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The Griffiths phase and beyond: A large deviations study of the magnetic susceptibility of the two-dimensional bond-diluted Ising model

Published 5 May 2024 in cond-mat.dis-nn, cond-mat.stat-mech, and physics.comp-ph | (2405.02889v2)

Abstract: The Griffiths phase in systems with quenched disorder occurs below the ordering transition of the pure system down to the ordering transition of the actual disordered system. While it does not exhibit long-range order, large fluctuations in the disorder degrees of freedom result in exponentially rare, long-range ordered states and hence the occurrence of broad distributions in response functions. Inside the Griffiths phase of the two-dimensional bond-diluted Ising model the distribution of the magnetic susceptibility is expected to have such a broad, exponential tail. A large-deviations Monte Carlo algorithm is used to sample this distribution and the exponential tail is extracted over a wide range of the support down to very small probabilities of the order of 10<sup>−30010<sup>{-300}. We study the behavior of the susceptibility distribution across the full phase diagram, from the paramagnetic state through the Griffiths phase to the ferromagnetically ordered system and down to the zero-temperature point. We extract the rate function of large-deviation theory as well as its finite-size scaling behavior and we reveal interesting differences and similarities between the cases. A connection between the fraction of ferromagnetic bonds in a given disorder sample and the size of the magnetic susceptibility is demonstrated numerically.

