Bell inequalities and entanglement at quantum phase transitions in the $\mathit{XXZ}$ model

Entanglement and violation of Bell inequalities are aspects of quantum nonlocality that have been often confused in the past. It is now known that this equivalence is only true for pure states. Even though almost all the studies of quantum correlations at quantum phase transitions deal only with entanglement, we here argue that Bell inequalities can also reveal a general quantum phase transition. This is also shown for a particular case of two spin-$\frac{1}{2}$ particles in an infinite one-dimensional chain described by the XXZ model. In this case, the Bell inequality is able to signal not only the first-order phase transition, but also the infinite-order Kosterlitz-Thouless quantum phase transition, which cannot be revealed either by the energy of the system nor by the bipartite entanglement. We also show that although the nearest-neighbor spins are entangled, they, unexpectedly, never violate the Bell inequality. This indicates that the type of entanglement which is relevant for quantum phase transition is not trivial, i.e., it cannot be revealed by the Bell inequality.

Figure 1

Ground-state energy per site of the $\mathit{XXZ}$ model as a function of the anisotropy $\Delta$. There is a first-order quantum phase transition at $\Delta =-1$ from a ferromagnetic to a gapless phase. At $\Delta =1$ there is an infinite-order Kosterlitz-Thouless phase transition from the gapless to the antiferromagnetic phase, which is not indicated by $e_0$.

Figure 2

Correlation function in the $\widehat{x}$ direction, $\langle {\sigma }_i^x{\sigma }_{i+r}^x\rangle$, for first, second, and third neighbors of the $\mathit{XXZ}$ model as a function of the anisotropy $\Delta$. It can be seen that the correlation is able to signal the first-order quantum phase transition at $\Delta =-1$, but not the infinite-order quantum phase transition at $\Delta =1$.

Figure 3

Correlation function in the $\widehat{z}$ direction, $\langle {\sigma }_i^z{\sigma }_{i+r}^z\rangle$, for first, second, and third neighbors of the $\mathit{XXZ}$ model as a function of the anisotropy $\Delta$. It can be seen that the correlation is able to signal the first-order quantum phase transition at $\Delta =-1$, but not the infinite-order quantum phase transition at $\Delta =1$.

Figure 4

Bell measurement (three upper curves) and concurrence (three lower curves) for first, second, and third neighbors of the $\mathit{XXZ}$ model as a function of the anisotropy $\Delta$. It can be seen that the Bell inequality is never violated, even when the two spins are entangled. While both the concurrence and the Bell measurement are able to signal the first-order quantum phase transition at $\Delta =-1$, only the Bell measurement reveals the infinite-order QPT at $\Delta =1$.

Figure 5

Parameter space of the reduced density matrix of two spins with the symmetries of the XXZ Hamiltonian. The greater triangle defines the physical states. The two smaller triangles, inside and on the lower vertices of the greater, are the entangled states (ENT). The two smallest regions, also on the lower vertices of the greatest triangle, are the states violating Bell inequality (NL). The curves are the “trajectory” of the ground state of the $\mathit{XXZ}$ Hamiltonian as the parameter $\Delta$ is varied for ${\rho }_{i,i+1}$ (solid blue curve), ${\rho }_{i,i+2}$ (dashed red curve), and ${\rho }_{i,i+3}$ (dotted yellow curve). For all the curves we marked fours points that correspond to $\Delta =-1$ (full circle), –0.999 (square), 0 (triangle), and 1 (diamond).

Figure 6

Derivative of the Bell measurement ${\mathcal{B}}_1$ and the concurrence $C_1$ in relation to $\Delta$ for nearest neighbors.