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arXiv:1612.02539 · Journal of Physics B Atomic Molecular and Optical Physics

Entanglement and coherence in a spin- s XXZ system under non-uniform fields

Elías Ríos, R. Rossignoli, N. Canosa

Published Apr 6, 2017• 8 citations
arXivHTMLPDF

Abstract

Abstract We investigate entanglement and coherence in an XXZ spin s pair immersed in a non-uniform transverse magnetic field. The ground state and thermal entanglement phase diagrams are analyzed in detail in both the ferromagnetic and antiferromagnetic cases. It is shown that a non-uniform field can control the energy levels and the entanglement of the corresponding eigenstates, making it possible to entangle the system for any value of the exchange couplings, both at zero and finite temperatures. Moreover, the limit temperature for entanglement is shown to depend only on the difference ∣ h 1 − h 2 ∣ between the fields applied at each spin, leading for T > 0 to a separability stripe in the ( h 1 , h 2 ) field plane such that the system becomes entangled above a threshold value of ∣ h 1 − h 2 ∣ . These results are demonstrated to be rigorously valid for any spin s . On the other hand, the relative entropy of coherence in the standard basis, which coincides with the ground state entanglement entropy at T = 0 for any s , becomes non-zero for any value of the fields at T > 0 , decreasing uniformly for sufficiently high T . A special critical point arising at T = 0 for non-uniform fields in the ferromagnetic case is also discussed.

DOI
10.1088/1361-6455/aa66c3

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Abstract We investigate entanglement and coherence in an XXZ spin s pair immersed in a non-uniform transverse magnetic field. The ground state and thermal entanglement phase diagrams are analyzed in detail in both the ferromagnetic and antiferromagnetic cases. It is shown that a non-uniform field can control the energy levels and the entanglement of the corresponding eigenstates, making it possible to entangle the system for any value of the exchange couplings, both at zero and finite temperatures. Moreover, the limit temperature for entanglement is shown to depend only on the difference ∣ h 1 − h 2 ∣ between the fields applied at each spin, leading for T > 0 to a separability stripe in the ( h 1 , h 2 ) field plane such that the system becomes entangled above a threshold value of ∣ h 1 − h 2 ∣ . These results are demonstrated to be rigorously valid for any spin s . On the other hand, the relative entropy of coherence in the standard basis, which coincides with the ground state entanglement entropy at T = 0 for any s , becomes non-zero for any value of the fields at T > 0 , decreasing uniformly for sufficiently high T . A special critical point arising at T = 0 for non-uniform fields in the ferromagnetic case is also discussed.
We evaluate the exact concurrence between any two spins in a cyclic XXXXXX chain of nnn spins placed in a uniform transverse magnetic field, at both zero and finite temperature, by means of the Jordan-Wigner transformation plus a number-parity-projected statistics. It is shown that, while at T=0T=0T=0 there is always entanglement between any two spins in a narrow field interval before the transition to the aligned state, at low but nonzero temperatures the entanglement remains nonzero for arbitrarily high fields, for any pair separation LLL, although its magnitude decreases exponentially with increasing field. It is also demonstrated that the associated limit temperatures approach a constant nonzero value in this limit, which decreases as ${L}^{\ensuremath{-}2}$ for L⪡nL⪡nL⪡n, but exhibit special finite-size effects for distant qubits $(L\ensuremath{\approx}n∕2)$. Related aspects such as the different behavior of even and odd antiferromagnetic chains, the existence of nnn ground-state transitions, and the thermodynamic limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ are also discussed.
We investigate the behavior of thermal quantum coherence in the Heisenberg XXX model for a two-qubit system placed in independently controllable Inhomogeneous magnetic fields applied to two qubits respectively. We discuss the behavior of quantum coherence by systematically varying the coupling parameter, magnetic field, and temperature for both ferromagnetic and antiferromagnetic cases. The results show the interesting behavior of quantum coherence in a certain range of parameters. Generally, it is observed that quantum correlations decay with temperature, but in the ferromagnetic case with uniform magnetic interaction, it rises with temperature up to a certain threshold value and ultimately it decreases its value to zero. Moreover, it is observed that preserving the quantum coherence for small temperatures is very hard with the increasing magnetic field because, at small temperatures, quantum coherence decays sharply with the increase in magnetic field whereas at larger temperatures it decays completely at fairly large values of the magnetic field. The variation of quantum coherence with uniform magnetic field in the antiferromagnetic case is observed to be Gaussian for larger temperature but at zero or nearly zero temperature, it behaves as a constant function for uniform magnetic field up to a threshold value and then decays to zero with an infinite slope. This shows the signature of quantum phase transition from quantum nature to classicality.
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.
We determine the conditions for the existence of nontransverse factorizing magnetic fields in general spin arrays with anisotropic XYZXYZXYZ couplings of arbitrary range. It is first shown that a uniform, maximally aligned, completely separable eigenstate can exist just for fields ${\mathbit{h}}_{s}$ parallel to a principal plane and forming four straight lines in the field space, with the alignment direction different from that of ${\mathbit{h}}_{s}$ and determined by the anisotropy. Such a state always becomes a nondegenerate ground state for sufficiently strong (yet finite) fields along these lines, in both ferromagnetic and antiferromagnetic-type systems. In antiferromagnetic chains, this field coexists with the nontransverse factorizing field ${\mathbit{h}}_{s}^{\ensuremath{'}}$ associated with a degenerate N\'eel-type separable ground state, which is shown to arise at a level crossing in a finite chain. It is also demonstrated for arbitrary spin that pairwise entanglement reaches full range in the vicinity of both ${\mathbit{h}}_{s}$ and ${\mathbit{h}}_{s}^{\ensuremath{'}}$, vanishing at ${\mathbit{h}}_{s}$ but approaching small yet finite side limits at ${\mathbit{h}}_{s}^{\ensuremath{'}}$, which are analytically determined. The behavior of the block entropy and entanglement spectrum in their vicinity is also analyzed.
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 numerically investigate thermal entanglement of the spins (1/2,1) and (1/2,1/2) in the three-mixed (1/2,1,1/2) anisotropic Heisenberg XXZ spin system on a simple triangular cell under an inhomogeneous magnetic field. We show that the external magnetic field induces strong plateau formation in pairwise thermal entanglement for fixed parameters of Hamiltonian in the case of the ferromagnetic and anti-ferromagnetic interactions. We also observe an unexpected critical point at finite temperature in the thermal entanglement of the spins (1/2,1) for antiferromagnetic case while the entanglement of the spins (1/2,1) for ferromagnetic case and the entanglement of the spins (1/2,1/2) for both ferromagnetic and antiferromagnetic cases almost decays exponentially to zero with increasing temperature. The critical point in entanglement of the spins (1/2,1) for antiferromagnetic case may be signature of the quantum phase transition at finite temperature.