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arXiv:0709.2357 · Physical Review A

Two-spin-subsystem entanglement in spin-1/2 rings with long-range interactions

Marcos E. Gaudiano, Omar Osenda, G. A. Raggio

Published Feb 15, 2008• 12 citations
arXivHTMLPDF

Abstract

We consider the two-spin-subsystem entanglement for eigenstates of the Hamiltonian $H={\ensuremath{\sum}}_{1\ensuremath{\le}j<k\ensuremath{\le}N}{(\frac{1}{{r}_{j,k}})}^{\ensuremath{\alpha}}{\ensuremath{\sigma}}_{j}\ensuremath{\cdot}{\ensuremath{\sigma}}_{k}$ for a ring of $N$ spin-1/2 particles with associated spin vector operator $(\ensuremath{\hbar}/2){\ensuremath{\sigma}}_{j}$ for the $j$th spin. Here ${r}_{j,k}$ is the chord distance between sites $j$ and $k$. The case $\ensuremath{\alpha}=2$ corresponds to the solvable Haldane-Shastry model whose spectrum has very high degeneracies not present for $\ensuremath{\alpha}\ensuremath{\ne}2$. Two-spin-subsystem entanglement shows high sensitivity and distinguishes $\ensuremath{\alpha}=2$ from $\ensuremath{\alpha}\ensuremath{\ne}2$. There is no entanglement beyond nearest neighbors for all eigenstates when $\ensuremath{\alpha}=2$. Whereas for $\ensuremath{\alpha}\ensuremath{\ne}2$ one has selective entanglement at any distance for eigenstates of sufficiently high energy in a certain interval of $\ensuremath{\alpha}$ which depends on the energy. The ground state (which is a singlet only for even $N$) does not have entanglement beyond nearest neighbors, and the nearest-neighbor entanglement is virtually independent of the range of the interaction controlled by $\ensuremath{\alpha}$.

