具有Dzyaloshinskii-Moriya相互作用的XY模型的量子相干性

伊天成; 丁悦然; 任杰; 王艺敏; 尤文龙

doi:10.7498/aps.67.20172755

摘要

研究了具有Dzyaloshinskii-Moriya（DM）相互作用的一维横场XY自旋链的量子相变和量子相干性.采用约旦-维格纳变换严格求解了哈密顿量，并描绘了体系的关联函数和相图，相图包含反铁磁相、顺磁相和螺旋相.利用相对熵和Jensen-Shannon熵讨论了XY模型的量子相干性.研究发现，相对熵与Jensen-Shannon熵所表现的行为都可以很好地表征该模型的量子相变.非螺旋相中量子相干性不依赖DM相互作用，而在螺旋相DM相互作用对量子相干性有显著影响.此外，指出了在带有DM相互作用的这一类反射对称破缺体系中关联函数计算的常见问题.

关键词:

Abstract

In this paper, we study the quantum coherence of one-dimensional transverse XY model with Dzyaloshinskii-Moriya interaction, which is given by the following Hamiltonian:HXY=∑i=1N((1+γ/2) σixσi+1x+(1-γ/2) σiyσi+1y-hσiz) ∑i=1ND(σixσi+1y-σiyσi+1x).(8)Here, 0 ≤ γ ≤ 1 is the anisotropic parameter, h is the magnitude of the transverse magnetic field, D is the strength of Dzyaloshinskii-Moriya (DM) interaction along the z direction. The limiting cases such as γ=0 and 1 reduce to the isotropic XX model and the Ising model, respectively. We use the Jordan-Winger transform to map explicitly spin operators into spinless fermion operators, and then adopt the discrete Fourier transform and the Bogoliubov transform to solve the Hamiltonian Eq.(8) analytically. When the DM interactions appear, the excitation spectrum becomes asymmetric in the momentum space and is not always positive, and thus a gapless chiral phase is induced. Based on the exact solutions, three phases are identified by varying the parameters:antiferromagnetic phase, paramagnetic phase, and gapless chiral phase. The antiferromagnetic phase is characterized by the dominant x-component nearest correlation function, while the paramagnetic phase can be characterized by the z component of spin correlation function. The two-site correlation functions Grxy and Gryx (r is the distance between two sites) are nonvanishing in the gapless chiral phase, and they act as good order parameters to identify this phase. The critical lines correspond to h=1, γ=2D, and h=√4D2 -γ2 + 1 for γ>0. When γ=0, there is no antiferromagnetic phase. We also find that the correlation functions undergo a rapid change across the quantum critical points, which can be pinpointed by the first-order derivative. In addition, Grxy decreases oscillatingly with the increase of distance r. The correlation function Grxy for γ=0 oscillates more dramatically than for γ=1. The upper boundary of the envelope is approximated as Grxy~r-1/2, and the lower boundary is approximately Grxy~r-3/2, so the long-range order is absent in the gapless chiral phase. Besides, we study various quantum coherence measures to quantify the quantum correlations of Eq.(8). One finds that the relative entropy CRE and the Jensen-Shannon entropy CJS are able to capture the quantum phase transitions, and quantum critical points are readily discriminated by their first derivative. We conclude that both quantum coherence measures can well signify the second-order quantum phase transitions. Moreover, we also point out a few differences in deriving the correlation functions and the associated density matrix in systems with broken reflection symmetry.

Keywords:

作者及机构信息

1.
苏州大学物理与光电·能源学院, 苏州 215006;

2.
常熟理工学院物理系, 常熟 215500;

3.
陆军工程大学通信工程学院, 南京 210007;

4.
苏州大学, 江苏省薄膜材料重点实验室, 苏州 215006

通信作者: 尤文龙, wlyou@suda.edu.cn

基金项目: 国家自然科学基金（批准号：11474211，61674110，11374043，11404407）资助的课题.

Authors and contacts

1.
College of Physics, Optoelectronics and Energy, Soochow University, Suzhou 215006, China;

2.
Department of Physics, Changshu Institute of Technology, Changshu 215500, China;

3.
College of Communications Engineering, The Army Engineering University of PLA, Nanjing 210007, China;

4.
Jiangsu Key Laboratory of Thin Films, Soochow University, Suzhou 215006, China

Corresponding author: You Wen-Long, wlyou@suda.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11474211, 61674110, 11374043, 11404407).

