Thermal robustness of multipartite entanglement of the one-dimensional spin-$\frac{1}{2}$ $XY$ model

We study the robustness of multipartite entanglement of the ground state of the one-dimensional spin-$\frac{1}{2}$ $XY$ model with a transverse magnetic field in the presence of thermal excitations by investigating a threshold temperature, below which the thermal state is guaranteed to be entangled. We obtain the threshold temperature based on the geometric measure of entanglement of the ground state. The threshold temperature reflects three characteristic lines in the phase diagram of the correlation function. Our approach reveals a region where multipartite entanglement at zero temperature is high but is thermally fragile and another region where multipartite entanglement at zero temperature is low but is thermally robust.

Figure 1

(Color online) The threshold temperature ${\overline{T}}_{\text{th}}\left(\kappa \delta \right)$ as a function of $\kappa$. We set Boltzmann constant $k_B$ as 1 for simplicity. Dashed line is the threshold temperature $T_{\text{th}}$ defined by Eq. (6). By definition, the gapped threshold temperature ${\overline{T}}_{\text{th}}\left(\kappa \delta \right)$ converges to the threshold temperature $T_{\text{th}}$ in the limit of $\kappa \to \infty$

Figure 2

(Color online) The phase diagram at zero temperature in the $XY$ model. These phases are characterized by two point correlation functions. The ground states degenerate in the shaded region. Dashed line ($\eta =0$ and $h>1$) in the phase diagram is not a phase-transition line but the correlation function on this line is constant with respect to the distance $r$ similar to the C line.

Figure 3

(Color online) The geometric measure of the ground state of the $XY$ model. The geometric measure is numerically calculated for the system of $N=80$. Note that the geometric measure in the region of $\eta ∊\left[-1,0\right)$ is the same as that in the region of $\eta ∊\left(0,1\right]$. The same result had been originally obtained for the region of $\eta ∊\left[0,1\right]$ in Ref. [16].

Figure 4

(Color online) The threshold temperature in the $XY$ model. It is calculated in the system with $N=80$. We set Boltzmann constant $k_B$ as 1. We can clearly see the V line, the H line, and the C line. We note that the line of $\eta =0$, $h\ge 1$ corresponds to the dashed line in the phase diagram (see Fig. 2), which is not a QPT line.

Figure 5

(Color online) The threshold temperature shown in Fig. 4. The (red) solid line with small squares, the (green) dashed line, the (blue) dotted line, and the (purple) solid line correspond to $\eta =0,0.4,0.9,1.0$, respectively. The threshold temperature has the singularity at $h=\sqrt{1-{\eta }^2}$ (on the C line) and at $h=1$ (on the H line).

Figure 6

(Color online) The threshold temperature in the $XY$ model calculated in the system with $N=80$. We set Boltzmann constant $k_B$ as 1. The (red) solid line with small squares, the (orange) dashed line, the (green) dotted dashed line, and the (blue) dotted line show the threshold temperature for $\eta =0,0.1,0.4,0.9$, respectively. The parameter $h$ is between 0 and 50. We can see that the threshold temperature is monotonically increasing with $h$ in the region of $h⪢1$. We note that the threshold temperature for the case of $\eta =0$ and $h\ge 1$ is zero since the ground state is separable in that case. We can see that the effect of nonzero $\eta$ to the threshold temperature is high although entanglement itself is small (see Fig. 3).

Figure 7

(Color online) The upper left figure is the numerical calculations of ${\varphi }_{N/2}^{+}\left(h,0.5\right)$ for $N=10,20,50,100,1000$. We can see the nonanalyticity at $h=1$. The upper right figure and the lower figure are the right-hand side of Eq. (C3) in the case of $h=0.9$ (upper right) and $h=1.1$ (lower) with respect to $\theta$. We fixed $\eta =0.5$. Inset is ${\varphi }_{\theta }\left(h,\eta \right)$ with respect to $\theta$, respectively.