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arXiv:1202.4324 · Physical Review A

Quantum correlations in collective spin systems

Chen Wang, Yuyu Zhang, Qing‐Hu Chen

Published May 15, 2012• 47 citations
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Abstract

Quantum and classical pairwise correlations in two typical collective spin systems (i.e., the Dicke model and the Lipkin-Meshkov-Glick model) are discussed. These correlations in the thermodynamical limit are obtained analytically and in a finite-size system are calculated numerically. Large-size scaling behavior for the quantum discord itself is observed, which has never been reported in another critical system. A logarithmic diverging behavior for the first derivative of the quantum discord is also found in both models, which might be universal in the second-order quantum phase transition. It is suggested that the pronounced maximum or minimum of first derivative of quantum discord signifies the critical point. Comparisons between the quantum discord and the scaled concurrence are performed. It is shown that the quantum discord is very small in one phase and robust in the other phase, while the scaled concurrence shows maximum at the critical point and decays rapidly when away from the the critical point.

DOI
10.1103/physreva.85.052112

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Quantum and classical pairwise correlations in two typical collective spin systems (i.e., the Dicke model and the Lipkin-Meshkov-Glick model) are discussed. These correlations in the thermodynamical limit are obtained analytically and in a finite-size system are calculated numerically. Large-size scaling behavior for the quantum discord itself is observed, which has never been reported in another critical system. A logarithmic diverging behavior for the first derivative of the quantum discord is also found in both models, which might be universal in the second-order quantum phase transition. It is suggested that the pronounced maximum or minimum of first derivative of quantum discord signifies the critical point. Comparisons between the quantum discord and the scaled concurrence are performed. It is shown that the quantum discord is very small in one phase and robust in the other phase, while the scaled concurrence shows maximum at the critical point and decays rapidly when away from the the critical point.
We discuss the behavior of quantum and classical pairwise correlations in critical systems, with the quantumness of the correlations measured by the quantum discord. We analytically derive these correlations for general real density matrices displaying Z2{Z}_{2}Z2​ symmetry. As an illustration, we analyze both the XXZXXZXXZ and the transverse field Ising models. Finite size as well as infinite chains are investigated and the quantum criticality is discussed. Moreover, we identify the spin functions that govern the correlations. As a further example, we also consider correlations in the Hartree-Fock ground state of the Lipkin-Meshkov-Glick model. It is then shown that both classical correlation and quantum discord exhibit signatures of the quantum phase transitions.
Abstract We study the global quantum discord (GQD) in the Lipkin–Meshkov–Glick (LMG) model at zero and finite temperatures, in which all spins are mutually interacted and introduced in an external magnetic field (denoted by h ). We confirm that the high coordinate number is one of the most distinguishing features of the LMG model, which directly results in the nontrivial behaviors of quantum correlations. We compare the GQD with other quantum correlations measures (such as concurrence, quantum discord, and global entanglement) and find the remarkable difference between them. For instance, we find that GQD spreads in the entire system and captures more information on quantum correlations when comparing with concurrence and quantum (pairwise) discord. We discover that GQD can characterize multipartite correlations in the both broken phase ( h < 1) and the symmetric phase ( h ⩾ 1), while global entanglement and its generalized fail. Moreover, we show that the ground-state GQD can identify second-order quantum phase transitions of the LMG model in the thermodynamic limit. By making the scaling behavior of the GQD in the LMG model analysis, we show that GQD (denoted by G ) scales as G ∼ k ⋅ N + c with k > 0 in the anisotropic cases for any fixed magnetic field. We further show that GQD behaves as G | s n ∼ k ⋅ 1 N + c with k < 0 in the isotropic cases for any Dicke state | s n ⟩. Herein k and c are the fitting parameters. We also find that the thermal stability of the GQD at low temperatures depends on the energy gap. We further reveal that the extraordinary behaviors of the thermal-state GQD in the isotropic LMG model are explained by the contribution theory of the energy levels.
We study the classical correlation (CC) and quantum discord (QD) between two spin subgroups of the Lipkin-Meshkov-Glick (LMG) model in both binary and trinary decompositions of spins. In the case of bipartition, we find that the classical correlations and all the quantum correlations including the QD, the entanglement of formation (EoF) and the logarithmic negativity (LN) are divergent in the same singular behavior at the critical point of the LMG model. In the case of tripartition, however, the classical correlation is still divergent but all the quantum correlation measures remain finite at the critical point. The present result shows that the classical correlation is very robust but the quantum correlation is much frangible to the environment disturbance. The present result may also lead to the conjecture that the classical correlation is responsible for the singularity behavior of physics quantities at critical points of a many-body quantum system.
In this work we explore the quantum correlation quantified by trace distance discord as a measure to analyze the quantum critical behaviors in the Ising-XXZ diamond structure at finite temperatures. It is found that the first-order derivative of the trace distance discord exhibits a maximum around the critical point at finite temperatures. By analyzing the finite-temperature scaling behavior, we show that such a quantum correlation can detect exactly the quantum phase transitions from the entan-gled state in ferrimagnetic phase to an unentangled state in ferrimagnetic phase or to an unentangled state in ferromagnetic phase. The results also indicate that the above two kinds of transitions can be distinguished by the different finite-temperature scaling behaviors. Moreover, we find that the trace distance discord, in contrast to other typical quantum correlations (e.g., concurrence, quantum discord and Hellinger distance), may be more reliable to exactly spotlight the critical points of this model at finite temperatures under certain situations.
The power-law scaling behaviours of the quantum discord and the scaled concurrence are presented in two coupled atom ensembles, which exhibit the Dicke-like quantum phase transition. In the super-excitation phase, the quantum discord reaches its maximum beyond the critical point and then decreases monotonically, which is ascribed to the double-degenerated ground state in the strong coupling limit. However, the scaled concurrence exhibits the maximum atomic quantum entanglement at the critical point. Moreover, we find that the persistence of the quantum discord is suppressed by different asymmetric effects in the super-excitation phase.
Pairwise quantum discord (QD) and classical correlation (CC) are studied in the XY\mathit{XY}XY spin chain with three-spin interaction. We analyze their capability in detecting quantum phase transitions (QPTs) at both zero and finite temperatures and find that the pairwise QD of two neighboring spins is more reliable than that of any other distances in identifying QPTs. Both the QD and CC detect quantum critical points associated with first- and higher-order QPTs caused by field and three-spin interactions at finite temperatures. In addition, we find a different finite-size scaling behavior for QD from previous reports for the transverse field Ising case and show some interesting phenomena of QD and entanglement of formation for finite temperatures.