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Constraining the symmetry energy at subsaturation densities using isotope binding energy difference and neutron skin thickness

We show that the neutron skin thickness $Δr_{np}$ of heavy nuclei is uniquely fixed by the symmetry energy density slope $L(ρ)$ at a subsaturation cross density $ρ_c \approx 0.11$ fm$^{-3}$ rather than at saturation density $ρ_0$, while the binding energy difference $ΔE$ between a heavy isotope pair is essentially determined by the magnitude of the symmetry energy $E_{\text{sym}}(ρ)$ at the same $ρ_c$. Furthermore, we find a value of $L({ρ_c})$ leads to a negative $E_{\text{sym}}({ρ_{0}})$-$L({ρ_{0}})$ correlation while a value of $E_{\text{sym}}({ρ_{c})}$ leads to a positive one. Using data on $Δr_{np}$ of Sn isotopes and $ΔE$ of a number of heavy isotope pairs, we obtain simultaneously $E_{\text{sym}}({ρ_{c})}=26.65\pm0.20$ MeV and $L({ρ_c})= 46.0\pm4.5$ MeV at 95% confidence level, whose extrapolation gives $E_{\text{sym}}({ρ_{0}})=32.3\pm1.0$ MeV and $L({ρ_{0}})=45.2\pm10.0$ MeV. The implication of these new constraints on the $Δr_{np}$ of $^{208}$Pb and the core-crust transition density in neutron stars is discussed.