Please wait a minute...
Chin. Phys. B, 2020, Vol. 29(11): 110308    DOI: 10.1088/1674-1056/abbbde
GENERAL Prev   Next  

Thermal entanglement in a spin-1/2 Ising–Heisenberg butterfly-shaped chain with impurities

Meng-Ru Ma(马梦如), Yi-Dan Zheng(郑一丹), Zhu Mao(毛竹), and Bin Zhou(周斌)
Department of Physics, Hubei University, Wuhan 430062, China
Abstract  

We investigate the effect of impurities on the thermal entanglement in a spin-1/2 Ising–Heisenberg butterfly-shaped chain, where four interstitial Heisenberg spins are localized on the vertices of a rectangular plaquette in a unit block. By using the transfer-matrix approach, we numerically calculate the partition function and the reduced density matrix of this model. The bipartite thermal entanglement between different Heisenberg spin pairs is quantified by the concurrence. We also discuss the fluctuations caused by the impurities through the uniform distribution and the Gaussian distribution. Considering the effects of the external magnetic field, temperature, Heisenberg and Ising interactions as well as the parameter of anisotropy on the thermal entanglement, our results show that comparing with the case of the clean model, in both the two-impurity model and the impurity fluctuation model the entanglement is more robust within a certain range of anisotropic parameters and the region of the magnetic field where the entanglement occurred is also larger.

Keywords:  thermal entanglement      Ising-Heisenberg butterfly-shaped chain      impurities      transfer-matrix approach  
Received:  13 August 2020      Revised:  01 September 2020      Accepted manuscript online:  01 January 1900
Fund: the National Natural Science Foundation of China (Grant No. 12074101), the Science Fund for the New Century Excellent Talents in University of the Ministry of Education of China (Grant No. NCET-11-0960), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20134208110001).
Corresponding Authors:  Corresponding author. E-mail: maozhu@hubu.edu.cn Corresponding author. E-mail: binzhou@hubu.edu.cn   

Cite this article: 

Meng-Ru Ma(马梦如), Yi-Dan Zheng(郑一丹), Zhu Mao(毛竹), and Bin Zhou(周斌) Thermal entanglement in a spin-1/2 Ising–Heisenberg butterfly-shaped chain with impurities 2020 Chin. Phys. B 29 110308

Fig. 1.  

A schematic representation of the spin-1/2 Ising–Heisenberg butterfly-shaped chain. The blue balls denote the Heisenberg particles and the red (orange) lines represent the Heisenberg interaction with (without) impurity, while the black balls denote the Ising spins and the green (black) lines represent the Ising interaction with (without) impurity.

Fig. 2.  

Concurrence C12 as a function of the impurity parameter η for different values of Δ. Here, JH = 2J, J = 1, T = 0.1, B = 4, α = 0, and γ = 0.5.

Fig. 3.  

Concurrences C12 and C13 as a function of the external magnetic field B for different values of temperature T. Here, JH = 2J, J = 1, and α = 0. In panels (a) and (c) we display concurrence for Δ = 0.4, panels (b) and (d) for Δ = 0.5. The solid lines and dot lines correspond to the clean model and the two-impurity model, respectively.

Fig. 4.  

Density plot of concurrence C12 as a function of the external magnetic field B and temperature T with JH = 2J and J = 1. In panels (a) and (b) we display C12 for the model with two impurities, where α = 0, γ = 0.5, and η = –0.5. In panels (c) and (d) we display C12 for the clean model, where α = 0, γ = 0, and η = 0. We set Δ = 0.4 in panels (a) and (c), Δ = 0.5 in panels (b) and (d), respectively. The red and blue solid curves are the contours of C12 = 0 which correspond to the two-impurity model and the clean model, respectively.

Fig. 5.  

Density plot of concurrence C13 as a function of the external magnetic field B and temperature T with JH = 2J and J = 1. In panels (a) and (b) we display C13 for the model with two impurities, where α = 0, γ = 0.5, and η = –0.5. In panels (c) and (d) we display C13 for the clean model, where α = 0, γ = 0, and η = 0. We set Δ = 0.4 in panels (a) and (c), Δ = 0.5 in panels (b) and (d), respectively. The red and blue solid curves are the contours of C13 = 0 which correspond to the two-impurity model and the clean model, respectively.

