Quantum coherence of XY model with Dzyaloshinskii-Moriya interaction

Yi Tian-Cheng; Ding Yue-Ran; Ren Jie; Wang Yi-Min; You Wen-Long

doi:10.7498/aps.67.20172755

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:

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).

References

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[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

Cited By

[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 物理学报 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 物理学报 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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Vol. 67, Issue 14 - 2018-07-20

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