  We measure isotope shifts for neutral Yb isotopes on an ultranarrow optical
clock transition $^{1}\text{S}_{0}-^{3}\text{P}_{0}$ with an accuracy of a few
Hz. The part-per-billion precise measurement was possible by loading the
ultracold atoms into a three-dimensional magic-wavelength optical lattice and
alternately interrogating the isotope pairs, thus minimizing the effects due to
the optical lattice light-shift and inter-atomic interaction as well as the
drifts of a clock laser frequency and a magnetic field. The determined isotope
shifts, combined with one of the recently reported isotope-shift measurements
of Yb$^+$ on two optical transitions, allow us to construct the King plots.
Extremely large nonlinearity with the corresponding $\chi^2$ on the order of
$10^4$ is revealed, and is not explained by a quadratic field shift. We further
carry out the generalized King plot for three optical transitions so that we
can eliminate the contribution arising from a higher-order effect within the
Standard Model which might explain the observed nonlinearity of King plots for
two transitions. Our analysis of the generalized King plot shows a deviation
from linearity at the 3$\sigma$ level, indicating that there exist at least two
higher order contributions in the measured isotope shifts. Then, under the
reasonable assumption to attribute them to higher-order field shifts within the
Standard Model, we obtain the upper bound of the product of the couplings for a
new boson mediating a force between electrons, and neutrons $|y_ey_n|/(\hbar
c)< 1\times10^{-10}$ for the mass less than 1 keV with the 95% confidence level
is derived, providing an important step towards probing new physics via
isotope-shift spectroscopy.