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References (48)
  1. S. G. Brush, History of the Lenz-Ising model, Rev. Mod. Phys. 39, 883 (1967).
  2. A. Coniglio and A. Fierro, Correlated percolation, in Complex Media and Percolation Theory, edited by M. Sahimi and A. G. Hunt (Springer, New York, 2021) p. 61.
  3. A. J. Page and R. P. Sear, Heterogeneous nucleation in and out of pores, Phys. Rev. Lett. 97, 065701 (2006).
  4. A. Grabowski and R. Kosiński, Ising-based model of opinion formation in a complex network of interpersonal interactions, Physica A 361, 651 (2006).
  5. J. P. Sethna, K. A. Dahmen, and C. R. Myers, Crackling noise, Nature 410, 242 (2001).
  6. D. L. Stein and C. M. Newman, Spin Glasses and Complexity (Princeton University Press, Princeton, 2013).
  7. H. Nishimori and G. Ortiz, Random systems, in Elements of Phase Transitions and Critical Phenomena (Oxford University Press, Oxford, 2011) p. 178.
  8. A. B. Harris, Effect of random defects on the critical behaviour of Ising models, J. Phys. C 7, 1671 (1974).
  9. I. A. Hadjiagapiou, Monte Carlo analysis of the critical properties of the two-dimensional randomly bond-diluted Ising model via Wang–Landau algorithm, Physica A 390, 1279 (2011).
  10. H. Ikeda, M. Suzuki, and M. T. Hutchings, Neutron scattering investigation of static critical phenomena in the two-dimensional antiferromagnets: Rb2⁢Coc⁢Mg1−c⁢F4subscriptRb2subscriptCo𝑐subscriptMg1𝑐subscriptF4\mathrm{Rb}_{2}\mathrm{Co}_{c}\mathrm{Mg}_{1-c}\mathrm{F}_{4}roman_Rb start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_Co start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT roman_Mg start_POSTSUBSCRIPT 1 - italic_c end_POSTSUBSCRIPT roman_F start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT , J. Phys. Soc. Jpn. 46, 1153 (1979).
  11. R. B. Griffiths, Nonanalytic behavior above the critical point in a random Ising ferromagnet, Phys. Rev. Lett. 23, 17 (1969).
  12. A. J. Bray, Nature of the Griffiths phase, Phys. Rev. Lett. 59, 586 (1987).
  13. A. J. Bray and M. A. Moore, On the eigenvalue spectrum of the susceptibility matrix for random spin systems, J. Phys. C 15, L765 (1982).
  14. S. Colborne and A. Bray, Monte Carlo study of Griffiths phase dynamics in dilute ferromagnets, J. of Phys. A 22, 2505 (1989).
  15. A. Bray and D. Huifang, Griffiths singularities in random magnets: results for a soluble model, Phys. Rev. B 40, 6980 (1989).
  16. Y. Matsuda, H. Nishimori, and K. Hukushima, The distribution of Lee-Yang zeros and Griffiths singularities in the ±Jplus-or-minus𝐽\pm J± italic_J model of spin glasses, J. Phys. A 41, 324012 (2008).
  17. T. Vojta, Quantum Griffiths effects and smeared phase transitions in metals: theory and experiment, Low Temp. Phys. 161, 299 (2010).
  18. D. S. Fisher, Random transverse field Ising spin chains, Phys. Rev. Lett. 69, 534 (1992).
  19. D. S. Fisher, Critical behavior of random transverse-field Ising spin chains, Phys. Rev. B 51, 6411 (1995).
  20. A. P. Young and H. Rieger, Numerical study of the random transverse-field Ising spin chain, Phys. Rev. B 53, 8486 (1996).
  21. K. Nishimura, H. Nishimori, and H. G. Katzgraber, Griffiths-McCoy singularity on the diluted Chimera graph: Monte Carlo simulations and experiments on quantum hardware, Phys. Rev. A 102, 042403 (2020).
  22. A. K. Hartmann, Big Practical Guide to Computer Simulations (World Scientific, Singapore, 2015).
  23. M. E. Newman and G. T. Barkema, Monte Carlo methods in statistical physics (Oxford University Press, Oxford, 1999).
  24. A. K. Hartmann, Sampling rare events: statistics of local sequence alignments, Phys. Rev. E 65, 056102 (2002).
  25. A. K. Hartmann, Large-deviation properties of largest component for random graphs, Eur. Phys. J. B 84, 627 (2011).
  26. K. Hukushima and Y. Iba, A Monte Carlo algorithm for sampling rare events: application to a search for the Griffiths singularity, J. Phys. Conf. Ser. 95, 012005 (2008).
  27. T. Neuhaus and J. S. Hager, Free-energy calculations with multiple Gaussian modified ensembles, Phys. Rev. E 74, 036702 (2006).
  28. M. Ohzeki, Locations of multicritical points for spin glasses on regular lattices, Phys. Rev. E 79, 021129 (2009).
  29. H. Nishimori, Conjecture on the exact transition point of the random Ising ferromagnet, J. Phys. C 12, L905 (1979).
  30. W. Zhong, G. T. Barkema, and D. Panja, Super slowing down in the bond-diluted Ising model, Phys. Rev. E 102, 022132 (2020).
  31. Note that in most cases we used Y𝑱=N⁢χ𝑱subscript𝑌𝑱𝑁subscript𝜒𝑱Y_{\bm{J}}=\sqrt{N\chi_{\bm{J}}}italic_Y start_POSTSUBSCRIPT bold_italic_J end_POSTSUBSCRIPT = square-root start_ARG italic_N italic_χ start_POSTSUBSCRIPT bold_italic_J end_POSTSUBSCRIPT end_ARG instead of χ𝑱subscript𝜒𝑱\chi_{\bm{J}}italic_χ start_POSTSUBSCRIPT bold_italic_J end_POSTSUBSCRIPT since this quantity is found to be easier to sample. The distribution of χ𝑱subscript𝜒𝑱\chi_{\bm{J}}italic_χ start_POSTSUBSCRIPT bold_italic_J end_POSTSUBSCRIPT can then be obtained by a change of variables.
  32. H. Schawe, Large Deviations of Convex Hulls of Random Walks and Other Stochastic Models, dissertation, Carl von Ossietzky Universität Oldenburg, Oldenburg (2019).
  33. P. L. Ebert, D. Gessert, and M. Weigel, Weighted averages in population annealing: analysis and general framework, Phys. Rev. E 106, 045303 (2022).
  34. R. H. Swendsen and J. S. Wang, Nonuniversal critical dynamics in monte carlo simulations, Phys. Rev. Lett. 58, 86 (1987).
  35. A. H. Kole, G. T. Barkema, and L. Fritz, Comparison of cluster algorithms for the bond-diluted Ising model, Phys. Rev. E 105, 015313 (2022).
  36. U. Wolff, Comparison between cluster Monte Carlo algorithms in the Ising model, Phys. Lett. B 228, 379 (1989).
  37. The FKCK clusters are also called FK clusters or CK droplets due to their historical origin [2], see also Ref. [57].
  38. K. Binder and D. W. Heermann, Some important recent developments of the Monte Carlo methodology, in Monte Carlo Simulation in Statistical Physics: An Introduction (Springer Berlin Heidelberg, Berlin, Heidelberg, 2010) p. 111.
  39. J. Roussenq, A. Coniglio, and D. Stauffer, Study of droplets for correlated site-bond percolation in three dimensions, Journal de Physique Lettres 43, 703 (1982).
  40. M. Weigel, W. Janke, and C. K. Hu, Random-cluster multihistogram sampling for the $q$-state potts model, Phys. Rev. E 65, 036109 (2002).
  41. A. P. Young, Everything You Wanted to Know About Data Analysis and Fitting but Were Afraid to Ask (Springer, Cham, 2015).
  42. H. Touchette, The large deviation approach to statistical mechanics, Phys. Rep. 478 (2009).
  43. S. Wiseman and E. Domany, Lack of self-averaging in critical disordered systems, Phys. Rev. E 52, 3469 (1995).
  44. S. Wiseman and E. Domany, Finite-size scaling and lack of self-averaging in critical disordered systems, Phys. Rev. Lett. 81 (1998).
  45. R. Fisch, Critical temperature for two-dimensional Ising ferromagnets with quenched bond disorder, J. Stat. Phys. 18, 111 (1978).
  46. O. Melchert, autoScale.py - a program for automatic finite-size scaling analyses: a user’s guide (2009), arXiv:0910.5403 .
  47. D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd ed. (Taylor & Francis, London, 1994).
  48. L. Münster and M. Weigel, Cluster percolation in the two-dimensional ising spin glass, Phys. Rev. E 107, 054103 (2023).

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