  Both the neutron skin thickness $\Delta R_{np}$ of atomic nuclei and the
low-energy neutrino-nucleon ($\nu N$) interactions are of fundamental
importance in nuclear and particle physics, astrophysics as well as new physics
beyond the standard model (SM) but largely uncertain currently, and the
coherent elastic neutrino-nucleus scattering (CE$\nu$NS) provides a clean way
to extract their information. New physics beyond the SM may cause effectively a
shift of the SM weak mixing angle $\theta_W$ in low-energy $\nu N$
interactions, leading to an effective weak mixing angle $\theta^*_W$. By
analyzing the CE$\nu$NS data of the COHERENT experiment, we find that while a
one-parameter fit to the COHERENT data by varying $\Delta R_{np}$ produces
$\Delta R^{\rm{CsI}}_{np} \simeq 0.68^{+0.91}_{-1.13}$ fm for CsI with an
unrealistically large central value by fixing $\sin^2 \theta^*_W$ at the
low-energy SM value of $\sin^2\theta_W^{\rm{SM}} = 0.23857$, a two-dimensional
fit by varying $\Delta R_{np}$ and $\sin^2 \theta^*_W$ leads to a strong
positive correlation between $\Delta R_{np}$ and $\sin^2 \theta^*_W$ with
significantly smaller central values of $\Delta R^{\rm {CsI}}_{np} \simeq
0.24_{-2.03}^{+2.30}$ fm and $\sin^2 \theta^*_W = 0.21_{-0.10}^{+0.13}$.
Although the uncertainty is too large to claim a determination of $\Delta
R^{\rm{CsI}}_{np}$ and $\sin^2 \theta^*_W$, the present study suggests that the
multi-dimensional fit is important in future analyses of high-precision
CE$\nu$NS data. The implication of the possible deviation of $\sin^2
\theta^*_W$ from $\sin^2\theta_W^{\rm{SM}}$ on new physics beyond the SM is
also discussed.
