  Focusing on the hole-doped two-dimensional systems of strongly correlated
electrons, we examine geometric phases acquired by electronic wave functions as
a result of polaron transport around a closed loop. For this study we apply the
Lanczos exact diagonalization method to Holstein-Hubbard, Holstein-$tJ$, and
Holstein-$tJ_z$ models in order to reveal various aspects of geometric phases.
We demonstrate that transverse spin fluctuations are responsible for the
generation of nontrivial geometric phases. From the exchange symmetry of
polarons we find that the statistical transmutation of polarons occurs
depending on the strength of electron correlations.