Fig. 6.  

The concurrence C12 as a function of the external magnetic field B under different values of temperature T with impurity fluctuations. Here, JH = 2J, J = 1 and α = 0. In panels (a) and (b) we present the uniform distribution for two impurities, and the Gaussian distribution in panels (c) and (d). For panels (a) and (c) Δ = 0.4, panels (b) and (d) Δ = 0.5. We take the average of 1000 calculations here.

Fig. 7.  

The concurrence C13 as a function of the external magnetic field B under different values of temperature T with impurity fluctuations. Here, JH = 2J, J = 1 and α = 0. In panels (a) and (b) we present the uniform distribution for two impurities, and the Gaussian distribution in panels (c) and (d). For panels (a) and (c) Δ = 0.4, panels (b) and (d) Δ = 0.5. We take the average of 1000 calculations here.

[1]
Bell J S 1987 Speakable and unspeakable in quantum mechanics Cambridge Cambridge University
[2]
Peres A 2006 Quantum theory: concepts and methods Springer Science & Business Media
[3]
Bennett C H Brassard G Crépeau C Jozsa R Peres A Wootters W K 1993 Phys. Rev. Lett. 70 1895 DOI: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.70.1895
[4]
Boschi D Branca S de Martini F Hardy L Popescu S 1998 Phys. Rev. Lett. 80 1121 DOI: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.80.1121
[5]
Venuti L C Giampaolo S Illuminati F Zanardi P 2007 Phys. Rev. A 76 052328 DOI: 10.1103/PhysRevA.76.052328
[6]
Bennett C H 1992 Phys. Rev. Lett. 68 3121 DOI: 10.1103/PhysRevLett.68.3121
[7]
Shor P W Preskill J 2000 Phys. Rev. Lett. 85 441 DOI: 10.1103/PhysRevLett.85.441
[8]
Lo H K Ma X Chen K 2005 Phys. Rev. Lett. 94 230504 DOI: 10.1103/PhysRevLett.94.230504
[9]
Gisin N Massar S 1997 Phys. Rev. Lett. 79 2153 DOI: 10.1103/PhysRevLett.79.2153
[10]
Scarani V Iblisdir S Gisin N Acin A 2005 Rev. Mod. Phys. 77 1225 DOI: 10.1103/RevModPhys.77.1225
[11]
Wang M H Cai Q Y 2019 Phys. Rev. A 99 012324 DOI: 10.1103/PhysRevA.99.012324
[12]
Wang X 2002 Phys. Rev. A 66 034302 DOI: 10.1103/PhysRevA.66.034302
[13]
Zhou L Song H S Guo Y Q Li C 2003 Phys. Rev. A 68 024301 DOI: 10.1103/PhysRevA.68.024301
[14]
Zhang G F Li S S 2005 Phys. Rev. A 72 034302 DOI: 10.1103/PhysRevA.72.034302
[15]
Vedral V Plenio M B Rippin M A Knight P L 1997 Phys. Rev. Lett. 78 2275 DOI: 10.1103/PhysRevLett.78.2275
[16]
Hill S Wootters W K 1997 Phys. Rev. Lett. 78 5022 DOI: 10.1103/PhysRevLett.78.5022
[17]
Plenio M B Vedral V 1998 Contemp. Phys. 39 431 DOI: 10.1080/001075198181766
[18]
Wootters W K 1998 Phys. Rev. Lett. 80 2245 DOI: 10.1103/PhysRevLett.80.2245
[19]
Coffman V Kundu J Wootters W K 1999 Phys. Rev. A 61 052306 DOI: 10.1103/physreva.61.052306
[20]
Gunlycke D Kendon V Vedral V Bose S 2001 Phys. Rev. A 64 042302 DOI: 10.1103/PhysRevA.64.042302
[21]
Wang X 2001 Phys. Rev. A 64 012313 DOI: 10.1103/PhysRevA.64.012313
