Abstract
In this paper a geometric phase is proposed to characterise the topological quantum phase transition of the Kitaev honeycomb model. The simultaneous rotation of two spins is crucial for generating the geometric phase for the multi-spin in a unit-cell unlike the one-spin case. It is found that the ground-state geometric phase, which is non-analytic at the critical points, possesses zigzagging behaviour in the gapless B phase of non-Abelian anyon excitations, but is a smooth function in the gapped A phase. Furthermore, the finite-size scaling behaviour of the non-analytic geometric phase along with its first- and second-order partial derivatives in the vicinity of critical points is shown to exhibit the universality. The divergent second-order derivative of the geometric phase in the thermodynamic limit indicates the typical second-order phase transition and thus the topological quantum phase transition can be well detected by the geometric phase.
Similar content being viewed by others
References
M.V. Berry, Proc. R. Soc. Lond. Ser. A 392, 45 (1984)
M.V. Berry, Phys. Today, 43, 34 (1990)
Geometric Phases in Physics, edited by A. Shapere, F. Wilczek (World Scientific, Singapore, 1989)
A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, J. Zwanziger, The Geometric Phase in Quantum Systems (Springer, New York, 2003)
A.C.M. Carollo, J.K. Pachos, Phys. Rev. Lett. 95, 157203 (2005)
S.L. Zhu, Phys. Rev. Lett. 96, 077206 (2006)
A. Hamma, arXiv:quant-ph/0602091v1 (2006)
S.-L. Zhu, Int. J. Mod. Phys. B 22, 561 (2008)
G. Chen, J. Li, J.-Q. Liang, Phys. Rev. A 74, 054101 (2006)
Y.-Q. Ma, S. Chen, Phys. Rev. A 79, 022116 (2009)
T. Hirano, H. Katsura, Y. Hatsugai, Phys. Rev. B 77, 094431 (2008)
J. Richert, Phys. Lett. A 372, 5352 (2008)
A.I. Nesterov, S.G. Ovchinnikov, Phys. Rev. E 78, R015202 (2008)
B. Basu, Phys. Lett. A 374, 1205 (2010)
S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, England, 1999)
S.L. Sondhi, S.M. Girvin, J.P. Carini, D. Shahar, Rev. Mod. Phys. 69, 315 (1997)
D.C. Tsui, H.L. Stormer, A.C. Gossard, Phys. Rev. Lett. 48, 1559 (1982)
R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983)
A. Kitaev, Ann. Phys. 303, 2 (2003)
X.G. Wen, Phys. Rev. Lett. 90, 016803 (2003)
J. Yu, S.P. Kou, X.G. Wen, Europhys. Lett. 84, 17004 (2008)
A. Kitaev, Ann. Phys. 321, 2 (2006)
X.G. Wen, Quantum Field Theory of Many-Body Systems (Oxford University Press, Oxford, 2004)
X.Y. Feng, G.M. Zhang, T. Xiang, Phys. Rev. Lett. 98, 087204 (2007)
H.C. Jiang, Z.Y. Weng, T. Xiang, Phys. Rev. Lett. 101, 090603 (2008)
H.D. Chen, J.P. Hu, Phys. Rev. B 76, 193101 (2007)
H.D. Chen, Z. Nussinov, J. Phys. A 41, 075001 (2008)
S. Yang, S.J. Gu, C.P. Sun, H.Q. Lin, Phys. Rev. A 78, 012304 (2008)
S.J. Gu, Int. J. Mod. Phys. B 24, 4371 (2010)
C. Nayak, S.H. Simon, A. Stern, M. Freedman, S.D. Sarma, Rev. Mod. Phys. 80, 1083 (2008)
K.P. Schmidt, S. Dusuel, J. Vidal, Phys. Rev. Lett. 100, 057208 (2008)
J. Vidal, K.P. Schmidt, S. Dusuel, Phys. Rev. B 78, 245121 (2008)
G. Kells, J.K. Slingerland, J. Vala, Phys. Rev. B 80, 125415 (2009)
J.-Q. Liang, H.J.W. Müller-Kirsten, Ann. Phys. 219, 42 (1992)
M.N. Barber, in Phase Transition and Critical Phenomena, edited by C. Domb, J.L. Lebowitz (Academic, New York, 1983), Vol. 8, p. 145
S.J. Gu, H.M. Kwok, W.Q. Ning, H.Q. Lin, Phys. Rev. B 77, 245109 (2008)
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Lian, J., Liang, J.Q. & Chen, G. Geometric phase in the Kitaev honeycomb model and scaling behaviour at critical points. Eur. Phys. J. B 85, 207 (2012). https://doi.org/10.1140/epjb/e2012-20901-1
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2012-20901-1