Spotlighting quantum critical points via quantum correlations at finite temperatures

We extend the program initiated by T. Werlang et al. [Phys. Rev. Lett. 105, 095702 (2010)] in several directions. Firstly, we investigate how useful quantum correlations, such as entanglement and quantum discord, are in the detection of critical points of quantum phase transitions when the system is at finite temperatures. For that purpose we study several thermalized spin models in the thermodynamic limit, namely, the $\mathit{XXZ}$ model, the $\mathit{XY}$ model, and the Ising model, all of which with an external magnetic field. We compare the ability of quantum discord, entanglement, and some thermodynamic quantities to spotlight the quantum critical points for several different temperatures. Secondly, for some models we go beyond nearest neighbors and also study the behavior of entanglement and quantum discord for second nearest neighbors around the critical point at finite temperature. Finally, we furnish a more quantitative description of how good all these quantities are in spotlighting critical points of quantum phase transitions at finite $T$, bridging the gap between experimental data and those theoretical descriptions solely based on the unattainable absolute zero assumption.

Figure 1

QD (black solid line) and EOF (red dashed line) as functions of the tuning parameter $\Delta$ and external field $h$ for the $\mathit{XXZ}$ model in the thermodynamic limit at $T=0$. (a) $h=0$, (b) $h=2$, (c) $h=6$, and (d) $h=12$. The dashed black vertical lines denote the CPs ${\Delta }_1$ and ${\Delta }_{\inf }$. Here and in the following graphics all quantities are dimensionless.

Figure 2

(a) TQD and (b) EOF as a function of $\Delta$ for $h=0$ and, from top to bottom when $\Delta <1$, $\mathit{kT}=0.02,0.1,0.5,1.0,2.0.$ The sudden change of TQD remains in the CP ${\Delta }_{\inf }=1$ as the temperature increases while the maximum of EOF is shifted to the right.

Figure 3

(a) TQD and (b) EOF as a function of $\Delta$ for $h=6$ and, for $\Delta \approx 2$, from top to bottom $\mathit{kT}=0.02,0.1,0.5,1.0,2.0$.

Figure 4

(a) TQD and (b) EOF as a function of $\Delta$ for $h=12$ and, for $\Delta \approx 3$, from top to bottom $\mathit{kT}=0.02,0.1,0.5,1.0,2.0.$

Figure 5

First-order derivative of TQD as a function of $\Delta$ for (a) $\mathit{kT}=0.02$, (b) $\mathit{kT}=0.1$, and (c) $\mathit{kT}=0.5$. Second-order derivative of TQD as a function of $\Delta$ for (d) $\mathit{kT}=0.02$, (e) $\mathit{kT}=0.1$, and (f) $\mathit{kT}=0.5$. The maximum of the first and second derivatives of TQD are very close to the CPs ${\Delta }_1=2$ and ${\Delta }_{\inf }=4.88$, respectively. Here we fixed $h=12$. Dashed vertical bars indicate the CPs.

Figure 6

Thermodynamic quantities for the $\mathit{XXZ}$ model in the thermodynamic limit for $h=12$ and $\mathit{kT}=0.02$ (black line), $\mathit{kT}=0.1$ (red line), $\mathit{kT}=0.5$ (blue line), $\mathit{kT}=1.0$ (green line), and $\mathit{kT}=2.0$ (orange line). Here the CPs are ${\Delta }_1=2$ and ${\Delta }_{\inf }=4.88$. For $\Delta <1$ and for the curves of the magnetization and the two-point correlations $\mathit{kT}$ increases from top to bottom while the opposite happens to the curves of the other quantities.

Figure 7

CPs determined by TQD (square), EOF (circle), $\langle {\sigma }_1^x{\sigma }_2^x\rangle$ (up arrow), and $\langle {\sigma }_1^z{\sigma }_2^z\rangle$ (down arrow) as a function of $T$. In (a) and (b) we have $h=6$ with a first (a) and an infinite-order (b) CP; in (c) and (d) we have $h=12$ with a first (c) and an infinite-order (d) CP. Dashed horizontal lines indicate the CPs. To determine the CPs at finite $T$ we used the derivatives of TQD and EOF. As stated in the text, for some temperature ranges, the derivatives of these quantities at finite $T$ retain some information about the divergences or discontinuities that exist only at $T=0$. Note that the estimated CPs given by TQD is always closer to the exact value than the CPs coming from other quantities.

Figure 8

TQD (bottom) and EOF (top) as a function of $T$ for different values of the tuning parameter $\Delta$. The insets show the regrowth of TQD and EOF. For $\Delta ⩽1$ (solid curves) we note that as we increase $\Delta$ we also increase TQD and EOF. For $\Delta >1$ (dashed curves) this only happens for EOF at high temperatures.

Figure 9

(a)–(c) TQD and (d)–(f) EOF as a function of $\lambda$ for $\mathit{kT}=0.01$ (black or solid line), $\mathit{kT}=0.1$ (red or dashed line), and $\mathit{kT}=0.5$ (blue or dotted line) for nearest neighbors. We use three values of $\gamma$ as shown in the graphs.

Figure 10

(a)–(c) TQD and (d)–(f) EOF as a function of $\lambda$ for $\mathit{kT}=0.01$ (black or solid line), $\mathit{kT}=0.1$ (red or dashed line), and $\mathit{kT}=0.5$ (blue or dotted line) for second nearest neighbors. We use three values of $\gamma$ as shown in the graphs.

Figure 11

CPs determined by TQD (black line with squares) and EOF (red line with circles) as a function of $\mathit{kT}$ for (a)–(c) first nearest neighbors and for (d)–(f) second nearest neighbors. Dashed horizontal lines indicate the CPs. The values of $\gamma$ used were $0$, $0.5$, and $1.0$, as shown in the graphs. The derivatives of TQD and EOF were used to estimate the CPs as explained in the text. Note that at panel (b) the curves for TQD and EOF coincide and that at panels (d) and (f) we only have two points for EOF since for greater $T$ it is zero.

Figure 12

Top panels TQD and bottom panels EOF as a function of $\gamma$. From top to bottom $\mathit{kT}=0.001,0.1,0.5,1.0$, and $2.0$. Left panels are first nearest neighbors and right panels second nearest neighbors. Here we fixed $\lambda =1.5$ and the CP is ${\gamma }_c=0$ (indicated by dashed vertical lines).