Fig. 6: Susceptibilities.
We plot a the disconnected susceptibility χdis rescaled by N, b the connected susceptibility χcon rescaled by N1/2, and c the ratio between χdis and \({\chi }_{{{\rm{con}}}}^{2}\), as functions of \((\varphi -{\varphi }_{{{\rm{J}}}}^{\infty }){N}^{1/2}\). The solid black lines in a–c are Eqs. (5), (6), and the ratio between Eq. (5) and the square of Eq. (6), respectively. The dashed black line in c is the second-order expansion Eq. (7). The dashed red line in c is the constant term in Eq. (7), i.e., \({\chi }_{{{\rm{dis}}}}/{\chi }_{{{\rm{con}}}}^{2}=\pi {\sigma }_{\varphi }^{2}/2\). To draw these lines, we use u = 0.041 and σφ = 0.043 obtained from the fitting in Fig. 5d, and \({Z}_{{{\rm{J}}}}^{ * }=4.1\) determined at \({\varphi }_{{{\rm{J}}}}^{\infty }=0.8432\) in Fig. 3d. No fitting is performed here. Symbols in a–c have the same meanings as indicated in the legend of a. d, e the susceptibilities of jammed states χJ are plotted as functions of N for a few different φ, and are fitted to Eq. (8). The open squares in d are χJ data obtained in a small pressure window 0.0009 < P < 0.0011, corresponding to ∆Z ≈ 0.15 (φ ≈ 0.843−0.844). The fitting parameter μ is plotted in f as a function of φ, where the error bar represents the fitting error.
