Journal of Statistical Physics

, Volume 163, Issue 6, pp 1339–1349 | Cite as

General Properties of Overlap Operators in Disordered Quantum Spin Systems



We study short-range quantum spin systems with Gaussian disorder. We obtain quantum mechanical extensions of the Ghirlanda–Guerra identities. We discuss properties of overlap spin operators with these identities.


Quenched disorder Spin glass Quantum spin systems Ghirlanda Guerra Identities 


  1. 1.
    Aizenman, M., Contucci, P.: On the stability of quenched state in mean-field spin glass models. J. Stat. Phys. 92, 765–783 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chatterjee, S.: Absence of replica symmetry breaking in the random field Ising model. Commun. Math. Phys. 337, 93–102 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chatterjee,S.: The Ghirlanda–Guerra identities without averaging. preprint, arXiv:0911.4520 (2009)
  4. 4.
    Chatterjee, S. : Disorder chaos and multiple valleys in spin glasses. preprint, arXiv:0907.3381 (2009)
  5. 5.
    Contucci, P., Giardinà, C.: Spin-glass stochastic stability: a rigorous proof. Ann. Henri Poincare 6, 915–923 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Contucci, P., Giardinà, C.: The Ghirlanda–Guerra identities. J. Stat. Phys. 126, 917–931 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Contucci, P., Giardinà, C.: Perspectives on Spin Glasses. Cambridge University Press, Cambridge (2012)CrossRefMATHGoogle Scholar
  8. 8.
    Contucci, P., Giardinà, C., Pulé, J.: The thermodynamic limit for finite dimensional classical and quantum disordered systems. Rev. Math. Phys. 16, 629–638 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Contucci, P., Lebowitz, J.L.: Correlation inequalities for quantum spin systems with quenched centered disorder. J. Math. Phys. 51, 023302 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Crawford, N.: Thermodynamics and universality for mean field quantum spin glasses. Commun. Math. Phys. 274, 821–839 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Edwards, S.F., Anderson, P.W.: Theory of spin glasses. J. Phys. F 5, 965–974 (1975)ADSCrossRefGoogle Scholar
  12. 12.
    Fortuin, C.M., Kasteleyn, P.W., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22, 89–103 (1971)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ghirlanda, S., Guerra, F.: General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity. J. Phys. A 31, 9149–9155 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Griffiths, R.B.: A proof that the free energy of a spin system is extensive. J. Math. Phys. 5, 1215–1222 (1964)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Panchenko, D.: The Ghirlanda–Guerra identities for mixed \(p\)-spin glass model. Compt. Read. Math. 348, 189–192 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Panchenko, D.: The Parisi ultrametricity conjecture. Ann. Math. 177, 383–393 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Parisi, G.: A sequence of approximate solutions to the S-K model for spin glasses. J. Phys. A 13, L-115 (1980)ADSCrossRefGoogle Scholar
  18. 18.
    Seiler, E., Simon, B.: Nelson’s symmetry and all that in Yukawa and \((\phi ^4)_3\) theories. Ann. Phys. 97, 470–518 (1976)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Sherrington, S., Kirkpatrick, S.: Solvable model of spin glass. Phys. Rev. Lett. 35, 1792–1796 (1975)ADSCrossRefGoogle Scholar
  20. 20.
    Talagrand, M.: The Parisi formula. Ann. Math. 163, 221–263 (2006)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Talagrand, M.: Mean Field Models for Spin Glasses. Springer, Berlin (2011)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Physics, GS & CSTNihon UniversityTokyoJapan

Personalised recommendations