DOI
10.1103/physreva.77.022109

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We consider the two-spin-subsystem entanglement for eigenstates of the Hamiltonian H=\ensuremath∑1\ensuremath≤j<k\ensuremath≤N(1rj,k)\ensuremathα\ensuremathσj\ensuremath⋅\ensuremathσkH={\ensuremath{\sum}}_{1\ensuremath{\le}j<k\ensuremath{\le}N}{(\frac{1}{{r}_{j,k}})}^{\ensuremath{\alpha}}{\ensuremath{\sigma}}_{j}\ensuremath{\cdot}{\ensuremath{\sigma}}_{k}H=\ensuremath∑1\ensuremath≤j<k\ensuremath≤N​(rj,k​1​)\ensuremathα\ensuremathσj​\ensuremath⋅\ensuremathσk​ for a ring of NNN spin-1/2 particles with associated spin vector operator (\ensuremathℏ/2)\ensuremathσj(\ensuremath{\hbar}/2){\ensuremath{\sigma}}_{j}(\ensuremathℏ/2)\ensuremathσj​ for the jjjth spin. Here rj,k{r}_{j,k}rj,k​ is the chord distance between sites jjj and kkk. The case \ensuremathα=2\ensuremath{\alpha}=2\ensuremathα=2 corresponds to the solvable Haldane-Shastry model whose spectrum has very high degeneracies not present for \ensuremathα\ensuremath≠2\ensuremath{\alpha}\ensuremath{\ne}2\ensuremathα\ensuremath=2. Two-spin-subsystem entanglement shows high sensitivity and distinguishes \ensuremathα=2\ensuremath{\alpha}=2\ensuremathα=2 from \ensuremathα\ensuremath≠2\ensuremath{\alpha}\ensuremath{\ne}2\ensuremathα\ensuremath=2. There is no entanglement beyond nearest neighbors for all eigenstates when \ensuremathα=2\ensuremath{\alpha}=2\ensuremathα=2. Whereas for \ensuremathα\ensuremath≠2\ensuremath{\alpha}\ensuremath{\ne}2\ensuremathα\ensuremath=2 one has selective entanglement at any distance for eigenstates of sufficiently high energy in a certain interval of \ensuremathα\ensuremath{\alpha}\ensuremathα which depends on the energy. The ground state (which is a singlet only for even NNN) does not have entanglement beyond nearest neighbors, and the nearest-neighbor entanglement is virtually independent of the range of the interaction controlled by \ensuremathα\ensuremath{\alpha}\ensuremathα.
Consider a ring of N qubits in a translationally invariant quantum state. We ask to what extent each pair of nearest neighbors can be entangled. Under certain assumptions about the form of the state, we find a formula for the maximum possible nearest-neighbor entanglement. We then compare this maximum with the entanglement achieved by the ground state of an antiferromagnetic ring consisting of an even number of spin-1/2 particles. We find that, though the antiferromagnetic ground state typically does not maximize the nearest-neighbor entanglement relative to all other states, it does so relative to other states having zero z component of spin.
We consider N initially disentangled spins, embedded in a ring or d-dimensional lattice of arbitrary geometry, which interact via some long-range Ising-type interaction. We investigate relations between entanglement properties of the resulting states and the distance dependence of the interaction in the limit N-->infinity. We provide a sufficient condition when bipartite entanglement between blocks of L neighboring spins and the remaining system saturates and determine S(L) analytically for special configurations. We find an unbounded increase of S(L) as well as diverging correlation and entanglement length under certain circumstances. For arbitrarily large N, we can efficiently calculate all quantities associated with reduced density operators of up to ten particles.
We investigate the entanglement properties of finite spin rings, with noncollinear Ising interaction between nearest neighbors. The orientations of the Ising axes are determined either by the spin position within the ring (model AAA) or by the direction of the bond (model BBB). In both cases, the considered spin Hamiltonians have a point-group symmetry instead of the translation invariance that characterizes spin rings with collinear Ising interaction. The ground state of these models exhibit remarkable entanglement properties, resembling Greenberger-Horne-Zeilinger (GHZ)-like states in the absence of an applied magnetic field (model BBB). Besides, the application of a homogeneous magnetic field allows to modify qualitatively the character of the ground-state entanglement, switching from multipartite to pairwise quantum correlations (both models AAA and BBB).
In this chapter, we discuss the static and dynamic properties of the spin chain which is often referred to as the Haldane–Shastry model [77, 161]. In contrast with more familiar spin chains with the nearest-neighbor exchange interaction, the Haldane–Shastry model has a particular form of the longrange exchange interaction. In spite of its peculiar shape, it has turned out that the Haldane–Shastry model is the most fundamental one-dimensional spin system. We shall start with a discussion of how the model is solved for the ground state in Sections 4.1 to 4.3, and proceed to static correlation functions in Sections 4.4 and 4.5. The excitation spectrum is interpreted by magnons in Section 4.6, and by spinons in Section 4.7. The spinon picture gives the correct degeneracy of the energy levels in Section 4.8, and leads to the derivation of thermodynamics in Section 2.4, and the dynamical correlation function in Section 4.11. Most results have been obtained in an analytic form without any approximation. There is a close connection between the spectrum of the spin chain and that of the Sutherland model. The most remarkable fact is that the spin chain has a correspondence with two different values of the coupling parameter λ in the Sutherland model. Namely, the spectrum is mapped to the case with either λ = 2 or λ = ∞. In the latter case, particles crystallize with equal spacing, and a small oscillation from the equilibrium, as well as exchange interactions, make up the spectrum. In both cases of λ, however, the degeneracy of each level in the spin chain is not reproduced. Hence, the mapping as it stands cannot be used for thermodynamics.
Most quantum system with short-ranged interactions show a fast decay of entanglement with the distance. In this Letter, we focus on the peculiarity of some systems to distribute entanglement between distant parties. Even in realistic models, like the spin-1 Heisenberg chain, sizable entanglement is present between arbitrarily distant particles. We show that long-distance entanglement appears for values of the microscopic parameters which do not coincide with known quantum critical points, hence signaling a transition detected only by genuine quantum correlations.