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arXiv:1104.5694 · Physical Review A

Thermal entanglement in fully connected spin systems and its random-phase-approximation description

J. M. Matera, R. Rossignoli, N. Canosa

Published Jul 10, 2008• 17 citations
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Abstract

We examine the thermal pairwise entanglement in a symmetric system of $n$ spins fully connected through anisotropic $XYZ$-type couplings embedded in a transverse magnetic field. We consider both the exact evaluation together with that obtained with the static $\text{path}+\text{random}$ phase approximation (RPA) and the ensuing mean $\text{field}+\text{RPA}$. The latter is shown to provide an accurate analytic description of both the parallel and antiparallel thermal concurrence in large systems. We also analyze the limit temperature for pairwise entanglement, which is shown to increase for large fields and to decrease logarithmically with increasing $n$. Special finite-size effects are also discussed.

DOI
10.1103/physreva.78.012316

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We examine the thermal pairwise entanglement in a symmetric system of nnn spins fully connected through anisotropic XYZXYZXYZ-type couplings embedded in a transverse magnetic field. We consider both the exact evaluation together with that obtained with the static path+random\text{path}+\text{random}path+random phase approximation (RPA) and the ensuing mean field+RPA\text{field}+\text{RPA}field+RPA. The latter is shown to provide an accurate analytic description of both the parallel and antiparallel thermal concurrence in large systems. We also analyze the limit temperature for pairwise entanglement, which is shown to increase for large fields and to decrease logarithmically with increasing nnn. Special finite-size effects are also discussed.
We discuss the application of the static path plus random-phase approximation (SPA+RPA)(\mathrm{SPA}+\mathrm{RPA})(SPA+RPA) and the ensuing mean field+RPA\text{field}+\mathrm{RPA}field+RPA treatment to the evaluation of entanglement in composite quantum systems at finite temperature. These methods involve just local diagonalizations and the determination of the generalized collective vibrational frequencies. As an illustration, we evaluate the pairwise entanglement in a fully connected XXZXXZXXZ chain of nnn spins at finite temperature in a transverse magnetic field bbb. It is shown that already the mean field+RPA\text{field}+\mathrm{RPA}field+RPA provides an accurate analytic description of the concurrence below the mean field critical region $(\ensuremath{\mid}b\ensuremath{\mid}<{b}_{c})$, exact for large nnn, whereas the full SPA+RPA\mathrm{SPA}+\mathrm{RPA}SPA+RPA is able to improve results for finite systems in the critical region. It is proven as well, that for T>0T>0T>0 weak entanglement also arises when the ground state is separable $(\ensuremath{\mid}b\ensuremath{\mid}>{b}_{c})$, with the limit temperature for pairwise entanglement exhibiting quite distinct regimes for $\ensuremath{\mid}b\ensuremath{\mid}<{b}_{c}$ and $\ensuremath{\mid}b\ensuremath{\mid}>{b}_{c}$.
We discuss a general mean field plus random phase approximation (RPA) for describing composite systems at zero and finite temperature. We analyze in particular its implementation in finite systems invariant under translations, where for uniform mean fields it requires just the solution of simple local-type RPA equations. As test and application, we use the method for evaluating the entanglement between two spins in cyclic spin--1/2 chains with both long- and short-range anisotropic XYXYXY-type couplings in a uniform transverse magnetic field. The approach is shown to provide an accurate analytic description of the concurrence for strong fields, for any coupling range, pair separation, or chain size, where it predicts an entanglement range which can be at most twice that of the interaction. It also correctly predicts the existence of a separability field together with full entanglement range in its vicinity. The general accuracy of the approach improves as the range of the interaction increases.
We examine the entanglement of thermal states of nnn spins interacting through different types of XYXYXY couplings in the presence of a uniform magnetic field, by evaluating the negativities of all possible bipartite partitions of the whole system and of subsystems. We consider both the case where every qubit interacts with all others and where just nearest neighbors interact in a one-dimensional chain. Limit temperatures for nonzero negativities are also evaluated and compared with the mean field critical temperature. It is shown that limit temperatures of global negativities are strictly independent of the magnetic field in all XXZXXZXXZ models, in spite of the quantum transitions that these models may exhibit at zero temperature, while in anisotropic models they always increase for sufficiently large fields. Results also show that these temperatures are higher than those limiting pairwise entanglement.
We discuss a general treatment based on the mean field plus random-phase approximation (RPA) for the evaluation of subsystem entropies and negativities in ground states of spin systems. The approach leads to a tractable general method that becomes straightforward in translationally invariant arrays. The method is examined in arrays of arbitrary spin with XYZ\mathit{XYZ}XYZ couplings of general range in a uniform transverse field, where the RPA around both the normal and parity-breaking mean-field state, together with parity-restoration effects, is discussed in detail. In the case of a uniformly connected XYZ\mathit{XYZ}XYZ array of arbitrary size, the method is shown to provide simple analytic expressions for the entanglement entropy of any global bipartition, as well as for the negativity between any two subsystems, which become exact for large spin. The limit case of a spin sss pair is also discussed.
We examine the entanglement of thermal states of n spins interacting through an XYZ type Heisenberg coupling in the presence of a uniform magnetic field, by evaluating the negativities of bipartite partitions of the whole system and subsystems. The corresponding limit temperatures for entanglement are also examined. Results indicate that limit temperatures for global entanglement depend on the type of partition and are higher than those limiting pairwise entanglement, and that their behavior with anisotropy and applied magnetic field may differ significantly from that of the corresponding mean field critical temperature.
The thermal pairwise entanglement (TE) of the S=1/2S=1/2S=1/2 XYXYXY chain in a transverse magnetic field is exactly resolved by means of the Jordan-Wigner transformation in the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. It is found that the TE vanishes at a fixed point with temperature ${T}_{c}\ensuremath{\simeq}0.484\text{ }3J$, which is independent of the magnetic field. A thermal quantity is proposed to witness the entangled state. Furthermore, the TE of the S=1/2S=1/2S=1/2 antiferromagnetic-ferromagnetic (AF-F) Heisenberg chain is studied by the transfer-matrix renormalization-group method. The TEs of the spins coupled by AF and F interactions are found to behave distinctively. The vanishing temperature of the field-induced TE of the spins coupled by F interactions is observed dependent on the magnetic field. The results are further confirmed and analyzed within a mean-field framework.