Advertisement

Numerical Tests of Schramm-Loewner Evolution in Random Lattice Spin Models

  • Christophe ChatelainEmail author
Part of the Lecture Notes in Physics book series (LNP, volume 853)

Abstract

Interfaces induced by symmetry-breaking boundary conditions in lattice spin models are possible realisations of SLE. For homogeneous systems with short-ranged interactions, it can be shown, even rigorously in some cases, that the assumptions behind SLE (Domain Markov property and conformal invariance) hold true. However, this is not the case for random systems. In recent years, numerical simulations gave some evidences of a possible description of interfaces in random systems by SLE. Here, different numerical tests will be reviewed, both of the basic assumptions which underlie SLE and of several derived consequences of the SLE description of critical phenomena, which have been tried up to now. A detailed exposition of these evidences in the cases of the random Potts model, the Solid-On-Solid model on a random substrate and the 2D Ising spin glass will be provided.

Keywords

Fractal Dimension Domain Wall Ising Model Conformal Transformation Conformal Invariance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Adams, D.A., Sander, L.M., Ziff, R.M.: Fractal dimensions of the Q-state Potts model for complete and external hulls. J. Stat. Mech. P03004 (2010) Google Scholar
  2. 2.
    Amoruso, C., Hartmann, A.K., Hastings, M.B., Moore, M.A.: Conformal invariance and stochastic Loewner evolution processes in two-dimensional Ising spin glasses. Phys. Rev. Lett. 97, 267202 (2006) ADSCrossRefGoogle Scholar
  3. 3.
    Bauer, M., Bernard, D.: Conformal field theories of stochastic Loewner evolutions. Commun. Math. Phys. 239, 493 (2003) MathSciNetADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Bauer, M., Bernard, D.: Conformal transformations and the SLE partition function martingale. Ann. Henri Poincaré 5, 289 (2004) MathSciNetADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Bauer, M., Bernard, D., Houdayer, J.: Dipolar SLEs. J. Stat. Mech. P03001 (2005) Google Scholar
  6. 6.
    Beffara, V.: Hausdorff dimensions for SLE6. Ann. Probab. 32, 2606 (2002) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bernard, D., Boffetta, G., Celani, A., Falkovich, G.: Inverse turbulent cascades and conformally invariant curves. Phys. Rev. Lett. 98, 024501 (2007) ADSCrossRefGoogle Scholar
  8. 8.
    Bernard, D., Doussal, P.L., Middleton, A.A.: Possible description of domain walls in two-dimensional spin glasses by stochastic Loewner evolutions. Phys. Rev. B 76, 020403 (2007) ADSCrossRefGoogle Scholar
  9. 9.
    Camia, F., Newman, C.M.: Critical percolation exploration path and SLE6: a proof of convergence. Probab. Theory Relat. Fields 139, 473 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cardy, J.L., Jacobsen, J.L.: Critical behaviour of random-bond Potts models. Phys. Rev. Lett. 79, 4063 (1997) MathSciNetADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Caselle, M., Lottini, S., Rajabpour, M.A.: Critical domain walls in the Ashkin-Teller model. J. Stat. Mech. P02039 (2011) Google Scholar
  12. 12.
    Chatelain, C.: Numerical study of Schramm-Loewner evolution in the random 3-state Potts model. J. Stat. Mech. P08004 (2010) Google Scholar
  13. 13.
    Chatelain, C., Berche, B.: Finite-size scaling study of the surface and sulk critical behaviour in the random-bond eight-states Potts model. Phys. Rev. Lett. 80, 1670 (1998) ADSCrossRefGoogle Scholar
  14. 14.
    Chatelain, C., Berche, B.: Tests of conformal invariance in randomness-induced second-order phase transitions. Phys. Rev. E 58, 6899 (1998) ADSCrossRefGoogle Scholar
  15. 15.
