# Help me check a fact about Berry phase

• andresB
In summary, the forum post correctly describes the phenomenon of Berry phase holonomy and its implications in quantum mechanics and condensed matter physics.
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?

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.

## 1. What is Berry phase?

Berry phase is a concept in quantum mechanics that describes the phase shift of a quantum state as it evolves along a closed path in a parameter space. It was first introduced by physicist Sir Michael Berry in 1984.

## 2. How is Berry phase calculated?

Berry phase can be calculated using the Berry connection, which is a mathematical expression that describes the relationship between the parameters and the quantum state. The Berry phase is then obtained by integrating the Berry connection along the closed path.

## 3. What is the significance of Berry phase?

Berry phase has important applications in many areas of physics, including condensed matter, quantum computation, and quantum information theory. It also plays a crucial role in understanding certain phenomena such as the Aharonov-Bohm effect and topological phases of matter.

## 4. Can Berry phase be observed experimentally?

Yes, Berry phase has been observed in various experiments, such as in the polarization of light passing through a birefringent material and in the spin dynamics of a quantum dot. However, it is often challenging to measure Berry phase directly, and indirect methods are often used.

## 5. Is Berry phase relevant in classical physics?

No, Berry phase is a purely quantum phenomenon and does not have a classical counterpart. It is a manifestation of the wave-like nature of quantum particles and is not observed in classical systems.

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