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arXiv:1410.6007 · Physical Review B

Generalized mean-field description of entanglement in dimerized spin systems

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

Published Feb 27, 2015• 5 citations
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Abstract

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-$1/2$ systems with anisotropic ferromagnetic-type $XY$ and $XYZ$ 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 $({S}_{z})$ 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.

DOI
10.1103/physrevb.91.064428

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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 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.
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 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 determine the conditions under which general dimer-type spin chains with XYZXYZXYZ couplings of arbitrary range in a general transverse field will exhibit an exactly separable parity-breaking eigenstate. We also provide sufficient conditions which ensure that it will be a ground state. We then examine the exact side limits at separability of the entanglement between any two spins in a finite chain, showing that in the vicinity of separability, the system will loose all signatures of dimerization, with pairwise entanglement approaching infinite range and becoming independent of separation and interaction range. The possibility of a nonuniform exactly separable ground state induced by an alternating field is also shown. As illustration, we examine the behavior of the pairwise entanglement in a finite XYXYXY dimer chain under a uniform as well as alternating field. Related aspects of the magnetization are also discussed.
We examine the entanglement of cyclic spin-1/2 chains with anisotropic XYZXYZXYZ Heisenberg couplings of arbitrary range at transverse factorizing magnetic fields. At these fields the system exhibits a degenerate symmetry-breaking separable ground state (GS). It is shown, however, that the side limits of the GS pairwise entanglement at these fields are actually nonzero in finite chains, corresponding such fields to a GS spin-parity transition. These limits exhibit universal properties like being independent of the pair separation and interaction range, and are directly related to the magnetization jump. Illustrative exact results are shown for chains with (I) full range and (II) nearest-neighbor couplings. Global entanglement properties at such points are also discussed.
We determine the conditions for the existence of a pair of degenerate parity breaking separable eigenstates in general arrays of arbitrary spins connected through XYZXYZXYZ couplings of arbitrary range and placed in a transverse field, not necessarily uniform. Sufficient conditions under which they are ground states are also provided. It is then shown that in finite chains, the associated definite parity states, which represent the actual ground state in the immediate vicinity of separability, can exhibit entanglement between any two spins regardless of the coupling range or separation, with the reduced state of any two subsystems equivalent to that of pair of qubits in an entangled mixed state. The corresponding concurrences and negativities are exactly determined. The same properties persist in the mixture of both definite parity states. These effects become specially relevant in systems close to the XXZXXZXXZ limit. The possibility of field induced alternating separable solutions with controllable entanglement side limits is also discussed. Illustrative numerical results for the negativity between the first and the jth{j}^{\text{th}}jth spin in an open spin sss chain for different values of sss and jjj are as well provided.