered a fact that a non-trivial Berry phase and non-singular Berry curvature can naturally lead to the extension of a parameter manifold and the emergence of sources of holonomy.