Quantum phase transitions driven by rhombic-type single-ion anisotropy in the $S=1$ Haldane chain

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, $D{\left(S^z\right)}^2$, and the rhombic-type, $E[{\left(S^x\right)}^2-{\left(S^y\right)}^2]$, 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 ${\mathbb{Z}}_2$ 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-$E_x$, large-$E_y$, 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, ${(D/J)}_c=0.9684713\left(1\right)$. Close to this critical point, a small rhombic single-ion anisotropy $\left|E\right|/J\ll 1$ can destroy the Haldane phase and bring the system into a $y$-Néel phase. We propose that the compound $\left[\mathrm{Ni}\left({\mathrm{HF}}_2\right){\left(3\text{−}\mathrm{Clpy}\right)}_4\right]{\mathrm{BF}}_4$ is a candidate system to search for the $y$-Néel phase.

Figure 1

Quantum phase diagram for the $S=1$ Haldane chain with both uniaxial and rhombic single-ion anisotropies. Topological quantum phase transitions occur through particular (red arrows) routes. The Haldane-large-$D$ critical point ${(D/J)}_c=0.9684713\left(1\right)$ is determined by the LS+DMRG method. A small rhombic anisotropy $\left|E\right|/J\ll 1$ at this point ${(D/J)}_c$ induces a transverse antiferromagnetic order.

Figure 2

(a) The excitation gap $\mathrm{\Delta }=E_0(m,p,t;\mathrm{PBC})-E_g$, where $E_g=E_0(0,1,1;\mathrm{PBC})$ is the ground state energy. The ground state in the $y$-Néel phase has double degeneracy. (b) The entanglement spectra of at least fourfold, double, and nondegeneracy characterize the Haldane, the $y$-Néel, and the large-$E_x$ phases, respectively. The vertical lines indicate two critical points ${(E/J)}_c\simeq 0.214$ and 1.717, respectively. DMRG data for $D/J=0$ and $L=80$ are presented. $K=1000$ states are kept.

Figure 3

Spin and quadrupole correlation functions at typical values of $E/J$ for $D/J=0$ and $L=200$. (a) ${(-1)}^r\langle S_0^y{S}_r^y\rangle$, (b) $\langle Q_0^{x^2-y^2}{Q}_r^{x^2-y^2}\rangle$, and (c) $\langle Q_0^{z^2}{Q}_r^{z^2}\rangle$. Note that $r=100$ for the most distant sites in the periodic chain of $L=200$.

Figure 4

(a) For a closed chain with even number of singlets, the quantum numbers for the Haldane phase are $(m,p,t)=(0,1,1)$. (b) Within TBC, the number of singlets become odd, and the quantum numbers change as $(m,p,t)=(0,-1,-1)$. (c) For large single-ion anisotropy, the quantum numbers are $(m,p,t)=(0,1,1)$.

Figure 5

(Left) Energy level crossing with different quantum numbers occurs at $D_c^{*}$ for $L=64$, $E/J=0$, within TBC. $\mathrm{\Delta }=E_0(m,p,t;\mathrm{TBC})-E_g$ and $E_g=E_0(0,1,1;\mathrm{PBC})$ is the ground state energy within PBC. (Right) Extrapolation of the critical point is performed by linear fitting. ${(D/J)}_c=0.9684713\left(1\right)$ is obtained. $K=2000$ states are kept.