Transition between critical antiferromagnetic phases in the $J_1$-$J_2$ spin chain

The $J_1$-$J_2$ spin chain is one of the canonical models of quantum magnetism, and has long been known to host a critical antiferromagnetic phase with power-law decay of spin correlations. We show in this Letter that there are, in fact, $\mathit{\text{two}}$ distinct critical antiferromagnetic phases, where the roles of the local dimer field and its dual field are interchanged: the Affleck-Haldane' phase near the Heisenberg point $J_2 = 0$, where the dimer field that parametrises local singlet order is gapless and part of a joint $O(4)$ Néel-singlet order parameter; and the Zirnbauer’ phase which appears at sufficiently large ferromagnetic $J_2$, where the dimer field is gapped out and its $\mathit{\text{dual}}$ field – the instanton density of the $O(3)$ Néel field – is critical instead. The phases are so-named because each realises one of the competing pictures for how the $O(3)$ non-linear sigma model with a topological theta term renormalises to the $\widehat{\mathfrak{s}\mathfrak{u}}(2{)}_1$ Wess-Zumino-Witten model. We support these predictions with density matrix renormalisation group calculations.