Investigation entanglemenet entropy and entanglement spectrum of the spin-1/2 chain with hexamer modulation of exchange

Document Type : Full length research Paper

Authors

1 Department of Physics, Faculty of Science, Islamic azad University, Takestan Branch, Takestan, Iran

2 Department of Physics, University of Guilan, 41335-1914 Rasht, Iran

Abstract

In this paper, we consider the spin-1/2 Heisenberg chain with alternating exchange in presence of hexamer modulation of exchange on bonds. We studied the effect of uniform external magnetic field on the ground state phase diagram of this model using numerical Lanczos method. The ground state phase diagram consists of three gapless Luttinger Liquid(LL), two gapped plateau phases and a singlet spin phase. It is specified in the presence of additional modulation of exchange, the magnetization curve of system has two plateaus in magnetization diagram that are equal to and of the saturation value, respectively. Here, numerically we compute the entanglement entropy and the entanglement spectrum for a chain with finite number of particle. Numerical results shows that the singlet gapped phase exhibits four times degenerate and the first and second gapped plateau phases indicate two times degenerate.

Keywords


[1] M.Z. Hasan, C.L. Kane, Topological Insulators, Reviews of modern Physics 82 (2010) 3045.
 ##
[2] X.L. Qi, S.C. Zhang, Topological insulators and superconductors, Reviews of modern Physics 83 (2010) 1057.
 ##
[3] C.K. Chiu, J.C.Y. Teo, A.P. Schnyderz, S. Ryu, Classification of topological quantum matter with symmetries, Reviews of modern Physics 88 (2016) 035005.
 ##
 [4] N. Read, D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect, Physical Review B 61 (2000)  10267.
 ##
[5] C. Kane, E. Mele, Quantum Spin Hall Effect in Graphene, Physical Review Letter 95 (2005) 226801.
 ##
[6] B.A. Bernevig, T.L. Hughes, S.C. Zhang, Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells, Science 314 (2006) 1757.
https://science.sciencemag.org/content/314/5806/1757.abstract
 ##
[7] F.D.M. Haldane, Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: semiclassically Quantiz-ed Solitons of the One-Dimensional Easy-Axis Neel State, Physical Review Letter 50 (1983) 1153.
 ##
[8] F. Pollmann, A.M. Turner, E. Berg, M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Physical Review B 81 (2010) 064439.
 ##
[9] A.M. Turner, F. Pollmann, E. Berg, Topological Phases of One-Dimensional Fermions: An Entanglement Point of View, Physical Review B 83 (2011) 075102.
 ##
[10] F. Pollmann, E. Berg, A.M. Turner, M. Oshikawa, Symmetry protection of topological phases in one-dimensional quantum spin systems Physical Review B 85 (2012) 075125.
 ##
[11] A. Kitaev, J. Preskill, Topological entanglement entropy, Physical Review Letter 96, (2006) 110404.
 ##
[12] M. Levin, X.G. Wen, Detecting topological order in a ground state wave function, Physical Review Letter 96 (2006) 110405.
 ##
[13] K. Hida, K. Takano, H. Suzuki, Topological Phases of the Spin-1/2 Ferro-
magnetic-Antiferromagnetic Alternating Heisenberg Chain with Frustrated Next-Nearest-Neighbour Interaction, Journal Of Physics Society Of Japan 82, (2013) 064703.
 ##
[14] H.H. Tu, R. Or'us, Intermediate Haldane phase in spin-2 quantum chains with uniaxial anisotropy, Physical Review B 84, (2011) 140407. https://doi.org/10.1103/PhysRevB.84.140407
 ##
 [15] W. Li, A. Weichselbaum, J.V. Delft, Identifying Symmetry-Protected Topological Order by Entanglement Entropy, Physical Review B 88 (2013) 245121.
 ##
[16] K. Hida, Topological Phases of Spin-1/2 Ferromagnetic–Antiferromagnetic Alternating Heisenberg Chains with Alter-nating Next-Nearest-Neighbour Intera-ction, Journal Of Physics Society Of Japan 85 (2016) 124712.
 ##
[17] G. Torlai, L. Tagliacozzo, G. De Chiara, Dynamics of the entanglement spectrum in spin chains, Journal Of Statistical Mechanics 2014 (2014)06001.
 ##
[18] M.S. Naseri, G.I Japaridze, S Mahdavifar, and SF Shayesteh, Magnetic properties of the spin S= 1/2 Heisenberg chain with hexamer modulation of exchange, Journal Of Physics: Condensed Matter (2012) 116002.
 ##
 [19] H. Li, F.D. M. Haldane, Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States,  Physical Review Letter 101 (2008) 010504.