Publication: Numerical test of the Cardy-Jacobsen conjecture in the site-diluted Potts model in three dimensions
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2012-11-16
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American Physical Society
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Abstract
We present a microcanonical Monte Carlo simulation of the site-diluted Potts model in three dimensions with eight internal states, partly carried out on the citizen supercomputer Ibercivis. Upon dilution, the pure model’s first-order transition becomes of the second order at a tricritical point. We compute accurately the critical exponents at the tricritical point. As expected from the Cardy-Jacobsen conjecture, they are compatible with their random field Ising model counterpart. The conclusion is further reinforced by comparison with older data for the Potts model with four states.
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© 2012 American Physical Society. We have been partly supported through Research Contracts No. FIS2009-12648-C03 and No. FIS2010-16587 (MICINN), No. GR10158 (Junta de Extremadura), and No. ACCVII-08 (UEX), and by UCM-Banco de Santander. We thank Ibercivis for the equivalent of 3 × 10^(6) CPU hours. The simulations were completed in the clusters Terminus (BIFI) and Horus (U. Extremadura). We also thank N. G. Fytas for a careful reading of the manuscript.
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1) E. Dagotto, Science, 309, 258 (2005)
-- J. Burgy, M. Mayr, V. Martín-Mayor, A. Moreo, E. Dagotto, Phys. Rev. Lett., 87, 277202 (2001)
-- J. Burgy, A. Moreo, E. Dagotto, ibid., 92, 097202 (2004)
-- C. Sen, G. Álvarez, E. Dagotto, ibid., 98, 127202 (2007).
2) See, e.g., G. Parisi, Field Theory, Disorder, and Simulations (World Scientific, 1994).
3) G. Xu, W. Jiang, M. Qian, X. Chen, Z. Li, G. Han, Crystal Growth and Design, 9, 13 (2009).
4) H. Koibuchi, Phys. Rev. E, 77, 021104 (2008).
5) Z.-N. Hu, V. C. Lo, Commun. Theor. Phys., 48, 183 (2007).
6) J. F. Scott, et al., J. Am. Ceram. Soc., 88, 1691 (2005).
7) H. Westfahl, J. Schmalian, Phys. Rev. E, 72, 011806 (2005).
8) I. I. Naumov, L. Bellaiche, H. X. Fu, Nature (London), 432, 737 (2004).
9) M. Freedman, C. Nayak, K. Shtengel, K. Walker, Z. H. Wang, Ann. Phys., 310, 428 (2004).
10) R. Berthet, A. Petrossian, S. Residori, B. Roman, S. Fauve, Physica D, 174, 84 (2003).
11) S. Residori, R. Berthet, B. Roman, S. Fauve, Phys. Rev. Lett., 88, 024502 (2001).
12) M. Aizenman, J. Wehr, Phys. Rev. Lett., 62, 2503 (1989)
-- K. Hui, A. N. Berker, ibid., 62, 2507 (1989).
13) J. Cardy, J. L. Jacobsen, Phys. Rev. Lett., 79, 4063 (1997)
-- J. L. Jacobsen, J. Cardy, Nucl. Phys. B, 515, 701 (1998)
-- J. Cardy, Physica A, 263, 215 (1999).
14) C. Chatelain, B. Berche, Phys. Rev. Lett., 80, 1670 (1998)
-- Phys. Rev E, 58, R6899 (1998)
--- ibid, 60, 3853 (1999).
15) T. Natherman, in Spin Glasses and Random Fields, edited by A. P. Young (World Scientific, Singapore, 1998).
16) D. P. Belanger, in Spin Glasses and Random Fields, edited by A. P. Young (World Scientific, Singapore, 1998).
17) F. Y. Wu, Rev. Mod. Phys., 54, 235 (1982).
18) J. Bricmont, A. Kupiainen, Phys. Rev. Lett. 59, 1829 (1987).
19) L. A. Fernández, A. Gordillo-Guerrero, V. Martín-Mayor, J. J. Ruiz-Lorenzo, Phys. Rev. Lett., 100, 057201 (2008).
20) M. T. Mercaldo, J-Ch. Angles d’Auriac, F. Igloi, Phys. Rev. E, 73, 026126 (2006).
21) One may fit spin-glass interactions into the same scheme, see Refs. 41 and 42.
22) Z. Slanic, D. P. Belanger, J. A. Fernández-Baca, Phys. Rev. Lett., 82, 426 (1999).
23) F. Ye, L. Zhou, S. Larochelle, L. Lu, D. P. Belanger, M. Greven, D. Lederman, Phys. Rev. Lett., 89, 157202 (2002).
24) L. A. Fernández, V. Martín-Mayor, D. Yllanes, Phys. Rev. B, 84, 100408(R) (2011).
25) A. T. Ogielski, Phys. Rev. Lett., 57, 1251 (1986).
26) M. Gofman, J. Adler, A. Aharony, A. B. Harris, M. Schwartz, Phys. Rev. Lett., 71, 1569 (1993).
27) M. E. J. Newman, G. T. Barkema, Phys. Rev. E, 53, 393 (1996).
28) M. R. Swift, A. J. Bray, A. Maritan, M. Cieplak, J. R. Banavar, Europhys. Lett., 38, 273 (1997).
29) H. Rieger, Phys. Rev. B, 52, 6659 (1995).