参考文献

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[2]	Alexander S, Gerardo A, Martin B P 2017 Rev. Mod. Phys. 89 041003
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[9]	Gu S J, Lin H Q, Li Y Q 2003 Phys. Rev. A 68 042330
[10]	Vidal G, Latorre G I, Rico E, Kitaev A 2003 Phys. Rev. Lett. 90 227902
[11]	Vidal J, Palacios G, Mosseri R 2004 Phys. Rev. A 69 022107
[12]	Ollivier H, Zurek W H 2001 Phys. Rev. Lett. 88 017901
[13]	Modi K, Brodutch A, Cable H, Paterek T, Vedral V 2012 Rev. Mod. Phys. 84 1655
[14]	You W L, Li Y W, Gu S J 2007 Phys. Rev. E 76 022101
[15]	Gu S J, Int J 2010 Mod. Phys. B 24 4371
[16]	Eisert J, Cramer M, Plenio M B 2010 Rev. Mod. Phys. 82 277
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[22]	Moriya T 1960 Phys. Rev. Lett. 4 288
[23]	Seki S, Yu X Z, Ishiwata S, Tokura Y 2012 Science 336 198
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[25]	Yang J H, Li Z L, Lu X Z, Whangbo M H, Wei S H, Gong X G, Xiang H J 2012 Phys. Rev. Lett. 109 107203
[26]	Matsuda M, Fishman R S, Hong T, Lee C H, Ushiyama T, Yanagisawa Y, Tomioka Y, Ito T 2012 Phys. Rev. Lett. 109 067205
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[29]	You W L, Dong Y L 2010 Eur. Phys. J. D 57 439
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[35]	Brockmann M, Klumper A, Ohanyan V 2013 Phys. Rev. B 87 054407
[36]	Derzhko O, Verkholyak T, Krokhmalskii T, Bttner H 2006 Phys. Rev. B 73 214407
[37]	Barouch E, McCoy B M 1970 Phys. Rev. A 2 1075
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[39]	Its A R, Izergin A G, Korepin V E, Slavnov N A 1993 Phys. Rev. Lett. 70 1704
[40]	Bunder J E, McKenzie R H 1999 Phys. Rev. B 60 344
[41]	Vedral V 2002 Rev. Mod. Phys. 74 197
[42]	Horodecki R, Horodecki P, Horodecki M, Horodecki K 2009 Rev. Mod. Phys. 81 865
[43]	Modi K, Brodutch A, Cable H, Paterek T, Vedral V 2012 Rev. Mod. Phys. 84 1655
[44]	Liu B Q, Shao B, Li J G, Zou J, Wu L A 2011 Phys. Rev. A 83 052112
[45]	Radhakrishnan C, Ermakov I, Byrnes T 2017 Phys. Rev. A 96 012341
[46]	You W L, Qiu Y C, Oleś A M 2016 Phys. Rev. B 93 214417
[47]	You W L, Zhang C J, Ni W, Gong M, Oleś A M 2017 Phys. Rev. B 95 224404
[48]	Lei S, Tong P 2015 Physica B 463 1
[49]	Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113 140401
[50]	Chen J, Cui J, Zhang Y, Fan H 2016 Phys. Rev. A 94 022112
[51]	Lamberti P W, Majtey A P, Borras A, Casas M, Plastino A 2008 Phys. Rev. A 77 052311

