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Physical Review A

Many-particle entanglement in the gaped antiferromagnetic Lipkin model

Razmik G. Unanyan, C. Ionescu, Michael Fleischhauer

Published Aug 18, 2005• 31 citations
Publisher

Abstract

Bipartite and global entanglement are analyzed for the ground state of a system of $N$ spin-$1∕2$ particles interacting via a collective spin-spin coupling described by the Lipkin-Meshkov-Glick Hamiltonian. Under certain conditions, which include the special case of supersymmetry, the ground state can be constructed analytically. In the case of antiferromagnetic coupling and for an even number of particles, the system has a finite energy gap and the ground state undergoes a smooth transition, as a function of the continuous anisotropy parameter $\ensuremath{\gamma}$, from a separable $(\ensuremath{\gamma}=\ensuremath{\infty})$ to a maximally entangled state $(\ensuremath{\gamma}=0)$. From the analytic expression for the ground state, the bipartite entanglement between two subsets of spins as well as the global entanglement are calculated. Despite the absence of a quantum phase transition a discontinuous change of the scaling of the bipartite entanglement with the number of spins in the subsystem is found at the isotropy point $\ensuremath{\gamma}=0$: While at $\ensuremath{\gamma}=0$ the entanglement grows logarithmically with the system size with no upper bound, it saturates for $\ensuremath{\gamma}\ensuremath{\ne}0$ at a level only depending on $\ensuremath{\gamma}$.

