Multipartite quantum nonlocality and Bell-type inequalities in a spin-1/2 model with topological quantum phase transitions

Multipartite quantum nonlocality and Bell-type inequalities are used to characterize quantum correlations in an infinite-size bond-alternating spin-$\frac{1}{2}$ Heisenberg chain with next-nearest-neighbor interactions. With the help of powerful tensor-network algorithms, both zero and finite temperatures are considered. First, at zero temperature, both in the even-Haldane phase and in the odd-Haldane phase, a high hierarchy of multipartite nonlocality is observed. Nevertheless, in the two phases, the spread of the multipartite nonlocality among the lattice is different. Thereby, the nonlocality measures are relatively large in one phase and vanish in the other phase, and provide quite sharp signals for the topological quantum phase transitions (QPTs) between these two phases. The influence of the next-nearest-neighbor coupling $\alpha$ upon the multipartite nonlocality in the model is also discussed. Second, we find that the footprints of the QPTs survive at low temperatures. Third, based upon the scaling behavior of the finite-temperature nonlocality measure, we propose a quantity $\mathcal{K}$ to characterize the finite-temperature nonlocality in the large-$n$ limit. We find that in high-temperature regions, $\mathcal{K}$ is reduced linearly as the temperature rises.

Figure 1

Illustration of multipartite nonlocality in five-site models. Only sites in the same group (pink shadow) share nonlocal correlations with each other. The leftmost figure illustrates a model with genuine multipartite nonlocality. The rightmost figure illustrates a model without any form of multipartite nonlocality. From left to right, the hierarchy of multipartite nonlocality decreases gradually. For general quantum states, the hierarchy of nonlocality can be investigated by Bell-type inequalities.

Figure 2

Structure of an infinite-size bond-alternating spin-$\frac{1}{2}$ Heisenberg chain with next-nearest-neighbor interactions. $J_1=1-\delta \ge 0$ and $J_2=1+\delta \ge 0$ denote bond-alternating nearest-neighbor coupling, and $\alpha \ge 0$ denotes next-nearest-neighbor coupling. For a finite fixed $\alpha$, when $\delta$ increases from $-1$ to 1, a topological quantum phase transition occurs at ${\delta }_c=0$.

Figure 3

(a), (b) Two concerned subchains for calculating the nonlocality measures, where the reduced states are denoted by ${\widehat{\rho }}_o$ and ${\widehat{\rho }}_e$, respectively. Since $J_1=1-\delta$ and $J_2=1+\delta$, we have ${\widehat{\rho }}_o\left(\delta \right)={\widehat{\rho }}_e(-\delta )$. (c)–(e) The nonlocality measures as a function of the dimerization parameter $\delta$ (with $n=10$ and $T=0$). ${\mathcal{S}}_o$ (dots) corresponds to the odd-bond subchain in (a), and ${\mathcal{S}}_e$ (triangles) corresponds to the even-bond subchain in (b). For any given $\delta$, we find ${\mathcal{S}}_o\left(\delta \right)={\mathcal{S}}_e(-\delta )$.

Figure 4

Multipartite nonlocality measure ${\mathcal{S}}_o$ as a function of the dimerization parameter $\delta$ for various next-nearest-neighbor coupling parameter $\alpha$, with $n=10$ (black dashed curves) and $n=20$ (red solid curves). The horizontal grid lines represent the threshold of the Bell-type inequalities $\mathcal{S}\le 1,2,4,8$ in Eq. (2). ${\delta }_c=0$ is the quantum phase transition point.

Figure 5

The $n$ dependence of the nonlocality measure ${\mathcal{S}}_o$ at zero temperature with various coupling constants. When $n$ is large enough, we have ${log}_2{\mathcal{S}}_o\sim \mathcal{K}n+b$. This behavior holds in the entire odd-Haldane phase and in a rather narrow region $\delta \approx {\delta }_c^{+}$ of the even-Haldane phase (in most regions of this phase, we simply have ${\mathcal{S}}_o=0$).

Figure 6

Two special situations with (a) $\alpha =0,\delta =-1$ and (b) $\alpha =0,\delta =1$. In (a), ${\widehat{\rho }}_o$ is a pure state, for which the nonlocality measure is ${\mathcal{S}}_o=\sqrt{2}$. In (b), ${\widehat{\rho }}_o$ is a mixed state, for which the nonlocality measure turns out to be ${\mathcal{S}}_o=0$ exactly. (c) A tensor network for $f_{s_1{s}_8}$, which is used to explain why ${\mathcal{S}}_o=0$ for the situation given in (b).

Figure 7

Multipartite nonlocality measure ${\mathcal{S}}_o$ at finite temperatures. (a),(b) The influence of the temperature $T$ upon the ${\mathcal{S}}_o\left(\delta \right)$ curves. The horizontal grid lines represent the threshold of the Bell-type inequalities $\mathcal{S}\le 1,2,4,\cdots$ in Eq. (2). (c), (d) The logarithm nonlocality measure as a function of the temperature for various coupling constants. The length of the subchain is $n=20$.

Figure 8

(a) The $n$ dependence of the nonlocality measure ${\mathcal{S}}_o$ at finite temperatures with $\{\alpha =0,\delta =-0.03\}$. The scaling behavior ${log}_2{\mathcal{S}}_o\sim \mathcal{K}n+b$ survives at finite temperatures. (b) As the temperature rises, the slope $\mathcal{K}$ presents an approximately linear decline.