Thermal entanglement between non-nearest-neighbor spins on fractal lattices

Abstract

We investigate the thermal entanglement between two end sites in spin chains and on fractal lattices by taking the negativity as a measure and using the decimation renormalization-group method. The effects of the temperature $T$, the anisotropy parameter $\ensuremath{\Delta}$, and the size of the system, $L$, on the entanglement are examined in detail. It is found that the entanglement decreases monotonically with increasing $T$ and vanishes beyond a critical value ${T}_{c}$. Our results also show that with increasing $\ensuremath{\Delta}$ from $\ensuremath{-}\ensuremath{\infty}$ to zero the entanglement first increases to the maximum and then decreases sharply to zero. Different from the cases of spin chains and Koch curves, the entanglement on the diamond-type hierarchical (DH) lattices presents some interesting behaviors. As the sizes of the DH lattices, $L$, become large, the entanglement is rather robust and there exists sizable entanglement between long-distance end sites. This result indicates that different fractal structures can result in various entanglement properties.