    Chatelain, C., Berche, B.: Magnetic critical behavior of two-dimensional random-bond Potts ferromagnets in confined geometries. Phys. Rev. E 60, 3853 (1999) ADSCrossRefGoogle Scholar
  16. 16.
    Coniglio, A.: Fractal structure of Ising and Potts clusters: exact results. Phys. Rev. Lett. 62, 3054 (1989) MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Dotsenko, V.S., Picco, M., Pujol, P.: Renormalisation-group calculation of correlation functions for the 2D random bond Ising and Potts models. Nucl. Phys. B 455, 701 (1995) ADSCrossRefGoogle Scholar
  18. 18.
    Dubail, J., Jacobsen, J.L., Saleur, H.: Bulk and boundary critical behaviour of thin and thick domain walls in the two-dimensional Potts model. J. Stat. Mech. P12026 (2010) Google Scholar
  19. 19.
    Dubail, J., Jacobsen, J.L., Saleur, H.: Critical exponents of domain walls in the two-dimensional Potts model. J. Phys. A, Math. Theor. 43, 482002 (2010) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Duplantier, B.: Conformally invariant fractals and potential theory. Phys. Rev. Lett. 84, 1363 (2000) MathSciNetADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Duxbury, P.M.: Exact computations test stochastic Loewner evolution and scaling in glassy systems. J. Stat. Mech. N09001 (2009) Google Scholar
  22. 22.
    Edwards, S.F., Anderson, P.W.: Theory of spin glasses. J. Phys. F, Met. Phys. 5, 965 (1975) ADSCrossRefGoogle Scholar
  23. 23.
    Fortuin, C.M., Kasteleyn, P.W.: On the random-cluster model. I. Introduction and relation to other models. Physica 57, 536 (1972) MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Gamsa, A., Cardy, J.L.: Schramm-Loewner evolution in the three-state Potts model: a numerical study. J. Stat. Mech. P08020 (2007) Google Scholar
  25. 25.
    Gliozzi, F., Rajabpour, M.A.: Conformal curves in the Potts model: numerical calculation. J. Stat. Mech. L05004 (2010) Google Scholar
  26. 26.
    Harris, A.B.: Effect of random defects on the critical behaviour of Ising models. J. Phys. C, Solid State Phys. 7, 1671 (1974) ADSCrossRefGoogle Scholar
  27. 27.
    Hartmann, A.K., Young, A.P.: Large-scale low-energy excitations in the two-dimensional Ising spin glass. Phys. Rev. B 66, 094419 (2002) ADSCrossRefGoogle Scholar
  28. 28.
    Ising, E.: Beitrag zur Theorie des Ferromagnetismus. Z. Phys. 31, 253 (1925) ADSCrossRefGoogle Scholar
  29. 29.
    Jacobsen, J.L., Cardy, J.L.: Critical behaviour of random-bond Potts models: a transfer matrix study. Nucl. Phys. B 515, 701 (1998) MathSciNetADSCrossRefzbMATHGoogle Scholar
  30. 30.
    Jacobsen, J.L., Doussal, P.L., Picco, M., Santachiara, R., Wiese, K.J.: Critical interfaces in the random-bond Potts model. Phys. Rev. Lett. 102, 070601 (2009) ADSCrossRefGoogle Scholar
  31. 31.
    Kennedy, T.: A fast algorithm for simulating the chordal Schramm-Loewner evolution. J. Stat. Phys. 128, 1125 (2007) MathSciNetADSCrossRefzbMATHGoogle Scholar
  32. 32.
    Kennedy, T.: Computing the Loewner driving process of random curves in the half plane. J. Stat. Phys. 131, 803 (2008) MathSciNetADSCrossRefzbMATHGoogle Scholar
  33. 33.
    Kennedy, T.: Numerical computations for the Schramm-Loewner evolution. J. Stat. Phys. 137, 839 (2009) MathSciNetADSCrossRefzbMATHGoogle Scholar
  34. 34.
    Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning tress. Ann. Probab. 32, 939 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lawler, G.F., Schramm, O., Werner, W.: On the scaling limit of planar self-avoiding walk. In: Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2, p. 339. Am. Math. Soc., Providence (2004) CrossRefGoogle Scholar
  36. 36.
