Question

How is the Geometric Phase measured in the experiment?

Answer 1

The geometric phase is a phase acquired over the course of a cycle. If you take the Aharonov Bohm effect, you need to make an interferometer, the two "branchs" ABDABD and ACDACD of the "cycle" ABDCAABDCA have a different phase, and so you can see an interference pattern at DD.
had read some papers that have mentioned the geometric phase (Berry phase) can be used to detect the quantum phase transitions in a quantum many-body system.
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Answer 2

Explanation:

The geometric phase is a phase acquired over the course of a cycle. If you take the Aharonov Bohm effect, you need to make an interferometer,