DOI
10.1103/physreva.72.022326

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Bipartite and global entanglement are analyzed for the ground state of a system of NNN spin-1∕21∕21∕2 particles interacting via a collective spin-spin coupling described by the Lipkin-Meshkov-Glick Hamiltonian. Under certain conditions, which include the special case of supersymmetry, the ground state can be constructed analytically. In the case of antiferromagnetic coupling and for an even number of particles, the system has a finite energy gap and the ground state undergoes a smooth transition, as a function of the continuous anisotropy parameter \ensuremathγ\ensuremath{\gamma}\ensuremathγ, from a separable (\ensuremathγ=\ensuremath∞)(\ensuremath{\gamma}=\ensuremath{\infty})(\ensuremathγ=\ensuremath∞) to a maximally entangled state (\ensuremathγ=0)(\ensuremath{\gamma}=0)(\ensuremathγ=0). From the analytic expression for the ground state, the bipartite entanglement between two subsets of spins as well as the global entanglement are calculated. Despite the absence of a quantum phase transition a discontinuous change of the scaling of the bipartite entanglement with the number of spins in the subsystem is found at the isotropy point \ensuremathγ=0\ensuremath{\gamma}=0\ensuremathγ=0: While at \ensuremathγ=0\ensuremath{\gamma}=0\ensuremathγ=0 the entanglement grows logarithmically with the system size with no upper bound, it saturates for \ensuremathγ\ensuremath≠0\ensuremath{\gamma}\ensuremath{\ne}0\ensuremathγ\ensuremath=0 at a level only depending on \ensuremathγ\ensuremath{\gamma}\ensuremathγ.
Bipartite and global entanglement are analyzed for the ground state of a system of NNN spin 1/2 particles interacting via a collective spin-spin coupling described by the Lipkin-Meshkov-Glick (LMG) Hamiltonian. Under certain conditions which includes the special case of a super-symmetry, the ground state can be constructed analytically. In the case of an anti-ferromagnetic coupling and for an even number of particles this state undergoes a smooth crossover as a function of the continuous anisotropy parameter γγγ from a separable ($γ=\infty )toamaximallyentangledmany−particlestate() to a maximally entangled many-particle state ()toamaximallyentangledmany−particlestate(γ=0$). From the analytic expression for the ground state, bipartite and global entanglement are calculated. In the thermodynamic limit a discontinuous change of the scaling behavior of the bipartite entanglement is found at the isotropy point γ=0γ=0γ=0. For % γ=0 the entanglement grows logarithmically with the system size with no upper bound, for γ≠0γ\neq 0γ=0 it saturates at a level only depending on γγγ. For finite systems with total spin J=N/2J=N/2J=N/2 the scaling behavior changes at γ=γcrit=1/Jγ=γ_{\mathrm{crit}}=1/Jγ=γcrit​=1/J.
The multiparticle entanglement in the Lipkin-Meshkov-Glick model has been discussed extensively in this paper. Measured by the global entanglement and its generalization, our calculation shows that the multiparticle entanglement can faithfully detect quantum phase transitions. For an antiferromagnetic case the multiparticle entanglement reaches the maximum at the transition point, whereas for ferromagnetic coupling, two different behaviors of multiparticle entanglement can be identified, dependent on the anisotropic parameter in the coupling.
The entanglement between noncomplementary blocks of a many-body system, where a part of the system forms an ignored environment, is a largely untouched problem without analytic results. We rectify this gap by studying the logarithmic negativity between two macroscopic sets of spins in an arbitrary tripartition of a collection of mutually interacting spins described by the Lipkin-Meshkov-Glick Hamiltonian. This entanglement measure is found to be finite and universal at the critical point for any tripartition whereas it diverges for a bipartition. In this limiting case, we show that it behaves as the entanglement entropy, suggesting a deep relation between the scaling exponents of these two independently defined quantities which may be valid for other systems.
We discuss the behaviour of the entanglement entropy of the ground state in various collective systems. Results for general quadratic two-mode boson models are given, yielding the relation between quantum phase transitions of the system (signalled by a divergence of the entanglement entropy) and the excitation energies. Such systems naturally arise when expanding collective spin Hamiltonians at leading order via the Holstein–Primakoff mapping. In a second step, we analyse several such models (the Dicke model, the two-level Bardeen–Cooper–Schrieffer model, the Lieb–Mattis model and the Lipkin–Meshkov–Glick model) and investigate the properties of the entanglement entropy over the whole parameter range. We show that when the system contains gapless excitations the entanglement entropy of the ground state diverges with increasing system size. We derive and classify the scaling behaviours that can be met.
Lipkin-Meshkov-Glick (LMG) models describe a set of NNN qubits which are located at the corners of an NNN-dimensional simplex and embedded in an external magnetic field. Because of its high coordinate number, quantum correlations in the models are quite nontrivial. For instance, in the large-NNN limit, two-site entanglement concurrence vanishes for any magnetic field. Multisite global entanglement and generalized global entanglement also tend to vanish in some quantum regions. We characterize the quantum correlations in the LMG model with multipartite nonlocality. In LMG models with anisotropy in the $x\ensuremath{-}y$ plane, for any fixed magnetic field, the ground-state nonlocality (denoted by S\mathcal{S}S) is found to scale as ${log}_{10}\mathcal{S}\ensuremath{\sim}aN+b$, with aaa and bbb the fitting parameters. For LMG models which are isotropic in the $x\ensuremath{-}y$ plane, nevertheless, the nonlocality for each Dicke state scales as ${log}_{10}\mathcal{S}\ensuremath{\sim}a\frac{1}{N}+b$, with b>0b>0b>0. Signals for the quantum phase transitions of the models will also be discussed. These results indicate that multipartite nonlocality captures some key ingredient of the ground-state quantum correlations in the models. In addition to the ground states, we also study multipartite nonlocality in the LMG models at finite temperatures. In the anisotropic models, the thermal stability of the nonlocality at low temperatures is determined by the energy gap. For the isotropic models, nevertheless, as the temperature increases, the nonlocality presents a sharp valley which is followed by a round peak. The behavior is far beyond the above-mentioned ``thermal stability vs energy gap'' picture. The underlying mechanics is the ``contribution inversion'' of low-lying energy states in the thermal-state nonlocality $\mathcal{S}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{T})$; that is, in a low-temperature region, the first-excited state ``defeats'' the ground state and plays a dominant role in thermal-state nonlocality.