施引文献

[1]	Alexander S, Uttam S, Himadri S D, Manabendra N B, Gerardo A 2015 Phys. Rev. Lett. 115 020403
[2]	Alexander S, Gerardo A, Martin B P 2017 Rev. Mod. Phys. 89 041003
[3]	Amico L, Fazio R, Osterloh A, Vedral V 2008 Rev. Mod. Phys. 80 517
[4]	Shan C J, Man Z X, Xia Y J, Liu T K 2007 Int. J. Quant. Inform. 5 335
[5]	Ekert A K 1991 Phys. Rev. Lett. 67 661
[6]	Wooters W K, Zurek W H 1982 Nature 299 802
[7]	Osterloh A, Amico L, Falci G, Fazio R 2002 Nature 416 608
[8]	Osborne T J, Nielsen M A 2002 Phys. Rev. A 66 032110
[9]	Gu S J, Lin H Q, Li Y Q 2003 Phys. Rev. A 68 042330
[10]	Vidal G, Latorre G I, Rico E, Kitaev A 2003 Phys. Rev. Lett. 90 227902
[11]	Vidal J, Palacios G, Mosseri R 2004 Phys. Rev. A 69 022107
[12]	Ollivier H, Zurek W H 2001 Phys. Rev. Lett. 88 017901
[13]	Modi K, Brodutch A, Cable H, Paterek T, Vedral V 2012 Rev. Mod. Phys. 84 1655
[14]	You W L, Li Y W, Gu S J 2007 Phys. Rev. E 76 022101
[15]	Gu S J, Int J 2010 Mod. Phys. B 24 4371
[16]	Eisert J, Cramer M, Plenio M B 2010 Rev. Mod. Phys. 82 277
[17]	Lieb E, Schultz T, Mattis D 1961 Ann. Phys. 16 407
[18]	Lorenzo C V, Marco R 2010 Phys. Rev. A 81 060101
[19]	Kenzelmann M, Coldea R, Tennant D A, Visser D, Hofmann M, Smeibidl P, Tylczynski Z 2002 Phys. Rev. B 65 144432
[20]	Toskovic R, van-den Berg R, Spinelli A, Eliens I S, van-den Toorn B, Bryant B, Caux J S, Otte A F 2016 Nat. Phys. 12 656
[21]	Dzyaloshinskii I 1958 J. Phys. Chem. Solids 4 241
[22]	Moriya T 1960 Phys. Rev. Lett. 4 288
[23]	Seki S, Yu X Z, Ishiwata S, Tokura Y 2012 Science 336 198
[24]	Adams T, Chacon A, Wagner M, Bauer A, Brandl G, Pedersen B, Berger H, Lemmens P, Pfleiderer C 2012 Phys. Rev. Lett. 108 237204
[25]	Yang J H, Li Z L, Lu X Z, Whangbo M H, Wei S H, Gong X G, Xiang H J 2012 Phys. Rev. Lett. 109 107203
[26]	Matsuda M, Fishman R S, Hong T, Lee C H, Ushiyama T, Yanagisawa Y, Tomioka Y, Ito T 2012 Phys. Rev. Lett. 109 067205
[27]	Povarov K Y, Smirnov A I, Starykh O A, Petrov S V, Shapiro A Y 2011 Phys. Rev. Lett. 107 037204
[28]	Zhang X F, Liu T Y, Flatté M E, Tang H X 2014 Phys. Rev. Lett. 113 037202
[29]	You W L, Dong Y L 2010 Eur. Phys. J. D 57 439
[30]	You W L, Dong Y L 2011 Phys. Rev. B 84 174426
[31]	You W L, Liu G H, Horsch P, Oleś A M 2014 Phys. Rev. B 90 094413
[32]	Shan C J, Cheng W W, Liu T K, Huang Y X, Li H 2008 Acta Phys. Sin. 57 2687 (in Chinese) [单传家, 程维文, 刘堂昆, 黄燕霞, 李宏 2008 57 2687]
[33]	Zhong M, Xu H, Liu X X, Tong P Q 2013 Chin. Phys. B 22 090313
[34]	Song J L, Zhong M, Tong P Q 2017 Acta Phys. Sin. 66 180302 (in Chinese) [宋加丽, 钟鸣, 童培庆 2017 66 180302]
[35]	Brockmann M, Klumper A, Ohanyan V 2013 Phys. Rev. B 87 054407
[36]	Derzhko O, Verkholyak T, Krokhmalskii T, Bttner H 2006 Phys. Rev. B 73 214407
[37]	Barouch E, McCoy B M 1970 Phys. Rev. A 2 1075
[38]	Barouch E, McCoy B M 1971 Phys. Rev. A 3 786
[39]	Its A R, Izergin A G, Korepin V E, Slavnov N A 1993 Phys. Rev. Lett. 70 1704
[40]	Bunder J E, McKenzie R H 1999 Phys. Rev. B 60 344
[41]	Vedral V 2002 Rev. Mod. Phys. 74 197
[42]	Horodecki R, Horodecki P, Horodecki M, Horodecki K 2009 Rev. Mod. Phys. 81 865
[43]	Modi K, Brodutch A, Cable H, Paterek T, Vedral V 2012 Rev. Mod. Phys. 84 1655
[44]	Liu B Q, Shao B, Li J G, Zou J, Wu L A 2011 Phys. Rev. A 83 052112
[45]	Radhakrishnan C, Ermakov I, Byrnes T 2017 Phys. Rev. A 96 012341
[46]	You W L, Qiu Y C, Oleś A M 2016 Phys. Rev. B 93 214417
[47]	You W L, Zhang C J, Ni W, Gong M, Oleś A M 2017 Phys. Rev. B 95 224404
[48]	Lei S, Tong P 2015 Physica B 463 1
[49]	Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113 140401
[50]	Chen J, Cui J, Zhang Y, Fan H 2016 Phys. Rev. A 94 022112
[51]	Lamberti P W, Majtey A P, Borras A, Casas M, Plastino A 2008 Phys. Rev. A 77 052311

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