Narrow Results

The four site and eight site 1D anti-ferromagnetic Heisenberg chains in the Jordan–Wigner representation are investigated within the standard Hartree–Fock and random phase approximation (RPA) approaches, both in the symmetry unbroken and in the symmetry broken phases. A translation invariant groundstate, obtained by the projection method as a linear combination of a symmetry-broken HF state and its image under reflection, is also considered, for each chain type. It is found that the projection method considerably improves the HF treatment for instance as far as the groundstate energy is concerned, but also with respect to the RPA energies. The results are furthermore confronted with the ones obtained within so-called SCRPA scheme.

Onsager reaction field theory is used to investigate the one-dimensional ferromagnetic long-range interacting spin chain with the antiferromagnetic nearest-neighbor interaction (NNI) $\tilde{J}$. The ferromagnetic long-range interactions considered in this paper decay as $J/r^p$ with the distance $r$ between lattice sites. It is found that both the zero temperature and finite-temperature phase diagrams of the system are strongly affected by the interplay between ferromagnetic long-range and antiferromagnetic NNIs. The critical temperature and the uniform susceptibility are obtained as a function of $\alpha =\tilde{J}/J$ and $p$. At finite temperatures and in the region $\alpha <{\alpha }_c(p)$ in which ${\alpha }_c(p)$ is dependent of $p$, the ferromagnetic-paramagnetic phase transition survives for $1<p<2$ and no phase transition exists for $p\ge 2$. At $\alpha ={\alpha }_c(p)$, the ferromagnetic-antiferromagnetic phase transition happens at zero temperature for $p>1$. The ground state of the system keeps ferromagnetic when $\alpha <{\alpha }_c(p)$. But for $\alpha >{\alpha }_c$, the system becomes antiferromagnetic at all temperatures.

The entanglement characteristics of the three-qutrit spin-one Heisenberg chain in the presence of one impurity site is investigated by using the concept of negativity. The results show that the existence of impurity can lead to an enhancement of both bipartite and pairwise entanglement for certain ranges of the impurity parameter, and the amount of entanglement can be controlled and tuned by varying the impurity parameter J₁. The results also show that the thermal fluctuations always suppress the entanglement and as a result there exists a threshold temperature T_th after which the entanglement vanishes. T_th increases as J₁ increases except the case for that of the pairwise negativity which measures entanglement between the two norm spins in the chain.

By using the concept of concurrence, we study pairwise entanglement between the two end spins in the open-ended Heisenberg XXX and XY chains up to ten spins. The results show that by introducing two boundary impurities, one can obtain maximum entanglement at the limit of the impurity parameter |J₁| ≪ J for the even-number qubits. When |J₁/J| > 0, the entanglement always decreases with the increase in the absolute value of J₁/J, and for the Heisenberg XXX chain, C disappears when J₁/J exceeds a certain critical point J_ic, and attains an asymptotic value C₀ when |J₁| ≫ J(J₁ < 0), while for the Heisenberg XY chain, C always disappears when |J₁/J| exceeds a certain critical point J_ic. Both C₀ and J_ic decrease with the increase of the length of the chain.

Dynamics of the nearest-neighbor concurrence and the block–block entanglement in closed Heisenberg chains with Dzyaloshinskii–Moriya (DM) interactions are simulated numerically. It is found that the nearest-neighbor concurrence and the block–block entanglement can be generated dynamically from an initial antiferromagnetic (AF) state, and DM interactions will enhance the amplitude of oscillation of concurrence and that of block–block entanglement. Furthermore, dynamics of block–block entanglement S_L in the chain with even number spins is coincident with the dynamics of S_{N - L} for D = 0. However, with the consideration of DM interactions, the evolution of S_L will not be consistent with that of S_{N - L} for the chain with odd number spins. In a given time interval, the maximal value of the generated nearest-neighbor concurrence will first decrease with the increase of chain's length, then oscillate with the increase of N. The maximal value and the minimal value of the oscillation becomes stable for a large N, and of the chain with odd spins is larger than that of the chain with even spins for a large N. The influence of DM interaction on the value of becomes unobvious with the increase of chain's length.

We review recent developments in the study of matrix product states in quantum integrable models. We first clarify the relation between the matrix product ansatz, proposed by Alcaraz and Lazo, and the algebraic Bethe ansatz. In the Heisenberg chain, this relation turns out to be related to the six-vertex model with domain wall boundary conditions. To obtain explicit finite-dimensional representations for the matrices, factorizing F-matrices are necessary. Secondly, we discuss matrix product states in the Lieb-Liniger model describing a one-dimensional Bose gas with delta-function interaction. Although the model is continuous in space, we can construct continuous but finite-dimensional matrix product states from the integrable version of the Bose-Hubbard model. The factorizing F-matrices are again indispensable in this construction.