Two-dimensional spin 1/2 antiferromagnetic Heisenberg models are numerically studied using Entanglement Perturbation Theory, where the ground state wave function is described by a product of local matrices defined at every site, and each matrix is optimized variationally to minimize the energy. We first apply this method for the spin 1/2 antiferromagnetic Heisenberg model on the square lattices to reproduce correctly the known ground state energy and the spin structure factors. Then, we study the spin 1/2 antiferromagnetic Heisenberg model on the triangular lattice with spatially anisotropic nearest neighbor couplings, J and J ', to establish the ground state phase diagram as a function of J'/ J.
The features of the concurrences between the nearest-neighbors and that of the next-nearest-neighbors for the one-dimensional Heisenberg model with next-nearest-neighbor coupling
J are studied as functions of temperature and
J. The two concurrences exhibit a different dependence on
J at the ground state, which could be interpreted from the point of view of the correlation functions. The threshold temperature at which the concurrence is zero and the temperature effect on the two concurrences for systems up to 12 sites are studied numerically.
Quantum entanglement bears some resemblance to ordinary classical correlation, but there are important differences. In particular, whereas any number of classical variables can in principle be perfectly correlated with each other–the temperature fluctuations in ten different cities could conceivably be exactly parallel–any entanglement that exists between two quantum objects has the effect of limiting the entanglement that either of these objects can have with the rest of the world. In this talk I will discuss a number of examples of quantum systems in which this restriction on the sharing of entanglement can be expressed quantitatively. In the special case of three spin-1/2 particles A, B, and C, there is a convenient measure of entanglement called the concurrence in terms of which the trade-off between the AB entanglement and the AC entanglement takes a particularly simple form: the sum of the squares cannot exceed unity. For larger spins, preliminary results suggest that the limitation on entanglement sharing is less stringent. Finally, for a collection of many spin-1/2 particles arranged in a line, the problem of sharing entanglement points to an interesting feature of antiferromagnetic chains, namely, that they favor states with high nearest-neighbor entanglement.
The entanglement and quantum correlation measures have been investigated for the ground state of a spin chain with a Kitaev-type exchange interactions on alternating bonds, along with a transverse magnetic field. There is a macroscopic degeneracy in the ground state for zero magnetic field, implying a quantum critical point. In this model, many entanglement and correlation measures, except the macroscopic multispecies entanglement measure, do not show true singular behavior in the vicinity of the critical point, as seen in the transverse-Ising model ground state and related models. We compute the pair concurrence measure of entanglement, the pair quantum discord to track the quantum correlations, and a global entanglement measure and a multispecies entanglement measure to investigate multiparty entanglement, both analytically and numerically. The nearest-neighbor concurrence shows a peak structure as a function of magnetic field, near the critical point, for various values of the ratio of interaction strengths, but its derivative does not show a singular behavior close to the critical point. A similar behavior is shown by the quantum discord and the global entanglement. The multispecies entanglement shows a clear signature of the critical point behavior, and the first-order derivative shows a divergence near zero magnetic field, for equal interaction strengths.
Exchange interactions in spin systems can give rise to quantum entanglement in the ground and thermal states of the systems. We consider a spin tetramer, with spins of magnitude
21, in which the spins interact via nearest-neighbor, diagonal, and four-spin interactions of strength
J1,
J2, and
K, respectively. The ground- and thermal-state entanglement properties of the tetramer are calculated analytically in the various limiting cases. Both bipartite and multipartite entanglements are considered and a signature of the quantum phase transition (QPT), in terms of the entanglement ratio, is identified. The first-order QPT is accompanied by discontinuities in the nearest-neighbor and diagonal concurrences. The magnetic properties of a
S=21 antiferromagnetic polyoxovanadate compound V12 are well explained by tetramers, with
J2=0,
K=0, in which the spins interact via the isotropic Heisenberg exchange interaction Hamiltonian. Treating the magnetic susceptibility $\ensuremath{\chi}$ as an entanglement witness (EW), an estimate of the lower bound of the critical entanglement temperature
Tc below which entanglement is present in the experimental compound, is determined. Two other cases considered include the symmetric tetramer---i.e., tetrahedron
(J1=J2,K=0)---and the symmetric trimer. In both the cases, there is no entanglement between a pair of spins in the thermal state but multipartite entanglement is present. A second EW based on energy provides an estimate of the entanglement temperature
TE below which the thermal state is definitely entangled. This EW detects bipartite entanglement in the case of the tetramer describing a square of spins (the case of V12 ) and multipartite entanglement in the cases of the tetrahedron and symmetric trimer.
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.