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D. Lee Department of Physics, Columbia University, New York, NY 10027, USA 李荫远 研究员, 院士 Prof. Academician Li Yin-Yuan 中国科学院物理研究所, 北京 100190 Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 丁肇中 教授, 院士 Prof. Academician Samuel C. C. Ting LEP3, CERN, CH-1211, Geneva 23, Switzerland 杨振宁 教授, 院士 Prof. Academician C. N. Yang Institute for Theoretical Physics, State University of New York, USA 杨福家 教授, 院士 Prof. Academician Yang Fu-Jia 复旦大学物理二系, 上海 200433 Department of Nuclear Physics, Fudan University, Shanghai 200433, China 周光召 研究员, 院士 Prof. Academician Zhou Guang-Zhao (Chou Kuang-Chao) 中国科学技术协会, 北京 100863 China Association for Science and Technology, Beijing 100863, China 王乃彦 研究员, 院士 Prof. Academician Wang Nai-Yan 中国原子能科学研究院, 北京 102413 China Institute of Atomic Energy, Beijing 102413, China 梁敬魁 研究员, 院士 Prof. Academician Liang Jing-Kui 中国科学院物理研究所, 北京 100190 Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China (Continued ) Chin. Phys. B Vol. 22, No. 9 (2013) 090313 The effects of the Dzyaloshinskii–Moriya interaction on the ground-state properties of the XY chain in a transverse field∗ Zhong Ming(钟 鸣)a), Xu Hui(徐 卉)a), Liu Xiao-Xian(刘小贤)a), and Tong Pei-Qing(童培庆)a)b)† a)Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing 210023, China b)Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing 210023, China (Received 19 December 2012; revised manuscript received 27 March 2013) The effects of the Dzyaloshinski–Moriya (DM) interaction on the ground-state properties of the anisotropic XY chain in a transverse field have been studied by means of correlation functions and entanglement. Different from the case without the DM interaction, the excitation spectra ek of this model are not symmetrical in the momentum space and are not always positive. As a result, besides the ferromagnetic (FM) and the paramagnetic (PM) phases, a gapless chiral phase is induced. In the chiral phase, the von Neumann entropy is proportional to log2 L (L is the length of a subchain) with the coefficient A ≈ 1=3, which is the same as that of the XY chain in a transverse field without the DM interaction for g = 0 and 0 < h ≤ 1. And in the vicinity of the critical point between the chiral phase and the FM (or PM) phase, the behaviors of the nearest- neighbor concurrence and its derivative are like those for the anisotropy transition. Keywords: Dzyaloshinskii–Moriya interaction, the XY chain in a transverse field, quantum entanglement, ground-state properties PACS: 03.67.Mn, 75.10.Jm, 73.43.Nq, 75.40.Cx DOI: 10.1088/1674-1056/22/9/090313 1. Introduction correlations[24] of the one-dimensional XY chain in a trans- verse field were studied. Unfortunately, the authors used in- In recent years, the antisymmetric Dzyaloshinskii– correct excitation spectra, which brought doubts to their re- Moriya (DM) interaction has received much attention. The sults. In fact, the excitation spectra and the critical points of DM interaction introduced by Dzyaloshinskii[1] and Moriya[2] the anisotropic XY chain with the DM interaction were firstly is a combination of super-exchange and spin–orbital in- found by Siskens et al.,[25] however the properties of the ex- teractions, which can provide the microscopic mechanism citation spectra did not arouse enough attention. It should be for the coexistence of ferroelectricity and incommensurate noticed that the excitation spectra ek of this model are not sym- magnetism in many multiferroics.[3–5] Some quantum anti- metrical in the momentum space and are not always greater ferromagnetic systems are described by the DM interac- than zero, which may cause some special behaviors of the sys- tion with underlying helical magnetic structures, such as Cu tem. benzoate,[6,7] Yb As ,[8] Cs CuCl ,[9] BaCu Si O ,[10] and 4 3 2 4 2 2 7 On the other hand, quantum entanglement is viewed as a [11,12] kagome antiferromagnets. Other materials, such as an Fe main ingredient in quantum information processes and has at- double layer and a wheel single-molecule magnet, have also tracted considerable interest in various fields of physics. In the [13,14] been studied with the DM interaction. last decade, much attention has been paid to the relationship Besides those real materials, one-dimensional spin between quantum entanglement and quantum phase transitions chains with the DM interaction have also been studied (QPTs) (see Ref. [26] for a review). There are two widely [15–23] extensively. For example, the phase diagram of the used measures of entanglement for spin systems. One is the Ising model (without external field) with the DM interaction concurrence,[27] and the other is the von Neumann entropy.[28] was studied with the quantum renormalization-group (QRG) By using the concurrence, Osterloh et al.[29] found that for method, in which a quantum critical point separates the anti- the anisotropic XY chain in a transverse field, the derivative ferromagnetic phase and the chiral phase.[15] The phase di- of the nearest-neighbor concurrence presents a logarithmic di- agram of the XXZ model with the DM interaction was also vergence for an infinite system in the vicinity of the critical obtained with the QRG approach; three phases (ferromag- point h = 1, which is related to the critical properties of the netic, spin-fluid, and Neel phases) were found.[16] The dy- Ising transition.[30] For the same model, we found that[31] the namic properties of the XXZ and the XY chains with the DM derivative of the nearest-neighbor concurrence changes little interaction have also been studied.[17,18] Very recently, the with the increasing system size N in the vicinity of the critical effects of the DM interaction on the quantum and classical point g = 0 (0 < h ≤ 1) corresponding to the anisotropy transi- ∗Project supported by the National Natural Science Foundation of China (Grant Nos. 11205090 and 11175087) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 12KJB140008). †Corresponding author. E-mail: [email protected] © 2013 Chinese Physical Society and IOP Publishing Ltd http://iopscience.iop.org/cpb http://cpb.iphy.ac.cn 090313-1 Chin. Phys. B Vol. 22, No. 9 (2013) 090313 tion. And Vidal et al. found that[32] the von Neumann entropy by the Bogoliubov transformation SL is a constant in the noncritical region and SL ∼ (clog2 L)=3 c = u + v∗ † : (5) at the critical points. The coefficient is given by the central k khk −kh−k p charge c in the conformal field theory,[33] c = 1=2 at h = 1 ak − bk uk = q = cosqk; while c = 1 at = 0 (0 < h ≤ 1). However, for the anisotropic p g 2(bk − ak bk) XY chain in a transverse field with the DM interaction, the for- ig sink mulas used to calculate the concurrence and the von Neumann vk = = i sinqk; (6) q p entropy should be both improved, which results from the char- 2(bk − ak bk) acteristics of the excitation spectra. We will study the effects 2 2 2 with ak = cosk + h and bk = a + g sin k. of the DM interaction on the entanglement of this model in k detail.