  Heavy nuclei exhibit a crossover from vibrational to rotational collectivity
as the number of neutrons or protons increases from shell closure towards
midshell, but the microscopic description of this crossover has been a major
challenge. We apply the shell model Monte Carlo approach to families of
even-even samarium and neodymium isotopes and identify a microscopic signature
of the crossover from vibrational to rotational collectivity in the
low-temperature behavior of $<\mathbf{J}^2>_T$, where $\bf J$ is the total spin
and $T$ is the temperature. This signature agrees well with its values
extracted from experimental data. We also calculate the state densities of
these nuclei and find them to be in very good agreement with experimental data.
Finally, we define a collective enhancement factor from the ratio of the total
state density to the intrinsic state density as calculated in the
finite-temperature Hartree-Fock-Bogoliubov approximation. The decay of this
enhancement factor with excitation energy is found to correlate with the
pairing and shape phase transitions in these nuclei.
