Deconfined quantum criticality in a frustrated Haldane chain with single-ion anisotropy

We report a phase diagram of the antiferromagnetic spin-1 chain with nearest-neighbor Heisenberg and three-site interactions in the presence of single-ion anisotropy. We show that the Gaussian and Ising transitions that separate the topological Haldane phase from the two anisotropic phases eventually fuse into a higher symmetry point characterized by the Wess-Zumino-Witten (WZW) SU(2)$_2$ critical theory providing a lattice realization of the conformal embedding. On the other side of the WZW multi-critical point, the Ising critical line reappears together with the eight-vertex transition. This transition is a one-dimensional realization of a deconfined quantum criticality separating the dimerized and Ising antiferromagnetic phases - two ordered phases with incompatible order parameters.