Skip to main content
SpringerLink
Log in

Criticality, factorization and Wigner–Yanase skew information in quantum spin chains

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We apply the Wigner–Yanase skew information approach to analyze criticality and factorization phenomenon in the one-dimensional anisotropy \(XY\) model with uniform coupling interaction and periodic-two one. Based on the exact solutions of the ground states, the Wigner–Yanase skew information between two nearest-neighbor lattices is obtained. For the uniform case, the first-order derivative of the Wigner–Yanase skew information is non-analytically around the critical point. The scaling behavior and the universality are verified numerically. In particular, such skew information can also detect the factorization transition in this model. For the periodic-two case, it is found that there exist more than one phase-transition point in some parameter region due to the competition between periodicity and anisotropy. Furthermore, two kinds of phase transitions, i.e., the Ising and anisotropy transitions, driven by external field \(\lambda \) and the anisotropy parameter \(\gamma \), are investigated carefully by the skew information. Our results state that quantum phase transition driven by the anisotropy parameter \(\gamma \) can belong to the same universality class as the one driven by external field \(\lambda \).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Sachdev, S.: Quantum Phase Transition. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  2. Lieb, E., Schultz, T., Mattis, D.: Soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407 (1961)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Pfeuty, P.: The one-dimensional Ising model with a transverse field. Ann. Phys. 57, 79 (1970)

    Article  ADS  Google Scholar 

  4. Giampaolo, S.M., Adesso, G., Illuminati, F.: Probing quantum frustrated systems via factorization of the ground state. Phys. Rev. Lett. 104, 207202 (2010)

    Article  ADS  Google Scholar 

  5. Gemmera, J., Mahler, G.: Entanglement and the factorization-approximation. Eur. Phys. J. D 17, 385 (2001)

    Article  ADS  Google Scholar 

  6. Fubini, A., Roscilde, T., Tognetti, V., Tusa, M., Verrucchi, P.: Reading entanglement in terms of spin configurations in quantum magnets. Eur. Phys. J. D 38, 563 (2006)

    Article  ADS  Google Scholar 

  7. Tomasello, B., Rossini, D., Hamma, A., Amico, L.: Ground-state factorization and correlations with broken symmetry. EPL 96, 27002 (2011)

    Article  ADS  Google Scholar 

  8. Campbell, S., Richens, J., Lo Gullo, N., Busch, T.: Criticality, factorization, and long-range correlations in the anisotropic XY model. Phys. Rev. A 88, 062305 (2013)

    Article  ADS  Google Scholar 

  9. Osterloh, A., Amico, L., Falci, G., Fazio, R.: Scaling of entanglement close to a quantum phase transition. Nature 416, 608 (2002)

    Article  ADS  Google Scholar 

  10. Amico, L., Osterloh, A., Plastina, F., Fazio, R., Palma, G.M.: Dynamics of entanglement in one-dimensional spin systems. Phys. Rev. A 69, 022304 (2004)

    Article  ADS  Google Scholar 

  11. Amico, L., Fazio, R., Osterloh, A., Vedral, V.: Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Osborne, T.J., Nielsen, M.A.: Entanglement in a simple quantum phase transition. Phys. Rev. A 66, 032110 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  13. Vidal, G., Latorre, J.I., Rico, E., Kitaev, A.: Entanglement in quantum critical phenomena. Phys. Rev. Lett. 90, 227902 (2003)

    Article  ADS  Google Scholar 

  14. Maziero, J., Guzman, H.C., Celeri, L.C., Sarandy, M.S., Serra, R.M.: Quantum and classical thermal correlations in the XY spin-\(\frac{1}{2}\) chain. Phys. Rev. A 82, 012106 (2010)

    Article  ADS  Google Scholar 

  15. Cheng, W.W., Liu, J.-M.: Fidelity susceptibility approach to quantum phase transitions in the \(XY\) spin chain with multisite interactions. Phys. Rev. A 82, 012308 (2010)

    Article  ADS  Google Scholar 

  16. Chen, H.D.: Two-point entanglement near a quantum phase transition. J. Phys. A Math. Theor. 40, 10215 (2007)

    Article  MATH  ADS  Google Scholar 

  17. Gu, S.J., Sun, C.P., Lin, H.Q.: Universal role of correlation entropy in critical phenomena. J. Phys. A Math. Theor. 41, 025002 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  18. Cheng, W.W., Du, Z.Z., Gong, L.Y., Zhao, S.M., Liu, J.-M.: Signature of topological quantum phase transitions via Wigner–Yanase skew information. EPL 108, 46003 (2014)

    Article  ADS  Google Scholar 

  19. Osborne, T.J., Nielsen, M.A.: Entanglement, quantum phase transitions, and density matrix renormalization. Quantum Inf. Process. 1, 45 (2002)

