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arXiv:1110.0787 · Physical Review A

Bell inequalities and entanglement at quantum phase transitions in the XXZ model

L. Justino, Thiago Rodrigues de Oliveira

Published May 30, 2012• 76 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.85.052128

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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-12\frac{1}{2}21​ 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.
For many years it was believed that entanglement and violation of the Bell inequalities were the same peculiar aspect of Quantum Mechanics: non-locality. Nowadays, it is known that this equivalence is not complete, being true only for pure states; in other words, there are entangled mixed states that do not violate the Bell inequalities. Thus it is interesting to analyze properties of states which exhibit such difference, in order to better understand it. Here we present such a study of the connection between entanglement and violation of the Bell inequalities in Quantum Phase Transitions. More pecifically, we analyze this relation for two particles in an infinity one-dimensional spin-12 chain described by the XXZ model. We show that the two particles do not violate the Bell-CHSH inequality for any value of anisotropy, even in regions where they are entangled. Therefore, the entanglement that is relevant in Quantum Phase Transitions is not a trivial one , since it is not revealed by the Bell-CHSH inequality. We also demonstrate that the study of the Bell-CHSH inequality can indicate not only the first order phase transition as the infinity order one, while the fundamental energy per particle and the concurrence can only indicate the first order phase transition.
In this paper, using the infinite time-evolving block decimation (iTEBD) algorithm and Bell-type inequalities, we investigate multipartite quantum nonlocality in an infinite one-dimensional quantum spin-12XXZ\frac{1}{2} XXZ21​XXZ system. High hierarchy of multipartite nonlocality can be observed in the gapless phase of the model, while only the lowest hierarchy of multipartite nonlocality is observed in most regions of the gapped antiferromagnetic phase. Thereby, Bell-type inequalities disclose different correlation structures in the two phases of the system. Furthermore, at the infinite-order quantum phase transition (QPT, or Kosterlitz-Thouless QPT) point of the model, the correlation measures always show a local minimum value, regardless of the length of the subchains. It indicates that relatively low hierarchy of multipartite nonlocality would be observed at the infinite-order QPT point in a Bell-type experiment. The result is in contrast to the existing results of the second-order QPT in the one-dimensional XYXYXY model, where multipartite nonlocality with the highest hierarchy has been observed. Thus, multipartite nonlocality provides an alternative perspective to distinguish between these two kinds of QPTs. Reliable clues for the existence of tripartite quantum entanglement have also been found.
The ground state of interacting spin chains in external magnetic fields can undergo a quantum phase transition (QPT) characterized by dramatic changes at a critical value of the magnetic field. In this paper, we use Bell-type inequalities to study the multipartite correlations (including multipartite entanglement and multipartite nonlocality in an nnn-spin subsystem) in the QPT of an infinite XYXYXY chain. An efficient numerical optimization procedure is proposed to figure out the violation measure Mn{M}_{n}Mn​ of the inequalities. For $n\ensuremath{\le}7$, the magnetic-field ($\ensuremath{\lambda}$) dependence of Mn{M}_{n}Mn​ is studied. We find the derivative of Mn{M}_{n}Mn​ is divergent exactly at the QPT point ${\ensuremath{\lambda}}_{c}=1$ for any nnn. In addition, with the increase of nnn, Mn{M}_{n}Mn​ converges quickly for $\ensuremath{\lambda}<{\ensuremath{\lambda}}_{c}$ and converges very slowly for $\ensuremath{\lambda}>{\ensuremath{\lambda}}_{c}$, which can be regarded as another signal for the QPT. Furthermore, in the vicinity of ${\ensuremath{\lambda}}_{c}$, high-order Bell-type inequalities will be violated as long as nnn is large enough. This indicates that high-level multipartite correlation will be present when the system is in the vicinity of the QPT point. Nevertheless, genuine nnn-partite entanglement or genuine nnn-partite nonlocality is not observed in the QPT.
In this paper, we use Bell inequality and nonlocality to study the bipartite correlation in an exactly soluble two-dimensional mixed spin system. Bell inequality turns out to be a valuable detector for phase transitions in this model. It can detect not only the quantum phase transition, but also the thermal phase transitions, of the system. The property of bipartite correlation in the system is also analyzed. In the quantum anti-ferromagnetic phase, the Bell inequality is violated thus nonlocality is present. It is interesting that the nonlocality is enhanced by thermal fluctuation, and similar results have not been observed in anti-ferromagnetic phase. In the ferromagnetic phase, the quantum correlation turns out to be very novel, which cannot be captured by entanglement or nonlocality.
We explore quantum and classical correlations along with coherence in the ground states of spin-1 Heisenberg chains, namely the one-dimensional XXZ model and the one-dimensional bilinear biquadratic model, with the techniques of density matrix renormalization group theory. Exploiting the tools of quantum information theory, that is, by studying quantum discord, quantum mutual information, and three recently introduced coherence measures in the reduced density matrix of two nearest neighbor spins in the bulk, we investigate the quantum phase transitions and special symmetry points in these models. We point out the relative strengths and weaknesses of correlation and coherence measures as figures of merit to witness the quantum phase transitions and symmetry points in the considered spin-1 Heisenberg chains. In particular, we demonstrate that, as none of the studied measures can detect the infinite-order Kosterlitz-Thouless transition in the XXZ model, they appear to be able to signal the existence of the same type of transition in the biliear biquadratic model. However, we argue that what is actually detected by the measures here is the SU(3) symmetry point of the model rather than the infinite-order quantum phase transition. Moreover, we show in the XXZ model that examining even single site coherence can be sufficient to spotlight the second-order phase transition and the SU(2) symmetry point.
Multipartite quantum nonlocality and Bell-type inequalities are used to characterize quantum correlations in an infinite-size bond-alternating spin-12\frac{1}{2}21​ 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 $\ensuremath{\alpha}$ 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 K\mathcal{K}K to characterize the finite-temperature nonlocality in the large-nnn limit. We find that in high-temperature regions, K\mathcal{K}K is reduced linearly as the temperature rises.