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In the present work, we study the fidelity susceptibility and the entanglement entropy in an antiferromagnetic spin-1 chain with additional next-nearest neighbor interactions and three-site interactions, which are given by H=(J1SiSi+1+ J2SiSi+2)+[J3(SiSi+1)(Si+1Si+2)+ h.c.]. By using the density matrix renormalization group method, the ground-state properties of the system are calculated with very high accuracy. We investigate the effect of the three-site interaction J3 on the fidelity susceptibility numerically, and then analyze its relation with the quantum phase transition (QPT). The fidelity measures the similarity between two states, and the fidelity susceptibility describes the associated changing rate. The QPT is intuitively accompanied by an abrupt change in the structure of the ground-state wave function, so generally a peak of the fidelity susceptibility indicates a QPT and the location of the peak denotes the critical point. For the case of J2=0, a peak of the fidelity susceptibility is found by varying J3, and the height of the peak grows as the system size increases. The location of the peak shifts to a slightly lower J3 up to a particular value as the system size increases. Through a finite size scaling, the critical point J3c=0.111 of the QPT from the Haldane spin liquid to the dimerized phase is identified. We also study the effect of the three-site interaction on the entanglement entropy between the right half part and the rest. It is noted that the peak of the entanglement entropy does not coincide with the critical point. Instead, the critical point is determined by the position at which the first-order derivative of the entanglement entropy takes its minimum, since a second-order QPT is signaled by the first derivative of density matrix element. Moreover, the entanglement entropy disappears when J3=1/6, which corresponds to the size-independent Majumdar-Ghosh point. The positions of quantum critical points extracted from these two quantum information observables agree well with those obtained by the string order parameters, which characterizes the topological order in the Haldane phase. Secondly, we also study the case of J20, and obtain the critical points by both the fidelity susceptibility and the entanglement entropy. Finally we provide a ground-state phase diagram of the system. To sum up, the quantum information observables are effective tools for detecting diverse QPTs in spin-1 models.
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Keywords:
- quantum phase transitions /
- fidelity susceptibility /
- entanglement entropy /
- density-matrix renormalization group
[1] Sachdev S 1999 Quantum Phase Transitions (Cambridge: Cambridge University Press) p133
[2] den Nijs M, Rommelse K 1989 Phys. Rev. B 40 4709
[3] Chen W, Hida K, Sanctuary B C 2003 Phys. Rev. B 67 104401
[4] Degli Esposti Boschi C, Ercolessi E, Ortolani F, Roncaglia M 2003 Eur. Phys. J. B 35 465
[5] Darriet J, Regnault L 1993 Solid State Commun. 86 409
[6] Buyers W J L, Morra R M, Armstrong R L, Hogan M J, Gerlach P, Hirakawa K 1986 Phys. Rev. Lett. 56 371
[7] Singh K, Basu T, Chowki S, Mahapotra N, Iyer K K, Paulose P L, Sampathkumaran E V 2013 Phys. Rev. B 88 094438
[8] Zheludev A, Tranquada J M, Vogt T, Buttrey D J 1996 Phys. Rev. B 54 7210
[9] Li W, Andreas W, Delft J V 2013 Phys. Rev. B 88 245121
[10] You W L, Li Y W, Gu S J 2007 Phys. Rev. E 76 022101
[11] Cozzini M, Ionicioiu R, Zanardi P 2007 Phys. Rev. B 76 104420
[12] Ren J, Zhu S Q 2008 Eur. Phys. J. D 50 103
[13] Ren J, Xu X F, Gu L P, Li J L 2012 Phys. Rev. A 86 064301
[14] Ren J, Liu G H, You W L 2015 J. Phys.: Condens. Matter 27 105602
[15] Ren J, Zhu S Q 2009 Phys. Rev. A 79 034302
[16] Liu G H, Wang H L, Tian G S 2008 Phys. Rev. B 77 214418
[17] Zhao J H 2012 Acta Phys. Sin. 61 220501 (in Chinese)[赵建辉 2012 物理学报 61 220501]
[18] White S R 1993 Phys. Rev. B 48 10345
[19] Schollwöck U 2005 Rev. Mod. Phys. 77 259
[20] Chepiga N, Affleck I, Mila F 2016 Phys. Rev. B 93 241108
[21] Michaud F, Vernay F, Manmana S R, Mila F 2012 Phys. Rev. Lett. 108 127202
[22] Wu L A, Sarandy M S, Lidar D A 2004 Phys. Rev. Lett. 93 250404
[23] Gu S J 2010 Int. J. Mod. Phys. B 24 4371
[24] You W L, Dong Y L 2011 Phys. Rev. B 84 174426
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[1] Sachdev S 1999 Quantum Phase Transitions (Cambridge: Cambridge University Press) p133
[2] den Nijs M, Rommelse K 1989 Phys. Rev. B 40 4709
[3] Chen W, Hida K, Sanctuary B C 2003 Phys. Rev. B 67 104401
[4] Degli Esposti Boschi C, Ercolessi E, Ortolani F, Roncaglia M 2003 Eur. Phys. J. B 35 465
[5] Darriet J, Regnault L 1993 Solid State Commun. 86 409
[6] Buyers W J L, Morra R M, Armstrong R L, Hogan M J, Gerlach P, Hirakawa K 1986 Phys. Rev. Lett. 56 371
[7] Singh K, Basu T, Chowki S, Mahapotra N, Iyer K K, Paulose P L, Sampathkumaran E V 2013 Phys. Rev. B 88 094438
[8] Zheludev A, Tranquada J M, Vogt T, Buttrey D J 1996 Phys. Rev. B 54 7210
[9] Li W, Andreas W, Delft J V 2013 Phys. Rev. B 88 245121
[10] You W L, Li Y W, Gu S J 2007 Phys. Rev. E 76 022101
[11] Cozzini M, Ionicioiu R, Zanardi P 2007 Phys. Rev. B 76 104420
[12] Ren J, Zhu S Q 2008 Eur. Phys. J. D 50 103
[13] Ren J, Xu X F, Gu L P, Li J L 2012 Phys. Rev. A 86 064301
[14] Ren J, Liu G H, You W L 2015 J. Phys.: Condens. Matter 27 105602
[15] Ren J, Zhu S Q 2009 Phys. Rev. A 79 034302
[16] Liu G H, Wang H L, Tian G S 2008 Phys. Rev. B 77 214418
[17] Zhao J H 2012 Acta Phys. Sin. 61 220501 (in Chinese)[赵建辉 2012 物理学报 61 220501]
[18] White S R 1993 Phys. Rev. B 48 10345
[19] Schollwöck U 2005 Rev. Mod. Phys. 77 259
[20] Chepiga N, Affleck I, Mila F 2016 Phys. Rev. B 93 241108
[21] Michaud F, Vernay F, Manmana S R, Mila F 2012 Phys. Rev. Lett. 108 127202
[22] Wu L A, Sarandy M S, Lidar D A 2004 Phys. Rev. Lett. 93 250404
[23] Gu S J 2010 Int. J. Mod. Phys. B 24 4371
[24] You W L, Dong Y L 2011 Phys. Rev. B 84 174426
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