Finite-temperature entanglement for low-dimensional quantum spin chains

Abstract

Based on transfer-matrix density matrix renormalization group (TMRG) method, a general procedure to calculate the finite-temperature pairwise entanglement of low-dimensional quantum chains is proposed. The reduced pairwise density matrix is reconstructed with TMRG, and measures of quantum entanglement can be calculated from the pairwise density matrix. The finite-temperature entanglement of the diamond chain model and the spin trimerized model, which are two typical models revealing $1∕3$ plateaus in the magnetization curves, is calculated. For the diamond chain model, the anisotropy coefficient $\ensuremath{\Delta}$ is found to have a great effect on the appearance of the magnetization plateau, and the plateau disappears when $\ensuremath{\Delta}=0.5$. Moreover, our results show that the pairwise entanglement can provide information complementary to that obtained from bulk properties. For the trimerized model, the temperature dependence of the pairwise entanglement is calculated, and the threshold temperature ${T}_{c}$, above which the thermal entanglement vanishes, is found to be independent of the external magnetic field $B$. In addition, the scaling behavior of the thermal entanglement is calculated in the Trotter space. With the augmentation of the system in the Trotter direction, we find that the low-temperature entanglement shows obvious variation in the vicinity of quantum phase transition (QPT) point ${B}_{c}$ and converges fast in noncritical regions, which provides another way to identify QPT of one-dimensional quantum systems.