Quantum phase transitions in spin-1 $XXZ$ chains with rhombic single-ion anisotropy

We explore numerically the inverse participation ratios in the ground state of one-dimensional spin-1 $XXZ$ chains with the rhombic single-ion anisotropy. By employing the techniques of density-matrix renormalization group, effects of the rhombic single-ion anisotropy on various information theoretical measures are investigated, such as the fidelity susceptibility, the quantum coherence, and the entanglement entropy. Their relations with the quantum phase transitions are also analyzed. The phase transitions from the $Y$-Néel phase to the large-$E_x$ or the Haldane phase can be well characterized by the fidelity susceptibility. The second-order derivative of the ground-state energy indicates all the transitions are of second order. We also find that the quantum coherence, the entanglement entropy, the Schmidt gap, and the inverse participation ratios can be used to detect the critical points of quantum phase transitions. Results drawn from these quantum information observables agree well with each other. Finally we provide a ground-state phase diagram as functions of the exchange anisotropy $\mathrm{\Delta }$ and the rhombic single-ion anisotropy $E$.

Figure 1

(a) Second-order derivative of the ground-state energy density, (b) the Schmidt gap $G$, and (c) the string order parameter are plotted as a function of the rhombic single-ion anisotropy $E$ for different system sizes with $\mathrm{\Delta }=1$.

Figure 2

(a) Inverse participation ratio $T$ of the ground state and (b) its first-order derivative are plotted as a function of the rhombic single-ion anisotropy $E$ for different system sizes $N$ with $\mathrm{\Delta }=1.0$. Inset in (b) shows finite-size scaling of $E_c$ of the derivative of the inverse participation ratios as a function of $N^{-1}$. The line is the numerical fitting.

Figure 3

Inverse participation ratio $T$ is plotted as a function of system size $N$. A set of selected parameters are located in one of four phases, respectively.

Figure 4

(a) Two-site QC $I({\rho }_{N/2,N/2+1},S_{N/2}^x), I({\rho }_{N/2,N/2+1},S_{N/2}^z)$ and (b) their first-order derivatives are plotted as a function of the rhombic single-ion anisotropy $E$ for $N=100$.

Figure 5

(a) Entanglement entropy is plotted as a function of the rhombic single-ion anisotropy $E$ for different system sizes $N$ with $\mathrm{\Delta }=1$. Inset: the first-order derivative of the entanglement entropy. (b) Finite-size scaling of $E^c$ of the first extreme point. (c) Finite-size scaling of $E^c$ obtained from the first-order derivative of entanglement entropy at the second extreme point. The lines are the fitted lines.

Figure 6

Fidelity susceptibility per site is plotted as a function of the rhombic single-ion anisotropy $E$ for different system sizes $N$ with $\mathrm{\Delta }=1$. Insets show the finite-size scaling of $E^c$ in terms of the fidelity susceptibility versus $N^{-1}$. The lines are the numerical fittings.

Figure 7

Inverse participation ratio $T$ of the ground state is plotted as a function of the rhombic single-ion anisotropy $E$ for different system sizes with $\mathrm{\Delta }=1.5$. Inset: finite-size scaling of $E^c$ of the extreme point of the inverse participation ratios.

Figure 8

(a) Entanglement entropy is plotted as a function of the rhombic single-ion anisotropy $E$ for different system sizes with $\mathrm{\Delta }=1.5$. (b) The first-order derivative of the two-site QC $I_{N/2,N/2+1}$ is plotted as a function of $E$ with $\mathrm{\Delta }=1.5$ for $N=100$.

Figure 9

(a) Fidelity susceptibility per site and (b) the second-order derivative of the ground-state energy density are plotted as a function of the rhombic single-ion anisotropy $E$ for different system sizes $N$ with $\mathrm{\Delta }=1.5$. Inset in (a) shows finite-size scaling behavior of $E_c$ in terms of the fidelity susceptibility.

Figure 10

Phase diagram of spin-1 $XXZ$ chain as functions of the rhombic single-ion anisotropy $E$ and the exchange anisotropy $\mathrm{\Delta }$.