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Absence of Replica Symmetry Breaking in the Transverse and Longitudinal Random Field Ising Model

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Abstract

It is proved that replica symmetry is not broken in the transverse and longitudinal random field Ising model. In this model, the variance of spin overlap of any component vanishes in any dimension almost everywhere in the coupling constant space in the infinite volume limit. The weak Fortuin–Kasteleyn–Ginibre property in this model and the Ghirlanda–Guerra identities in artificial models in a path integral representation based on the Lie–Trotter–Suzuki formula enable us to extend Chatterjee’s proof for the random field Ising model to the quantum model.

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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., Greenblatt, 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. 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 

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

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

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Contucci, P., Giardinà, C.: Perspectives on Spin Glasses. Cambridge University Press, Cambridge (2012)

    Book  MATH  Google Scholar 

  8. 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 

  9. Campanino, M., Klein, A.: Decay of two-point functions for (d + 1)-dimensional percolation, ising and potts models with d-dimensional disorder. Commun. Math. Phys. 135, 483–497 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Campanino, M., Klein, A., Pelez, J.F.: Localization in the ground state of the ising model with a random transverse field. Commun. Math. Phys. 135, 499–515 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Contucci, P., Lebowitz, J.L.: Correlation inequalities for quantum spin systems with quenched centered disorder. J. Math. Phys. 51, 023302-1 -6 (2010)

    Article  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. Fortuin, C.M., Kasteleyn, P.W., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22, 89–103 (1971)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. 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 

  16. 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 

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

    Article  ADS  Google Scholar 

  18. 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 

  19. Itoi, C.: Universal nature of replica symmetry breaking in quantum systems with Gaussian disorder. J. Stat. phys. 163, 1339–1349 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Nishimori, H., Sherrington, D.: Absence of replica symmetry breaking in a region of the phase diagram of the Ising spin glass. In: AIP Conference Proceedings, vol. 553, p. 67 (2001)

  21. Nishimori, H.: Statistical Physics of Spin Glasses and Information Processing: An Introduction. Oxford university press, Oxford (2001)

    Book  MATH  Google Scholar 

  22. Panchenko, D.: The Ghirlanda-Guerra identities for mixed \(p\)-spin glass model. Compt. Read. Math. 348, 189–192 (2010)

    Article  MATH  Google Scholar 

  23. Parisi, G.: A sequence of approximate solutions to the S-K model for spin glasses. J. Phys. A 13, L-115 (1980)

    Article  ADS  Google Scholar 

  24. 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 

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

    Article  ADS  Google Scholar 

  26. Suzuki, M.: Relationship between d-dimensional quantal spin systems and (d+1)-dimensional Ising systems. Prog. Theor. Phys. 56, 1454–1468 (1976)

    Article  ADS  MathSciNet  MATH  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 

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Acknowledgements

It is pleasure to thank R. M. Woloshyn for reading the manuscript and helpful suggestions. I am grateful to M. Aoyagi for discussions in early stage of this work. I would like to thank the anonymous referees for essential comments.

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  1. Department of Physics, GS & CST, Nihon University, 1-8-14 Kanda-Surugadai, Chiyoda, Tokyo, 101-8308, Japan

    C. Itoi

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Itoi, C. Absence of Replica Symmetry Breaking in the Transverse and Longitudinal Random Field Ising Model. J Stat Phys 170, 684–699 (2018). https://doi.org/10.1007/s10955-017-1950-4

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