While the theory of composite Fermions (CFs) achieves tremendous successes in understanding the fractional quantum Hall effect and related phenomena, the true nature of CFs is still open to debate. The conventional interpretation, as explicated in Halperin-Lee-Read theory of the composite Fermi-liquid (CFL), regards the CF as an ordinary Newtonian particle. However, the theory suffers from difficulties, which motivate Son to propose that the CF should be a massless Dirac particle. An alternative interpretation, i.e., the CF is neither a Newtonian particle nor a Dirac particle, but a particle subject to a uniformly distributed Berry curvature in the momentum space and the Sundaram-Niu dynamics, is also put forward by us. Which interpretation is the correct one?

In this talk, I will answer this question by determining the Berry phase of adiabatic transport of a CF in the momentum space, directly from microscopic wave functions of the composite Fermi-liquid (CFL). We derive the definition of the Berry phase for the CFL. It is found to be different from that adopted in previous investigations by Geraedts and collaborators. With our definition, in contrast to the previous investigations, the numerical evaluation of the Berry phase is robust and free of extraneous phase factors. We show that the two forms of microscopic wave-functions of the CFL, i.e., the Jain-Kamilla type wave function and the standard form prescribed by the CF theory, yield different Berry phases/Berry curvatures in the momentum space. For the former, the Berry curvature for CFs has a continuous distribution inside the Fermi sea and vanishes outside, whereas for the latter, it is uniform in the whole momentum space. We also reveal the simple analytic structure of the standard CF wave function by establishing its connection to the Segal-Bargmann transform, and show that its Berry phase can be determined analytically. We conclude that the CF is nota massless Dirac particle. For the standard CF wave function, the uniform-Berry-curvature picture is the correct interpretation.