    Article  MathSciNet  Google Scholar 

  20. Song, X.K., Wu, T., Ye, L.: The monogamy relation and quantum phase transition in one-dimensional anisotropic \(XXZ\) model. Quantum Inf. Process. 12, 3305 (2013)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Ma, Y.Q., Chen, S.: Geometric phase and quantum phase transition in an inhomogeneous periodic \(XY\) spin-\(\frac{1}{2}\) model. Phys. Rev. A 79, 022116 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  22. Zhang, L.F., Tong, P.Q.: Entanglement of periodic anisotropic XY chains. J. Phys. A Math. Theor. 38, 7377 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  23. Zhong, M., Tong, P.Q.: The Ising and anisotropy phase transitions of the periodic \(XY\) model in a transverse field. J. Phys. A Math. Theor. 43, 505302 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  24. Sun, K.W., Zhang, Y.Y., Chen, Q.H.: Quantum phase transition in the one-dimensional period-two and uniform compass model. Phys. Rev. B 79, 104429 (2009)

    Article  ADS  Google Scholar 

  25. Amico, L., Baroni, F., Fubini, A., Patanè, D., Tognetti, V., Verrucchi, P.: Divergence of the entanglement range in low-dimensional quantum systems. Phys. Rev. A 74, 022322 (2006)

    Article  ADS  Google Scholar 

  26. Çakmak, B., Karpat, G., Gedik, Z.: Critical point estimation and long-range behavior in the one-dimensional \(XY\) model using thermal quantum and total correlations. Phys. Lett. A 376, 2982 (2012)

    Article  ADS  Google Scholar 

  27. Karpat, G., Çakmak, B., Fanchini, F.F.: Quantum coherence and uncertainty in the anisotropic \(XY\) chain. Phys. Rev. B 90, 104431 (2014)

    Article  ADS  Google Scholar 

  28. Çakmak, B., Karpat, G., Fanchini, F.F.: Factorization and criticality in the anisotropic \(XY\) chain via correlations. Entropy 17, 790 (2015)

    Article  ADS  Google Scholar 

  29. Cheng, W.W., Shan, C.J., Sheng, Y.B., Gong, L.Y., Zhao, S.M., Zheng, B.Y.: Geometric discord approach to quantum phase transition in the anisotropy \(XY\) spin model. Phys. E 44, 1320 (2012)

    Article  Google Scholar 

  30. Ohya, M., Petz, D.: Quantum Entropy and Its Use. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  31. Wehrl, A.: General properties of entropy. Rev. Mod. Phys. 50, 221 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  32. Wigner, E.P., Yanase, M.M.: Information contents of distributions. Proc. Natl. Acad. Sci. USA 49, 910 (1963)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  33. Luo, S.L.: Wigner–Yanase skew information and uncertainty relations. Phys. Rev. Lett. 91, 180403 (2003)

    Article  ADS  Google Scholar 

  34. Luo, S.L.: Quantum uncertainty of mixed states based on skew information. Phys. Rev. A 73, 022324 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  35. Luo, S.L., Fu, S.S., Oh, C.H.: Quantifying correlations via the Wigner–Yanase skew information. Phys. Rev. A 85, 032117 (2012)

    Article  ADS  Google Scholar 

  36. Chen, Z.Q.: Wigner–Yanase skew information as tests for quantum entanglement. Phys. Rev. A 71, 052302 (2005)

    Article  ADS  Google Scholar 

  37. Girolami, D., Tufarelli, T., Adesso, G.: Characterizing nonclassical correlations via local quantum uncertainty. Phys. Rev. Lett. 110, 240402 (2013)

    Article  ADS  Google Scholar 

  38. Huang, Z., Osenda, O., Kais, S.: Entanglement of formation for one-dimensional magnetic systems with defects. Phys. Lett. A 322, 137 (2004)

    Article  MATH  ADS  Google Scholar 

  39. Tong, P.Q., Zhong, M.: Quantum phase transitions of periodic anisotropic \(XY\) chain in a transverse field. Phys. B 304, 91 (2001)

    Article  ADS  Google Scholar 

  40. Chang, L.N., Luo, S.L.: Remedying the local ancilla problem with geometric discord. Phys. Rev. A 87, 062303 (2013)

    Article  ADS  Google Scholar 

Download references

Acknowledgments

W.W. is supported by the Natural Science Foundation of China (NSFC) (Grant Nos. 11105049, 11474168). J.X. and C.J. thank the support from the NSFC (Grant No. 11404108). L.Y. and S.M. acknowledge the financial support from the NSFC (Grant Nos. 61271238, 61475075).

Author information

Authors and Affiliations

  1. Institute of Signal Processing and Transmission, Nanjing University of Posts and Telecommunication, Nanjing, 210003, China

    W. W. Cheng, L. Y. Gong & S. M. Zhao

  2. Department of Physics, Hubei Normal University, Huangshi, 435002, China

    W. W. Cheng, J. X. Li & C. J. Shan

  3. National Laboratory of Solid State Microstructures, Nanjing University, Nanjing, 210093, China

    W. W. Cheng

Authors
  1. W. W. Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. X. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. J. Shan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. L. Y. Gong
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. S. M. Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to W. W. Cheng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, W.W., Li, J.X., Shan, C.J. et al. Criticality, factorization and Wigner–Yanase skew information in quantum spin chains. Quantum Inf Process 14, 2535–2549 (2015). https://doi.org/10.1007/s11128-015-1009-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-015-1009-x

Keywords

Search

Navigation