Kaleidoscope of Quantum Phases in a Frustrated Spin-1 Chain with Single-ion Anisotropy

We study the phases and phase transitions of a spin-1 chain with next-nearest neighbor (NNN) interaction J₂ and uniaxial single-ion anisotropy D by density-matrix renormalization group method. We reconfirm the topological first-order transition of the J₁-J₂ chain by the bond reversal method [Q. Luo, et al., Phys. Rev. B 100, 121111(R) (2019)], and we show accurately that the transition occurs at J_2,t = 0.7607(2). When considering a negative D, there is a spontaneously dimerized phase so as to avoiding the direct transition between the Neel phase and the NNN-Neel phase. The Neel--Dimer transition is of Gaussian type with central charge c = 1, while the NNN-Neel--Dimer transition belongs to the 2D Ising universality class. For the positive D case, there is a gapless chiral phase with a broken parity symmetry. It separates the NNN-Haldane phase and the large-D phase before the direct transition between the two. We also determine the topological critical points between the Haldane phase and the large-D phase by the level spectroscopy technique supplemented by the Richardson method. The critical point is obtained with spectacular accuracy, D_c = 0.96847256(2), for the frustration-free case.