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arXiv:1012.4762 · Physical Review A

Description of thermal entanglement with the static path plus random-phase approximation

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

Published Aug 13, 2007• 16 citations
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Abstract

We discuss the application of the static path plus random-phase approximation $(\mathrm{SPA}+\mathrm{RPA})$ and the ensuing mean $\text{field}+\mathrm{RPA}$ 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 $XXZ$ chain of $n$ spins at finite temperature in a transverse magnetic field $b$. It is shown that already the mean $\text{field}+\mathrm{RPA}$ provides an accurate analytic description of the concurrence below the mean field critical region $(\ensuremath{\mid}b\ensuremath{\mid}<{b}_{c})$, exact for large $n$, whereas the full $\mathrm{SPA}+\mathrm{RPA}$ is able to improve results for finite systems in the critical region. It is proven as well, that for $T>0$ 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 $\ensuremath{\mid}b\ensuremath{\mid}>{b}_{c}$.

DOI
10.1103/physreva.76.022310

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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∣b\ensuremath∣<bc)(\ensuremath{\mid}b\ensuremath{\mid}<{b}_{c})(\ensuremath∣b\ensuremath∣<bc​), 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∣b\ensuremath∣>bc)(\ensuremath{\mid}b\ensuremath{\mid}>{b}_{c})(\ensuremath∣b\ensuremath∣>bc​), with the limit temperature for pairwise entanglement exhibiting quite distinct regimes for \ensuremath∣b\ensuremath∣<bc\ensuremath{\mid}b\ensuremath{\mid}<{b}_{c}\ensuremath∣b\ensuremath∣<bc​ and \ensuremath∣b\ensuremath∣>bc\ensuremath{\mid}b\ensuremath{\mid}>{b}_{c}\ensuremath∣b\ensuremath∣>bc​.
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 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.
The static path plus random phase approximation (SPA+RPA) to the partition function is formulated exactly in a canonical ensemble, for the case of a density decomposition of an arbitrary two-body Hamiltonian. The canonical mean field plus RPA approach is also discussed. Numerical results are shown for a quadrupole interaction, where excellent agreement with the exact canonical results for a light nucleus is obtained. An effective canonical mean field plus RPA approach is in this case also developed, which avoids the sharp phase transition of the ordinary mean field and provides a reliable estimate of full SPA+RPA results.
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.