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arXiv:1104.3853 · Physical Review A

Evaluation of pairwise entanglement in translationally invariant systems with the random phase approximation

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

Published Oct 20, 2008• 7 citations
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Abstract

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 $XY$-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.

DOI
10.1103/physreva.78.042319

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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 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 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 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 discuss a generalized self-consistent mean-field (MF) treatment, based on the selection of an arbitrary subset of operators for representing the system density matrix, and its application to the problem of entanglement evaluation in composite quantum systems. As a specific example, we examine in detail a pair MF approach to the ground state (GS) of dimerized spin-1/21/21/2 systems with anisotropic ferromagnetic-type XYXYXY and XYZXYZXYZ couplings in a transverse field, including chains and arrays with first neighbor and also longer range couplings. The approach is fully analytic and able to capture the main features of the GS of these systems, in contrast with the conventional single-spin MF. Its phase diagram differs significantly from that of the latter, exhibiting (Sz)({S}_{z})(Sz​) parity breaking just in a finite field window if the coupling between pairs is sufficiently weak, together with a fully dimerized phase below this window and a partially aligned phase above it. It is then shown that through symmetry restoration, the approach is able to correctly predict not only the concurrence of a pair, but also its entanglement with the rest of the chain, which shows a pronounced peak in the parity breaking window. Perturbative corrections allow to reproduce more subtle observables like the entanglement between weakly coupled spins and the low lying energy spectrum. All predictions are tested against exact results for finite systems.
We examine the entanglement entropy of the even half of a translationally invariant finite chain or lattice in its ground state. This entropy measures the entanglement between the even and odd halves (each forming a ``comb'' of n/2n/2n/2 sites) and can be expected to be extensive for short-range couplings away from criticality. We first consider bosonic systems with quadratic couplings, where analytic expressions for arbitrary dimensions can be provided. The bosonic treatment is then applied to finite spin chains and arrays by means of the random-phase approximation. Results for first-neighbor anisotropic XY\mathit{XY}XY couplings indicate that, while at strong magnetic fields this entropy is strictly extensive, at weak fields important deviations arise, stemming from parity-breaking effects and the presence of a factorizing field (in the vicinity of which it becomes size-independent and identical to the entropy of a contiguous half). Exact numerical results for small spin sss chains are shown to be in agreement with the bosonic random-phase approximation prediction.
We examine the pair entanglement in the ground state of finite dimerized spin-sss chains interacting through anisotropic XYXYXY couplings immersed in a transverse magnetic field by means of a self-consistent pair mean-field approximation. The approach, which makes no a priori assumptions on the pair states, predicts, for sufficiently low coupling between pairs, 2s2s2s distinct dimerized phases for increasing fields below the pair factorizing field, separated by spin-parity-breaking phases. The dimerized phases lead to approximate magnetization and pair entanglement plateaus, while the parity-breaking phases are characterized by weak pair entanglement but non-negligible entanglement of the pair with the rest of the system. These predictions are confirmed by the exact results obtained in finite s=1s=1s=1 and s=3/2s=3/2s=3/2 chains. It is also shown that for increasing values of the spin sss, the entanglement of an isolated pair, as measured by the negativity, rapidly saturates in the anisotropic XYXYXY case but increases as s1/2{s}^{1/2}s1/2 in the XXXXXX case, reflecting a distinct single-spin entanglement spectrum.