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arXiv:1204.2631

Singularity of classical and quantum correlations at critical points of the Lipkin-Meshkov-Glick model in bipartition and tripartition of spins

Xiu-Xing Zhang, Li Fu-Li

Published Apr 12, 2012• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1204.2631

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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.
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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 investigate the dynamics of quantum discord and entanglement for two central spin qubits coupled to an isotropic Lipkin-Meshkov-Glick bath. It is found that both quantum discord and entanglement have quite distinct behaviors with respect to the two different phases of the bath. In the case of the symmetry broken phase bath, quantum discord and entanglement can remain as constant. In the case of the symmetric phase bath, quantum discord and entanglement always periodically oscillate with time. The critical point of quantum phase transition of the bath can be revealed clearly by the distinct behaviors of quantum correlations. Furthermore, it is observed that quantum discord is significantly enhanced during the evolution while entanglement periodically vanishes.
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.
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.
We examine the ability of quantum discord (QD) and entanglements (concurrence, EoF and negativity) to detect the critical points associated to quantum phase transitions (QPTs) for XY models, i.e., the isotropic XY model with three-spin interactions at zero temperature, and the anisotropic XY model in a transverse magnetic field h at finite temperatures. For the case of zero temperature, we found that both entanglements and QD can spotlight the critical points of QPTs for these two models. Moreover, QD versus distance M exhibits the long-range behavior of quantum correlation for the anisotropic XY model, while entanglement is short-ranged. For the case of finite temperatures, we found that negativity has the same behaviors with concurrence at or near transition points. Moreover, QD for the anisotropic XY model can increase with temperature even in the absence of a magnetic field.