FIG. 2.

(a) Bulk gap Δ_α and ground-state correlation length ξ_α in the L → ∞ limit, obtained by finite-size scaling for 200 ⩽ L ⩽ 500. (b) Bulk gap ${\Delta }_{\alpha }^{\prime }$ and ${\xi }_{\alpha }^{\prime }$ with L = 100 and L = 300. (c) Ground-state string-ordered correlation function ${\mathcal{S}}_{\mathit{ij}}$ for H_α and $H_{\alpha }^{\prime }$ with α = 1.5 and L = 300. For various α and 200 ⩽ L ⩽ 500, we consistently find that ${\mathcal{S}}_{\mathit{ij}}$ quickly saturates to a finite value for H_α at all α > 0, but vanishes at large distance for $H_{\alpha }^{\prime }$ at α ≲ 3. (d) Ground-state spin-spin correlation ${\mathcal{C}}_{\mathit{ij}}$ for α = 0.5 and L = 500. This choice of α = 0.5 is arbitrary, but assists in a clear presentation of the coexisting exponential and 1/r^α+4 power-law decays.