具有Dzyaloshinskii-Moriya相互作用的XY模型的量子相干性
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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.
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:H_(XY) =sum((1+γ)/2σ_i~∞σ_(i+1)~∞+(1-γ)/2σ_i~yσ_(i+1)~y-hσ_i~z) from i=1 to N+sum D(σ_i~∞σ_(i+1)~y-σ_i~yσ_(i+1)~∞) from i=1 to N.(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 G_r~(xy) and G_r~(yx)(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 = 2 D,and h =(4D~2-γ~2+1)~(1/2) 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, G_r~(xy) decreases oscillatingly with the increase of distance r. The correlation function G_r~(xy) for γ = 0 oscillates more dramatically than for γ = 1. The upper boundary of the envelope is approximated as G_r~(xy) ～ r~(-1/2), and the lower boundary is approximately G_r~(xy) ～ 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 C_(JS) 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.
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:H_(XY) =sum((1+γ)/2σ_i~∞σ_(i+1)~∞+(1-γ)/2σ_i~yσ_(i+1)~y-hσ_i~z) from i=1 to N+sum D(σ_i~∞σ_(i+1)~y-σ_i~yσ_(i+1)~∞) from i=1 to N.(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 G_r~(xy) and G_r~(yx)(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 = 2 D,and h =(4D~2-γ~2+1)~(1/2) 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, G_r~(xy) decreases oscillatingly with the increase of distance r. The correlation function G_r~(xy) for γ = 0 oscillates more dramatically than for γ = 1. The upper boundary of the envelope is approximated as G_r~(xy) ～ r~(-1/2), and the lower boundary is approximately G_r~(xy) ～ 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 C_(JS) 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.
引文
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[41]Vedral V 2002 Rev.Mod.Phys.74 197
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[43]Modi K,Brodutch A,Cable H,Paterek T,Vedral V 2012Rev.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,Ole?A M 2016 Phys.Rev.B 93214417
[47]You W L,Zhang C J,Ni W,Gong M,Ole?A M 2017Phys.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 94022112
[51]Lamberti P W,Majtey A P,Borras A,Casas M,Plastino A 2008 Phys.Rev.A 77 052311
[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 416608
[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 69022107
[12]Ollivier H,Zurek W H 2001 Phys.Rev.Lett.88 017901
[13]Modi K,Brodutch A,Cable H,Paterek T,Vedral V 2012Rev.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.B65 144432
[20]Toskovic R,van-den Berg R,Spinelli A,Eliens I S,vanden 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 336198
[24]Adams T,Chacon A,Wagner M,Bauer A,Brandl G,Pedersen B,Berger H,Lemmens P,Pfleiderer C 2012Phys.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,FlattéM 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,Ole?A M 2014 Phys.Rev.B 90 094413
[32]Shan C J,Cheng W W,Liu T K,Huang Y X,Li H 2008Acta 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.66180302(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,Büttner H2006 Phys.Rev.B 73 214407
[37]Barouch E,Mc Coy B M 1970 Phys.Rev.A 2 1075
[38]Barouch E,Mc Coy B M 1971 Phys.Rev.A 3 786
[39]Its A R,Izergin A G,Korepin V E,Slavnov N A 1993Phys.Rev.Lett.70 1704
[40]Bunder J E,Mc Kenzie 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 K2009 Rev.Mod.Phys.81 865
[43]Modi K,Brodutch A,Cable H,Paterek T,Vedral V 2012Rev.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,Ole?A M 2016 Phys.Rev.B 93214417
[47]You W L,Zhang C J,Ni W,Gong M,Ole?A M 2017Phys.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 94022112
[51]Lamberti P W,Majtey A P,Borras A,Casas M,Plastino A 2008 Phys.Rev.A 77 052311