Transition between critical antiferromagnetic phases in the J₁-J₂ spin chain

The J₁-J₂ 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 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) Néel-singlet order parameter; and the 'Zirnbauer' phase, where the dimer field is gapped out and its 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 su(2)₁ Wess-Zumino-Witten model.  We support these predictions with density matrix renormalisation group calculations.