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

Abstract: 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, two distinct critical phases: the Affleck-Haldane' phase, where the dimer field that parametrises local singlet order is part of a joint $O(4)$ N\'eel-singlet order parameter; and theZirnbauer' phase, where the dimer field is gapped out and the critical theory involves only an $O(3)$ N\'eel order parameter. 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 $\mathfrak{\hat{su}}(2)_1$ Wess-Zumino-Witten model.