Skip to main content
SpringerLink
Log in

Universal Nature of Replica Symmetry Breaking in Quantum Systems with Gaussian Disorder

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We study quantum spin systems with quenched Gaussian disorder. We prove that the variance of all physical quantities in a certain class vanishes in the infinite volume limit. We study also replica symmetry breaking phenomena, where the variance of an overlap operator in the other class does not vanish in the replica symmetric Gibbs state. On the other hand, it vanishes in a spontaneous replica symmetry breaking Gibbs state defined by applying an infinitesimal replica symmetry breaking field. We prove also that the finite variance of the overlap operator in the replica symmetric Gibbs state implies the existence of a spontaneous replica symmetry breaking.

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

Similar content being viewed by others

References

  1. Aizenman, M., Contucci, P.: On the stability of quenched state in mean-field spin glass models. J. Stat. Phys. 92, 765–783 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Aizenman, M., Greenbatt, R.L., Lebowitz, J.L.: Proof of rounding by quenched disorder of first order transitions in low-dimensional quantum systems. J. Math. Phys. 53(2), 023301 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Aizenman, M., Wehr, J.: Rounding effects of quenched randomness on first-order phase transitions. Commun. Math. Phys. 130, 489–528 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Chatterjee, S.: Absence of replica symmetry breaking in the random field Ising model. Commun. Math. Phys. 337, 93–102 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Chatterjee, S.: The Ghirlanda-Guerra identities without averaging. preprint, arXiv:0911.4520 (2009)

  6. Chatterjee, S.: Disorder chaos and multiple valleys in spin glasses. preprint, arXiv:0907.3381 (2009)

  7. Contucci, P., Giardinà, C.: Spin-glass stochastic stability: a rigorous proof. Ann. Henri Poincare 6, 915–923 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Contucci, P., Giardinà, C.: The Ghirlanda–Guerra identities. J. Stat. Phys. 126, 917–931 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Contucci, P., Giardinà, C.: Perspectives on spin glasses. Cambridge University Press, Cambridge (2012)

    Book  MATH  Google Scholar 

  10. Contucci, P., Giardinà, C., Pulé, J.: The infinite volume limit for finite dimensional classical and quantum disordered systems. Rev. Math. Phys. 16, 629–638 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Contucci, P., Lebowitz, J.L.: Correlation inequalities for quantum spin systems with quenched centered disorder. J. Math. Phys. 51(2), 023302 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Crawford, N.: Thermodynamics and universality for mean field quantum spin glasses. Commun. Math. Phys. 274, 821–839 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Edwards, S.F., Anderson, P.W.: Theory of spin glasses. J. Phys. F 5, 965–974 (1975)

    Article  ADS  Google Scholar 

  14. Ghirlanda, S., Guerra, F.: General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity. J. Phys. A 31, 9149–9155 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Goldschmidt, C., Ueltschi, D., Windridge, P.: Quantum Heisenberg models and their probabilistic representations entropy and the quantum II. Contemp. Math. 562, 177–224 (2011)

    Article  MATH  Google Scholar 

  16. Greenbatt, R.L., Aizenman, M., Lebowitz, J.L.: Rounding first order transitions in low-dimensional quantum systems with quenched disorder. Phys. Rev.Lett. 103(19), 197201 (2009)

    Article  ADS  Google Scholar 

  17. Griffiths, R.B.: Spontaneous magnetization in idealized ferromagnets. Phys. Rev. 152, 240–246 (1964)

    Article  ADS  Google Scholar 

  18. Guerra, E.: The phenomenon of spontaneous replica symmetry breaking in complex statistical mechanics systems. J. Phys. 442(1), 012013 (2013)

    MathSciNet  Google Scholar 

  19. Guerra, F.: Spontaneous replica symmetry breaking and interpolation methods for complex statistical mechanical systems. Lecture notes in mathematics, pp. 45–70. Springer, New York (2013)

    MATH  Google Scholar 

  20. Harris, A.B.: Bounds for certain thermodynamic averages. J. Math. Phys. 8, 1044–1045 (1967)

    Article  ADS  Google Scholar 

  21. Itoi, C.: General properties of overlap operators in disordered quantum spin systems. J. Stat. phys. 163, 1339–1349 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Koma, T., Tasaki, H.: Symmetry breaking in Heisenberg antiferromagnets. Commun. Math. Phys. 158, 198–214 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Mukaida, H.: Non-differentiability of the effective potential and the replica symmetry breaking in the random energy model. J. Phys. A 49(45002), 1–15 (2016)

    MathSciNet  MATH  Google Scholar 

  24. Parisi, G.: A sequence of approximate solutions to the S–K model for spin glasses. J. Phys. A 13(4), L115 (1980)

    Article  ADS  Google Scholar 

  25. Seiler, E., Simon, B.: Nelson’s symmetry and all that in Yukawa and \((\phi ^4)_3\) theories. Ann. Phys. 97, 470–518 (1976)

    Article  ADS  Google Scholar 

  26. Sherrington, S., Kirkpatrick, S.: Solvable model of spin glass. Phys. Rev. Lett. 35, 1792–1796 (1975)

    Article  ADS  Google Scholar 

  27. Talagrand, M.: The Parisi formula. Ann. Math. 163, 221–263 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Talagrand, M.: Mean field models for spin glasses. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

It is pleasure to thank I. Affleck, T. Koma and R. M. Woloshyn for helpful discussions in early stage of this study. I would like to thank I. Affleck for kind hospitality at UBC. Also, I am grateful to R. M. Woloshyn for careful reading of the manuscript and for kind hospitality at TRIUMF.

Author information

Authors and Affiliations

  1. Department of Physics, GS & CST, Nihon University, Tokyo, Japan

    C. Itoi

Authors
  1. C. Itoi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Itoi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Itoi, C. Universal Nature of Replica Symmetry Breaking in Quantum Systems with Gaussian Disorder. J Stat Phys 167, 1262–1279 (2017). https://doi.org/10.1007/s10955-017-1778-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-017-1778-y

Keywords

Search

Navigation