The spin-1 Haldane chain is an example of the symmetry-protected-topological (SPT) phase in one dimension. Experimental realization of the spin chain materials usually involves both the uniaxial-type,
D(Sz)2, and the rhombic-type, $E[{({S}^{x})}^{2}\ensuremath{-}{({S}^{y})}^{2}]$, single-ion anisotropies. Here, we provide a precise ground-state phase diagram for a spin-1 Haldane chain with these single-ion anisotropies. Using quantum numbers, we find that the
Z2 symmetry breaking phase can be characterized by double degeneracy in the entanglement spectrum. Topological quantum phase transitions take place on particular paths in the phase diagram, from the Haldane phase to the large-
Ex, large-
Ey, or large-
D phases. The topological critical points are determined by the level spectroscopy method with a newly developed parity technique in the density matrix renormalization group [Phys. Rev. B 86, 024403 (2012)], and the Haldane-large-
D critical point is obtained with an unprecedented precision,
(D/J)c=0.9684713(1). Close to this critical point, a small rhombic single-ion anisotropy $|E|/J\ensuremath{\ll}1$ can destroy the Haldane phase and bring the system into a
y-N\'eel phase. We propose that the compound $[\mathrm{Ni}({\mathrm{HF}}_{2}){(3\text{\ensuremath{-}}\mathrm{Clpy})}_{4}]{\mathrm{BF}}_{4}$ is a candidate system to search for the
y-N\'eel phase.
Low-lying excited states as well as the ground state of the spin-1 antiferromagnetic Heisenberg chain with a spin-1/2 impurity are studied by means of a method of numerical diagonalization. The isotropic nearest-neighbor exchange coupling plus the uniaxial single-ion-type anisotropy energy is assumed for the host-system Hamiltonian. The energy differences between the ground state and the low-lying excited states, which appear in the Haldane gap, are estimated in the thermodynamic limit. The results are used to analyze the electron-spin-resonance experimental data on Ni(C
2H
8N
2)
2NO
2(ClO
4), abbreviated NENP, containing a small amount of spin-1/2 Cu
2+ impurities. It is found that in this system, the impurity-host
(Cu
2+-Ni
2+) coupling is ferromagnetic and its magnitude is about 5
\% of the magnitude of the host-host
(Ni
2+-Ni
2+) coupling.
The ground-state phase diagram of a frustrated S=1 Heisenberg spin chain with uniaxial single-ion-type anisotropy is studied using the infinite-system density-matrix renormalization-group method. Particular attention is paid to "chiral" phases, in which the long-range order parameter is a vector chirality. It is found that, in addition to the Haldane, large-D (LD), and double-Haldane phases, the phase diagram includes three different chiral phases, i.e., the gapless chiral, chiral-Haldane, and chiral-LD phases. These chiral phases are distinguished from each other by the energy gap and the string long-range order.