How is the Geometric Phase measured in the experiment? I 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. My question is: How is it measured in the experiment?
 A: At least in condensed matter systems (i.e., all solid materials you can think of), there is no direct way to measure the electronic Berry phase. What people do instead is to measure quantities which are impacted by the Berry phase (at least in the non-interacting band picture), such as quantum Hall transport coefficients [1], number and character of band crossings in photoemission [2], quantized Kerr effect [3], etc. Unfortunately, none of these are bulletproof methods, as there are many other contributions besides the Berry phase that determines these quantities.
In cold atom systems ($N\lesssim 10^{6}$, where $N$ is the number of particles), the Berry phase can be measured directly because you have fine control of the many-body wavefunction itself. This situation is unlike in solids where there are just way too many constituents ($N \sim 10^{23}$) to even talk about coherent control of the true many body wavefunction.
Coming up with a direct way to measure the Berry phase for electronic condensed matter systems is a big open question. Moreover, if you take interactions between electrons into account, it is unclear what what kinds of Berry phases are relevant and if any of them can be concretely linked to any experimental signature. The point being that the usual concept of Berry phase in condensed matter is related to the single electron picture, but that is just an approximation once electron-electron interactions are taken into account.

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*https://iopscience.iop.org/book/978-1-6817-4117-8/chapter/bk978-1-6817-4117-8ch9


*https://www.science.org/doi/10.1126/sciadv.aay2730


*https://physics.aps.org/articles/v3/62
A: To add to @KFGauss answer. The anomalous Hall effect is believed to be proportional to Berry phase if I understand some of the conclusions in the 2010 review paper "Anomalous Hall Effect" correctly.
This is rather interesting as it can be easily determined if the anomalous Hall effect is zero or finite.
A: The geometric phase of a topological many-body quantum system has indeed been directly measured, in ultracold atomic systems. The first demonstration of this that I am aware of is by the Bloch group in Munich.
There are a few variants of this experimental protocol, but the general idea is the following: first, you create a Bose-Einstein condensate in a periodic standing wave of light (an optical lattice), which gives it a band structure. The BEC will occupy the lowest energy state and thus have some definite quasimomentum. Then, you take it in a superposition of paths along the bands, which might involve driving Bloch oscillations or applying momentum kicks with Raman processes (as in atom interferometry). Then you recombine the two parts of the superposition. Taking advantage of the macroscopic coherence of a BEC, this results in some interference with a phase that reflects the accumulated relative Berry phase between the two branches.
