Fidelity approach to Gaussian transitions

The fidelity approach to the Gaussian transitions in spin-one $\mathit{XXZ}$ spin chains with three different values of Ising-type anisotropy $\lambda$ is analyzed by means of the density matrix renormalization group technique for systems of large sizes. We find that, despite the success in the cases of $\lambda =2.59$ and 1, the fidelity susceptibility fails to detect the Gaussian transition for $\lambda =0.5$. Thus our results demonstrate the limitation of the fidelity susceptibility in characterizing quantum phase transitions, which was proposed recently in general frameworks.

Figure 1

(Color online) (a) Fidelity susceptibility $\mathcal{S}$. (b) The second derivative of the ground-state energy density $-{\partial }^{2}e/\partial D^2$ as functions of $\mathit{D}$ for various sizes $\mathit{L}$ with $\lambda =2.59$. The inset in the top panel shows the fidelity $\mathcal{F}\left(D,D+\delta \right)$ as functions of $\mathit{D}$ for the corresponding sizes. Here we take $\delta ={10}^{-3}$.

Figure 2

Finite-size scaling of $D_{\text{max}}$ of $\mathcal{S}$ $(\times )$ and $-{\partial }^{2}e/\partial D^2$ $(△)$ vs $L^{-1/\nu }$ for $\lambda =2.59$. The lines are least square fits, where $\nu \simeq 0.79$ $\left(\nu \simeq 0.77\right)$ for those $D_{\text{max}}$’s corresponding to the local maxima in the curves of $S$ $\left(-{\partial }^{2}e/\partial D^2\right)$.

Figure 3

The log-log plot of (a) ${\mathcal{S}}_{\text{max}}$ and (b) ${\left[-{\partial }^{2}e/\partial D^2\right]}_{\text{max}}$ for various sizes $\mathit{L}$ with $\lambda =2.59$. The lines are least square straight line fits.

Figure 4

(Color online) (a) Fidelity susceptibility $\mathcal{S}$. (b) The second derivative of the ground-state energy density $-{\partial }^{2}e/\partial D^2$ as functions of $\mathit{D}$ for various sizes $\mathit{L}$ with $\lambda =1$. The inset in the top panel shows the fidelity $\mathcal{F}\left(D,D+\delta \right)$ as functions of $\mathit{D}$ for the corresponding sizes. Here we take $\delta ={10}^{-3}$.

Figure 5

(Color online) (a) Fidelity susceptibility $\mathcal{S}$. (b) The second derivative of the ground-state energy density $-{\partial }^{2}e/\partial D^2$ as functions of $\mathit{D}$ for various sizes $\mathit{L}$ with $\lambda =0.5$. The inset in the top panel shows the fidelity $\mathcal{F}\left(D,D+\delta \right)$ as functions of $\mathit{D}$ for the corresponding sizes. Here we take $\delta ={10}^{-3}$.

Figure 6

The log-log plot of ${\mathcal{S}}_{\text{max}}$ for various sizes $\mathit{L}$ with $\lambda =0.5$. The dashed curve is the least square fit by using the formula ${\mathcal{S}}_{\text{max}}=S_{\infty }-a{L}^{-b}$, with ${\mathcal{S}}_{\infty }=0.073$, $a=0.23$, and $b=0.75$.

Figure 7

(Color online) (a) Entanglement entropy $\mathcal{E}$ as functions of $\mathit{D}$ for various sizes $\mathit{L}$. (b) The semilogarithmic plot of ${\mathcal{E}}_{\text{max}}$ for various sizes $\mathit{L}$. The line is the least square straight line fit. (c) Finite-size scaling of $D_{\text{max}}$ of $\mathcal{E}$ vs $L^{-1/\nu }$. The line is the least square fit with $\nu \simeq 1.51$. Here $\lambda =0.5$.

Figure 8

The configuration of wave functions in the superblock.

Figure 9

The basis transformation between a block and an enlarged block.