Scaling properties of fidelity in the spin-1 anisotropic model

By means of the density matrix renormalization group technique, the scaling relation of the fidelity susceptibility proposed recently is verified for the spin-one $XXZ$ spin chain with an on-site anisotropic term. Moreover, from the results of both the fidelity susceptibility and the entanglement entropy, the critical points and some of the corresponding critical exponents are determined through a proper finite-size scaling analysis, and these values agree with the findings in the literature. Thus our work provides a numerical support of the use of the fidelity in detecting quantum phase transitions.

Figure 1

(Color online) Fidelity susceptibility $\mathcal{S}\left(D\right)$ for the spin-1 $XXZ$ spin chain in Eq. (2) as functions of $D$ for various sizes $L$ with $\lambda =1$. The inset shows the fidelity $\mathcal{F}\left(D,D+\delta \right)$ as functions of $D$ for the corresponding sizes. Here we take $\delta ={10}^{-3}$.

Figure 2

(Color online) Entanglement entropy $\mathcal{E}\left(D\right)$ for the spin-1 $XXZ$ spin chain in Eq. (2) as functions of $D$ for various sizes $L$ with $\lambda =1$.

Figure 3

Finite-size scaling of $D_{\text{max}}$ versus $L^{-1/\nu }$. The full lines are least square straight line fits for sizes with $L\ge 100$. Top: the Haldane–large-$D$ transition, where $\nu \simeq 1.42$ $\left(\nu \simeq 1.45\right)$ for those $D_{\text{max}}$’s corresponding to the local maxima in the curves of $\mathcal{E}$ $\left(\mathcal{S}\right)$. Bottom: the Haldane-Néel transition, where $\nu \simeq 0.90$ $\left(\nu \simeq 1.05\right)$ for those $D_{\text{max}}$’s corresponding to the local maxima in the curves of $\mathcal{E}$ $\left(\mathcal{S}\right)$.

Figure 4

The log-log plot of ${\mathcal{S}}_{\text{max}}$ for various sizes $L$. The full lines are least square straight line fits for sizes with $L\ge 100$.