  Multiparticle azimuthal correlations of prompt D$^0$ mesons are measured in
PbPb collisions at a nucleon-nucleon center-of-mass energy of
$\sqrt{s_\mathrm{NN}} =$ 5.02 TeV. For the first time, a four-particle cumulant
method is used to extract the second Fourier coefficient of the azimuthal
distribution ($v_2$) of D$^0$ mesons as a function of event centrality and the
D$^0$ transverse momentum. The ratios of the four-particle $v_2$ values to
previously measured two-particle cumulant results provide direct experimental
access to event-by-event fluctuations of charm quark azimuthal anisotropies.
These ratios are also found to be comparable to those of inclusive charged
particles in the event. However, hints of deviations are seen in the most
central and peripheral collisions. To investigate the origin of flow
fluctuations in the charm sector, these measurements are compared with models
implementing fluctuations of charm quark energy loss via collisional or
radiative processes in the quark-gluon plasma. These models cannot
quantitatively describe the data over the full transverse momentum and
centrality ranges, although the calculations with collisional energy loss
provide a better description of the data.
