Robust linear correlations related to neutron skin thickness

We observe various robust linear correlations related to neutron skin thickness ($\Delta R_{\rm np}$) within different interaction ensembles, including newly proposed random Skyrme ensemble. The robust linear correlation between $\Delta R_{\rm np}$, or charge radius difference of mirror nuclei ($\Delta R_{\rm mirr}$), and the isospin asymmetry ($I=\frac{N-Z}{A}$) becomes apparent as the model space is enlarged. Shape coexistence, or shape effect on charge radius, is considered to explain the experimental deviation of ${}^{18}$O/Ne and some odd-$A$ $\Delta R_{\rm mirr}$s from the $\Delta R_{\rm mirr}-I$ linearity. The slopes of the linear $\Delta R_{\rm mirr}-I$ and $\Delta R_{\rm np}-I$ correlations ($C_{\rm np}$ and $C_{\rm mirr}$, respectively) are also robustly and linearly correlated to the slope of the symmetry energy ($L$). These linear correlations are further understood with the similar formulation between between $L$ and the symmetry energy coefficient ($J$). The linear correlations between $C_{\rm np}-L$ and $C_{\rm mirr}-L$ are also adopted to constrain $L$ to $20\sim36$ MeV with 1$\sigma$ confidence. Considering the deviation of ${}^{18}$O/Ne $\Delta R_{mirr}$ due to shape coexistence, the 1$\sigma$ range for $L$ is further narrowed to $28\sim36$ MeV, suggesting a relatively soft equation of state for nuclear matter.