Mutual information across the commensurate--incommensurate crossover and near the Kosterlitz--Thouless point of the spin-$\frac{1}{2}$ $J_1$--$J_2$ Heisenberg chain

What quantum mutual information adds to conventional correlation functions is often unclear. We study the spin-1/2 $J_1$-$J_2$ Heisenberg chain, using DMRG on infinite chains and periodic rings and an SU(2) decomposition of the mutual information into correlation channels. For singlet ground states the two-site mutual information is an exact function of the spin correlator, so any additional content must come from blocks. In the dimerized phase, blocks resolve bond-energy correlations in the SU(2)-scalar channel without an independent asymptotic length: singlet and triplet correlation lengths agree to about 1%. The scalar channel dominates block mutual information at large distances in the commensurate regime up to $J_2\mathrm{/}{J}_1=0.40$, but falls to a few percent beyond the disorder point, which coincides numerically with the Majumdar-Ghosh point $J_2\mathrm{/}{J}_1=1\mathrm{/}2$; the structure-factor peak splits only at the Lifshitz point $J_2\mathrm{/}{J}_1=0.520622(2)$. On the gapless side we compare block mutual information on rings extrapolated to infinite size with the two-interval prediction of SU(2)${}_1$ conformal field theory across the Kosterlitz-Thouless point $J_2\mathrm{/}{J}_1=0.241167$. There, for block lengths $l\le 6$, the correction decays as $l^{-p}$ with $p=1.96$-$2.07$, and any logarithmic term is at least 60 times smaller than in the Heisenberg chain. Expanding in the marginal coupling (from level spectroscopy) the first-order coefficient is consistent with $l^{-2}$ scaling and the second-order coefficient is positive for $x\le 1\mathrm{/}4$, suggesting that the mutual information relative to the field-theory value becomes stationary at the KT point for large blocks. The scalar-channel share approaches 1/4 at small cross ratio $x$, as expected from SO(4) symmetry; under two stated assumptions its first-order response to the coupling is $(3\mathrm{/}8)\ln x-5\mathrm{/}8$ at small $x$, matched by the $l=4$ data within 1-3% at $x\le 1\mathrm{/}9$.