[22]
Kargarian M Jafari R Langari A 2008 Phys. Rev. A 77 032346 DOI: 10.1103/PhysRevA.77.032346
[23]
Amico L Fazio R Osterloh A Vedral V 2008 Rev. Mod. Phys. 80 517 DOI: 10.1103/RevModPhys.80.517
[24]
Bethe H 1931 Z. Phys. 71 205 DOI: 10.1007/BF01341708
[25]
Baxter R J 1982 Exactly solved models in statistical mechanics New York Academic Press
[26]
Yeomans J M 1992 Statistical mechanics of phase transitions Clarendon Press
[27]
Rommer S Eggert S 1999 Phys. Rev. B 59 6301 DOI: 10.1103/PhysRevB.59.6301
[28]
Osenda O Huang Z Kais S 2003 Phys. Rev. A 67 062321 DOI: 10.1103/PhysRevA.67.062321
[29]
Santos L F 2003 Phys. Rev. A 67 062306 DOI: 10.1103/PhysRevA.67.062306
[30]
Apollaro T J Plastina F 2006 Phys. Rev. A 74 062316 DOI: 10.1103/PhysRevA.74.062316
[31]
Huang Z Osenda O Kais S 2004 Phys. Lett. A 322 137 DOI: 10.1016/j.physleta.2004.01.022
[32]
Cai J M Zhou Z W Guo G C 2006 Phys. Lett. A 352 196 DOI: 10.1016/j.physleta.2005.11.072
[33]
Hoyos J A Rigolin G 2006 Phys. Rev. A 74 062324 DOI: 10.1103/PhysRevA.74.062324
[34]
Rule K C Reehuis M Gibson M C R Ouladdiaf B Gutmann M J Hoffmann J U Gerischer S Tennant D A Süllow S Lang M 2011 Phys. Rev. B 83 104401 DOI: 10.1103/PhysRevB.83.104401
[35]
Jeschke H Opahle I Kandpal H Valentí R Das H Saha-Dasgupta T Janson O Rosner H Brühl A Wolf B Lang M Richter J Hu S Wang X Peters R Pruschke T Honecker A 2011 Phys. Rev. Lett. 106 217201 DOI: 10.1103/PhysRevLett.106.217201
[36]
Qiao J Zhou B 2015 Chin. Phys. B 24 110306 DOI: 10.1088/1674-1056/24/11/110306
[37]
Zheng Y D Mao Z Zhou B 2017 Chin. Phys. B 26 070302 DOI: 10.1088/1674-1056/26/7/070302
[38]
Rojas O Rojas M Ananikian N S de Souza S M 2012 Phys. Rev. A 86 042330 DOI: 10.1103/PhysRevA.86.042330
[39]
Rojas O Rojas M de Souza S M Torrico J Strečka J Lyra M L 2017 Physica A 486 367 DOI: 10.1016/j.physa.2017.05.099
[40]
Verkholyak T Strečka J 2013 Phys. Rev. B 88 134419 DOI: 10.1103/PhysRevB.88.134419
[41]
Rojas O Strečka J Lyra M L 2013 Phys. Lett. A 377 920 DOI: 10.1016/j.physleta.2013.02.013
[42]
Strečka J Rojas O Verkholyak T Lyra M L 2014 Phys. Rev. E 89 022143 DOI: 10.1103/PhysRevE.89.022143
[43]
Karl’ová K Strečka J Lyra M L 2018 Phys. Rev. B 97 104407 DOI: 10.1103/PhysRevB.97.104407
[44]
Carvalho I M Rojas O de Souza S M Rojas M 2019 Quantum Inf. Process. 18 134 DOI: 10.1007/s11128-019-2253-2
[45]
Freitas M Filgueiras C Rojas M 2019 Ann. Phys. 531 1900261 DOI: 10.1002/andp.201900261
[46]
Zad H A Ananikian N Kenna R 2019 J. Phys.: Condens. Matter 31 445802 DOI: 10.1088/1361-648x/ab3136
[47]
Zheng Y D Mao Z Zhou B 2019 Chin. Phys. B 28 120307 DOI: 10.1088/1674-1056/ab53cc
[1] Entanglement teleportation via a couple of quantum channels in Ising-Heisenberg spin chain model of a heterotrimetallic Fe-Mn-Cu coordination polymer
Yi-Dan Zheng(郑一丹), Zhu Mao(毛竹), Bin Zhou(周斌). Chin. Phys. B, 2019, 28(12): 120307.
