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On Random Field Induced Ordering in the Classical XY Model

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Abstract

Consider the classical XY model in a weak random external field pointing along the Y axis with strength ε. We study the behavior of this model as the range of the interaction is varied. We prove that in any dimension d≥2 and for all ε sufficiently small, there is a range L=L(ε) so that whenever the inverse temperature β is larger than some β(ε), there is strong residual ordering along the X direction.

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References

  1. Abanin, D.A., Lee, P.A., Levitov, L.S.: Randomness-induced XY ordering in a graphene quantum hall ferromagnet. Phys. Rev Lett. 98(15), 156801 (2007)

    Article  ADS  Google Scholar 

  2. Aharony, A.: Spin-flop multicritical points in systems with random fields and in spin glasses. Phys. Rev. B 18, 3328–3336 (1978)

    Article  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. van Enter, A.C.D., Ruszel, W.M.: Gibbsianness versus Non-Gibbsianness of time-evolved planar rotor models. Stoch. Proc. Appl. 119, 1866–1888 (2009)

    Article  MATH  Google Scholar 

  5. van Enter, A.C.D., Ruszel, W.M.: Loss and recovery of Gibbsianness for xy spins in a small external field. J. Math. Phys. 49, 125208–125208-8 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  6. van Enter, A.C.D., Külske, C., Opoku, A.A., Ruszel, W.M.: Gibbs-non-Gibbs properties for n-vector lattice and mean-field models. Braz. J. Prob. Stat. 24, 226–255 (2010). Special Proceedings volume of XII Brazilian school of Probability

    Article  MATH  Google Scholar 

  7. Chayes, J.T., Chayes, L., Fröhlich, J.: The low-temperature behavior of disordered magnets. Commun. Math. Phys. 100(3), 399–437 (1985)

    Article  ADS  Google Scholar 

  8. Imry, Y., Ma, S.-K.: Random-field instability for the ordered state of continuous symmetry. Phys. Rev. Lett. 35, 1399 (1975)

    Article  ADS  Google Scholar 

  9. Kesten, H., Schonmann, R.H.: Behavior in large dimensions of the Potts and Heisenberg models. Rev. Math. Phys. 1, 147–182 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  10. Mermin, D., Wagner, H.: Absence of ferromagnetism or anti-ferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. 17, 1133 (1966)

    ADS  Google Scholar 

  11. Presutti, E.: Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics. Springer, Berlin (2008)

    Google Scholar 

  12. Sanchez-Palencia, L., Lewenstein, M.: Disordered quantum gases under control. Nat. Phys. 6, 87–95 (2010)

    Article  Google Scholar 

  13. Sen(De), A., Sen, U., Wehr, J., Lewenstein, M.: Classical spin models with broken continuous symmetry: random field induced order and persistence of spontaneous magnetization (2007). arXiv:0712.0829

  14. Wehr, J., Niederberger, A., Sanchez-Palencia, L., Lewenstein, M.: Disorder versus the Mermin-Wagner-Hohenberg effect: From classical spin systems to ultracold atomic gases. Phys. Rev. B 74, 224448 (2006)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Industrial Engineering, The Technion, Haifa, Israel

    Nicholas Crawford

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  1. Nicholas Crawford
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Correspondence to Nicholas Crawford.

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Supported in part at the Technion by an Marilyn and Michael Winer Fellowship and by Binational Science Foundation Grant 2008421.

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Crawford, N. On Random Field Induced Ordering in the Classical XY Model. J Stat Phys 142, 11–42 (2011). https://doi.org/10.1007/s10955-010-0094-6

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  • DOI: https://doi.org/10.1007/s10955-010-0094-6

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