The relationship of the Berry phase with the topological aspects of a fermion and chiral anomaly is studied here. It is observed that this phase reflects the effect of quantum geometry when the quantization procedure of a Fermi field is considered from a classical system and may be considered as the manifestation of the multiply connected nature of quantum space-time.
We have calculated the Berry phase of Hall particles which are found to remain constant over the filling factor ν in a special arrangement. It has been found that the probability of getting this phase vanishes at ν > 1/5. This is the implication of the crystallization in the Hall fluid after the filling factor ν = 1/6.
It is shown here that a particle in an intense magnetic field may acquire the Berry phase and the topological features associated with this phase may be taken to be responsible for both the integrally and fractionally quantized Hall effect. The two different manifestations of quantum Hall effect have been realized in a unified scheme where the electrons associated with the fractional quantum Hall effect are found to be in an excited state having higher angular momentum.
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