Entanglement in spin-one Heisenberg chains.pdf

a r X i v : q u a n t - p h / 0 5 0 1 0 3 2 v 1 7 J a n 2 0 0 5 Entanglement in spin-one Heisenberg chains XiaoGuang Wang1, HaiBin Li2, Zhe Sun1, and You-Quan Li1 1, Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, HangZhou 310027, China and 2, Department of Applied Physics, Zhejiang University of technology, HangZhou 310014, China (Dated: February 1, 2008) By using the concept of negativity, we study entanglement in spin-one Heisenberg chains. Both the bilinear chain and the bilinear-biquadratic chain are considered. Due to the SU(2) symmetry, the negativity can be determined by two correlators, which greatly facilitate the study of entanglement properties. Analytical results of negativity are obtained in the bilinear model up to four spins and the two-spin bilinear-biquadratic model, and numerical results of negativity are presented. We determine the threshold temperature before which the thermal state is doomed to be entangled. PACS numbers: 03.65.Ud, 03.67.-a, 75.10.Jm I. INTRODUCTION Since Haldane predicts that the one-dimensional Heisenberg chain has a spin gap for integer spins [1], the physics of quantum spin chains has been the subject of many theoretical and experimental studies. In these studies, the bilinear spin-one Heisenberg model and the bilinear-biquadratic Heisenberg model have played im- portant roles [2, 3, 4]. The corresponding Hamiltonians are given by H1 = N∑ i=1 JSi · Si+1, (1) H2 = N∑ i=1 [ JSi · Si+1 + γ(Si · Si+1)2 ] , (2) respectively. Here, we have assumed the periodic bound- ary condition, and obviously, these two Hamiltonians exhibits a SU(2) symmetry. Moreover, the bilinear- biquadratic model exhibits very rich phase diagram [5]. Recently, the study of entanglement properties in Heisenberg systems have received much attention [6]-[39]. Quantum entanglement lies at the heart of quantum me- chanics, and can be exploited to accomplish some phys- ical tasks such as quantum teleportation [40]. Spin-hal