[2] Thermal quantum correlations of a spin-1/2 Ising-Heisenberg diamond chain with Dzyaloshinskii-Moriya interaction
Yidan Zheng(郑一丹), Zhu Mao(毛竹), Bin Zhou(周斌). Chin. Phys. B, 2018, 27(9): 090306.
[3] Envelope solitary waves and their reflection and transmission due to impurities in a granular material
Wen-Qing Du(杜文青), Jian-An Sun(孙建安), Yang-Yang Yang(杨阳阳), Wen-Shan Duan(段文山). Chin. Phys. B, 2018, 27(1): 014501.
[4] Thermal entanglement of the spin-1 Ising–Heisenberg diamond chain with biquadratic interaction
Yi-Dan Zheng(郑一丹), Zhu Mao(毛竹), Bin Zhou(周斌). Chin. Phys. B, 2017, 26(7): 070302.
[5] Comparative study on beryllium and magnesium as a co-doping element for ZnO:N
Yu-Quan Su(苏宇泉), Ming-Ming Chen(陈明明), Long-Xing Su(苏龙兴), Yuan Zhu(祝渊), Zi-Kang Tang(汤子康). Chin. Phys. B, 2016, 25(6): 066106.
[6] Influence of vacuum degree on growth of Bi2Te3 single crystal
Tang Yan-Kun, Zhao Wen-Juan, Zhu Hua-Qiang, Huang Yong-Chao, Cao Wei-Wei, Yang Qian, Yao Xiao-Yan, Zhai Ya, Dong Shuai. Chin. Phys. B, 2015, 24(7): 078101.
[7] Thermal entanglement of the Ising–Heisenberg diamond chain with Dzyaloshinskii–Moriya interaction
Qiao Jie, Zhou Bin. Chin. Phys. B, 2015, 24(11): 110306.
[8] A thermal entangled quantum refrigerator based on a two-qubit Heisenberg model with Dzyaloshinskii-Moriya interaction in an external magnetic field
Wang Hao, Wu Guo-Xing. Chin. Phys. B, 2013, 22(5): 050512.
[9] Entangled quantum heat engine based on two-qubit Heisenberg XY model
He Ji-Zhou,He Xian,Zheng Jie. Chin. Phys. B, 2012, 21(5): 050303.
[10] Teleportation and thermal entanglement in two-qubit Heisenberg XYZ spin chain with the Dzyaloshinski–Moriya interaction and the inhomogeneous magnetic field
Gao Dan, Zhao Zhen-Shuang, Zhu Ai-Dong, Wang Hong-Fu, Shao Xiao-Qiang, Zhang Shou. Chin. Phys. B, 2010, 19(9): 090313.
[11] Thermal entanglement in two-qutrit spin-1 anisotropic Heisenberg model with inhomogeneous magnetic field
Erhan Albayrak. Chin. Phys. B, 2010, 19(9): 090319.
[12] Quantum phase transition and entanglement in Heisenberg XX spin chain with impurity
Chen Shi-Rong, Xia Yun-Jie, Man Zhong-Xiao. Chin. Phys. B, 2010, 19(5): 050304.
[13] Influence of annealing conditions on impurity species in arsenic-doped HgCdTe grown by molecular beam epitaxy
Yue Fang-Yu, Chen Lu, Li Ya-Wei, Hu Zhi-Gao, Sun Lin, Yang Ping-Xiong, Chu Jun-Hao. Chin. Phys. B, 2010, 19(11): 117106.
[14] Thermal entanglement in molecular spin rings
Hou Jing-Min, Du Long, Ding Jia-Yan, Zhang Wen-Xin. Chin. Phys. B, 2010, 19(11): 110313.
[15] Pairwise thermal entanglement in a three-qubit Heisenberg XX model with a nonuniform magnetic field and Dzyaloshinski–Moriya interaction
Ren Jin-Zhong, Shao Xiao-Qiang, Zhang Shou, Yeon Kyu-Hwang. Chin. Phys. B, 2010, 19(10): 100307.
No Suggested Reading articles found!