30) I. Dukovski, J. Machta, Phys. Rev. B, 67, 014413 (2003).
31) J.-C. Angles d’Auriac, N. Sourlas, Europhys. Lett., 39, 473 (1997).
32) U. Nowak, K. D. Usadel, J. Esser, Physica A, 250, 1 (1998).
33) A. K. Hartmann, U. Nowak, Eur. Phys. J. B, 7, 105 (1999).
34) A. A. Middleton, D. S. Fisher, Phys. Rev. B, 65, 134411 (2002).
35) A. K. Hartmann, A. P. Young, Phys. Rev. B, 64, 214419 (2001).
36) N. Sourlas, Comp. Phys. Commun., 121, 183 (1999).
37) A. Maiorano, V. Martín-Mayor, J. J. Ruiz-Lorenzo, A. Tarancón, Phys. Rev. B, 76, 064435 (2007).
38) H. G. Ballesteros, L. A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, G. Parisi, J. J. Ruiz-Lorenzo, Phys. Rev. B, 61, 3215 (2000).
39) C. Chatelain, B. Berche, W. Janke, P.-E. Berche, Phys. Rev. E, 64, 036120 (2001).
40) C. Chatelain, B. Berche, W. Janke, P.-E. Berche, Nucl. Phys. B, 719, 275 (2005).
41) M. Paoluzzi, L. Leuzzi, A. Crisanti, Phys. Rev. Lett., 104, 120602 (2010).
42) L. Leuzzi, M. Paoluzzi, A. Crisanti, Phys. Rev. B, 83, 014107 (2011).
43) A. Malakis, A. N. Berker, N. G. Fytas, T. Papakonstantinou, Phys. Rev. E, 85, 061106 (2012).
44) V. Martín-Mayor, Phys. Rev. Lett., 98, 137207 (2007).
45) Ibercivis (http://www.ibercivis.net) provides nowadays as much as 105 computing cores (this quantity fluctuates, depending on the number of volunteers on-line). The current count of volunteers is 3 × 105 and growing.
46) D. Amit, V. Martín-Mayor, Field Theory, the Renormalization Group, and Critical Phenomena, 3rd ed. (World-Scientific, Singapore, 2005).
47) Remember that the parameter that plays the role of the reduced temperature in the RFIM is τ ≈ hR/J (Ref. 15). In the Cardy-Jacobsen mapping, the disorder strength in the Potts model, w, corresponds exactly with τ (Ref. 13). Hence, yp = 1/νRFIM. However, keep in mind that this relation was obtained by reasoning in the limit Q » 1.
48) L. A. Fernández, A. Gordillo-Guerrero, V. Martín Mayor, J. J. Ruiz-Lorenzo, Phys. Rev. E, 80, 051105 (2009).
49) W. Janke, Nucl. Phys. B, Proc. Suppl., 63, 631 (1998).
50) M. S. S. Challa, D. P. Landau, K. Binder, Phys. Rev. B, 34, 1841 (1986)
-- J. Lee, J. M. Kosterlitz, Phys. Rev. Lett., 65, 137 (1990).
51) C. Borgs, R. Kotecký, Phys. Rev. Lett., 68, 1734 (1992).
52) In a system with periodic boundary conditions, geometrical transitions arise when the energy e varies from the ordered to the disordered phase. In fact, the system struggles to minimize the surface energy while respecting the global constraint for e. Depending on the fraction of ordered phase, which is fixed by e, the minimizing geometry can be either a bubble, a cylinder, or a slab of ordered phase in a disordered matrix (or vice versa). As e varies, the minimizing geometry changes at definite e values. This phenomenon is named geometric transition (Ref. 59), and it is the reason underlying the steps and the flat central region for p = 1, shown in Fig. 3. On the other hand, the two cusps (one lies near to eo while the other is close to ed) are sometimes named size- dependent spinodal points. They are due to the condensation transition (Ref. 60).
53) R. A. Baños, et al., (Janus Collaboration), Phys. Rev. Lett., 105, 177202 (2010).
54) H. G. Ballesteros, L. A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, Phys. Lett. B, 378, 207 (1996)
-- ibid, 387, 125 (1996)
-- Nucl. Phys. B, 483, 707 (1997).
55) M. P. Nightingale, Physica A, 83, 561 (1976).
56) M. Tissier, G. Tarjus, Phys. Rev. B, 85, 104203 (2012).
57) D. P. Belanger, A. R. King, V. Jaccarino, J. L. Cardy, Phys. Rev. B, 28, 2522 (1983)
-- D. P. Belanger, Z. Slanic, J. Magn. Magn. Mater., 186, 65 (1998).
58) L. A. Fernández, V. Martín-Mayor, Phys. Rev. E, 79, 051109 (2009).
59) K. Leung, R. K. P. Zia, J. Phys. A: Math. Gen., 23, 4593 (1990)
-- L. G. MacDowell, V. K. Shen, J. R. Errington, J. Chem. Phys., 125, 034705 (2006).
60) M. Biskup, L. Chayes, R. Kotecký, Europhys. Lett., 60, 21 (2002)
-- K. Binder, Physica A, 319, 99 (2003)
-- L. G. MacDowell, P. Virnau, M. Müller, K. Binder, J. Chem. Phys., 120, 5293 (2004)
-- A. Nußbaumer, E. Bittner, T. Neuhaus, W. Janke, Europhys. Lett., 75, 716 (2006).