    Ludwig, A.W.W.: Critical behaviour of the two-dimensional random q-state Potts model by expansion in (q − 2). Nucl. Phys. B 285, 97 (1987) MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    Ludwig, A.W.W., Cardy, J.L.: Perturbative evaluation of the conformal anomaly at new critical points with applications to random systems. Nucl. Phys. B 285, 687 (1987) MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    Melchert, O., Hartmann, A.K.: Fractal dimension of domain walls in two-dimensional Ising spin glasses. Phys. Rev. B 76, 174411 (2007) ADSCrossRefGoogle Scholar
  39. 39.
    Melchert, O., Hartmann, A.K.: Scaling behavior of domain walls at the T = 0 ferromagnet to spin-glass transition. Phys. Rev. B 79, 184402 (2009) ADSCrossRefGoogle Scholar
  40. 40.
    Picco, M., Santachiara, R.: Numerical study on Schramm-Loewner evolution in nonminimal conformal field theories. Phys. Rev. Lett. 100, 015704 (2008) ADSCrossRefGoogle Scholar
  41. 41.
    Picco, M., Santachiara, R.: Critical interfaces of the Ashkin-Teller model at the parafermionic point. J. Stat. Mech. P07027 (2010) Google Scholar
  42. 42.
    Picco, M., Santachiara, R., Sicilia, A.: Geometrical properties of parafermionic spin models. J. Stat. Mech. P04013 (2009) Google Scholar
  43. 43.
    Potts, R.B.: Some generalized order-disorder transformations. Math. Proc. Camb. Philos. Soc. 48, 106 (1952) MathSciNetADSCrossRefzbMATHGoogle Scholar
  44. 44.
    Risau-Gusman, S., Romá, F.: Fractal dimension of domain walls in the Edwards-Anderson spin glass model. Phys. Rev. B 77, 134435 (2008) ADSCrossRefGoogle Scholar
  45. 45.
    Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math. 161, 879 (2005) MathSciNetCrossRefGoogle Scholar
  46. 46.
    Rushkin, I., Bettelheim, E., Gruzberg, I.A., Wiegmann, P.: Critical curves in conformally invariant statistical systems. J. Phys. A, Math. Theor. 40, 2165 (2007) MathSciNetADSCrossRefzbMATHGoogle Scholar
  47. 47.
    Saberi, A.A.: Thermal behaviour of spin clusters and interfaces in the two-dimensional Ising model on a square lattice. J. Stat. Mech. P07030 (2009) Google Scholar
  48. 48.
    Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Schramm, O.: A percolation formula. Electron. Commun. Probab. 6, 115 (2001) MathSciNetCrossRefGoogle Scholar
  50. 50.
    Schwarz, K., Karrenbauer, A., Schehr, G., Rieger, H.: Domain walls and chaos in the disordered SOS model. J. Stat. Mech. P08022 (2009) Google Scholar
  51. 51.
    Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris 333, 239 (2001) ADSCrossRefzbMATHGoogle Scholar
  52. 52.
    Smirnov, S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. 172, 1435 (2010) CrossRefzbMATHGoogle Scholar
  53. 53.
    Stanley, H.E.: Cluster shapes at the percolation threshold: and effective cluster dimensionality and its connection with critical-point exponents. J. Phys. A, Math. Gen. 10, 211 (1977) ADSCrossRefGoogle Scholar
  54. 54.
    Stevenson, J.D., Weigel, M.: Domain walls and Schramm-Loewner evolution in the random-field Ising model. Europhys. Lett. 95, 40001 (2011) ADSCrossRefGoogle Scholar
  55. 55.
    Stevenson, J.D., Weigel, M.: Percolation and Schramm-Loewner evolution in the 2D random-field Ising model. Comput. Phys. Commun. 182, 1879 (2011) ADSCrossRefzbMATHGoogle Scholar
  56. 56.
    Swendsen, R.H., Wang, J.S.: Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58, 86 (1987) ADSCrossRefGoogle Scholar
  57. 57.
    Wieland, B., Wilson, D.B.: Winding angle variance of Fortuin-Kasteleyn contours. Phys. Rev. E 68, 056101 (2003) ADSCrossRefGoogle Scholar
  58. 58.
    Wolff, U.: Collective Monte Carlo updating for spin systems. Phys. Rev. Lett. 62, 361 (1989) ADSCrossRefGoogle Scholar
  59. 59.
    Zatelepin, A., Shchur, L.: Duality of critical interfaces in Potts model: numerical check. arXiv:1008.3573 (2010)

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.GPS—DP2M—IJL (CNRS UMR 7198)Université de Lorraine NancyVandœuvre-lès-Nancy CedexFrance

Personalised recommendations