Dynamic critical behavior of model a in films: Zero-mode boundary conditions and expansion near four dimensions

Physical Review B - Condensed Matter and Materials Physics

Volumn 79, Issue 10, 2009, Pages

  Diehl, H W ^a,b   Chamati, H ^a,c  

^a UNIVERSITY OF DUISBURG ESSEN   (Germany)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c INSTITUTE OF SOLID STATE PHYSICS   (Bulgaria)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 63249121486     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.79.104301     Document Type: Article

References (110)

1. For background on the RG and references, see, e.g., Refs..
2. K. G. Wilson and J. Kogut, Phys. Rep. 12, 75 (1974). 10.1016/0370- 1573(74)90023-4
3. F. J. Wegner, in Phase Transitions and Critical Phenomena, edited by, C. Domb, and, M. S. Green, (Academic, London, 1976), Vol. 6, Chap., pp. 7-124.
4. M. E. Fisher, Rev. Mod. Phys. 70, 653 (1998). 10.1103/RevModPhys.70.653
5. P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977). 10.1103/RevModPhys.49.435
6. R. Folk and G. Moser, J. Phys. A 39, R207 (2006). 10.1088/0305-4470/39/ 24/R01
7. H. W. Diehl, in Phase Transitions and Critical Phenomena, edited by, C. Domb, and, J. L. Lebowitz, (Academic, London, 1986), Vol. 10, pp. 75-267.
8. H. W. Diehl, Int. J. Mod. Phys. B 11, 3503 (1997). 10.1142/ S0217979297001751
9. The application of the field-theoretic RG to the study of finite-size effects started with Symanzik's seminal paper (Ref.) on systems confined between parallel plates under Dirichlet boundary conditions, Brézin's work on the large- n limit (Ref.), and two papers (Refs.) developing a small ∈=4-d expansion for the study of finite-size effects in d -dimensional systems that are finite in all, or in all but one, of the d fundamental directions and satisfy periodic boundary conditions. For background and references on finite-size effects, see Refs..
10. K. Symanzik, Nucl. Phys. B 190, 1 (1981). 10.1016/0550-3213(81)90482-X
11. E. Brézin, J. Phys. (Paris) 43, 15 (1982). 10.1051/jphys: 0198200430101500
12. E. Brézin and J. Zinn-Justin, Nucl. Phys. B 257, 867 (1985). 10.1016/0550-3213(85)90379-7
13. J. Rudnick, H. Guo, and D. Jasnow, J. Stat. Phys. 41, 353 (1985). 10.1007/BF01009013
14. M. N. Barber, in Phase Transitions and Critical Phenomena, edited by, C. Domb, and, J. L. Lebowitz, (Academic, London, 1983), Vol. 8, pp. 145-266.
15. V. Privman, in Finite Size Scaling and Numerical Simulation of Statistical Systems, edited by, V. Privman, (World Scientific, Singapore, 1990), Chap..
16. J. G. Brankov, D. M. Dantchev, and N. S. Tonchev, Theory of Critical Phenomena in Finite-Size Systems-Scaling and Quantum Effects (World Scientific, Singapore, 2000).
17. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, International Series of Monographs on Physics, 4th ed. (Oxford University Press, Oxford, 2002).
18. Y. Y. Goldschmidt, Nucl. Phys. B 280, 340 (1987). 10.1016/0550-3213(87) 90152-0
19. J. C. Niél and J. Zinn-Justin, Nucl. Phys. B 280, 355 (1987). 10.1016/0550-3213(87)90153-2
20. H. W. Diehl, Z. Phys. B: Condens. Matter 66, 211 (1987). 10.1007/BF01311657
21. V. Dohm, Z. Phys. B: Condens. Matter 75, 109 (1989). 10.1007/BF01313573
22. V. Dohm, Phys. Scr. T T49A, 46 (1993). 10.1088/0031-8949/1993/T49A/007
23. M. Barmatz, I. Hahn, J. A. Lipa, and R. V. Duncan, Rev. Mod. Phys. 79, 1 (2007). 10.1103/RevModPhys.79.1
24. M. Calvo and R. A. Ferrell, Phys. Rev. A 31, 2570 (1985). 10.1103/PhysRevA.31.2570
25. M. Calvo, Phys. Rev. A 31, 2588 (1985). 10.1103/PhysRevA.31.2588
26. J. K. Bhattacharjee, Phys. Rev. Lett. 77, 1524 (1996). 10.1103/PhysRevLett.77.1524
27. S. Dietrich and H. W. Diehl, Z. Phys. B: Condens. Matter 51, 343 (1983) 10.1007/BF01319217
28. S. Dietrich and H. W. Diehl, Z. Phys. B: Condens. Matter 52, 171 (E) (1983). 10.1007/BF01445298
29. Y. Y. Goldschmidt, Nucl. Phys. B 285, 519 (1987). 10.1016/0550-3213(87) 90352-X
30. H. W. Diehl and S. Dietrich, in Festkörperprobleme, Advances in Solid State Physics Vol. 25, edited by, P. Grosse, (Vieweg, Braunschweig, 1985), pp. 39-52.
31. D. Frank and V. Dohm, Phys. Rev. Lett. 62, 1864 (1989). 10.1103/PhysRevLett.62.1864
32. G. M. Xiong and C. D. Gong, Z. Phys. B: Condens. Matter 74, 379 (1989). 10.1007/BF01307887
33. G. M. Xiong and C. D. Gong, J. Phys.: Condens. Matter 1, 8673 (1989). 10.1088/0953-8984/1/44/037
34. D. Frank and V. Dohm, Z. Phys. B: Condens. Matter 84, 443 (1991). 10.1007/BF01314020
35. K. Binder and H. L. Frisch, Z. Phys. B: Condens. Matter 84, 403 (1991). 10.1007/BF01314015
36. H. W. Diehl and U. Ritschel, J. Stat. Phys. 73, 1 (1993). 10.1007/BF01052748
37. H. W. Diehl and H. K. Janssen, Phys. Rev. A 45, 7145 (1992). 10.1103/PhysRevA.45.7145
38. H. W. Diehl, Phys. Rev. B 49, 2846 (1994). 10.1103/PhysRevB.49.2846
39. F. Wichmann and H. W. Diehl, Z. Phys. B: Condens. Matter 97, 251 (1995). 10.1007/BF01307476
40. U. Ritschel and P. Czerner, Phys. Rev. Lett. 75, 3882 (1995). 10.1103/PhysRevLett.75.3882
41. U. Ritschel and H. W. Diehl, Nucl. Phys. B 464, 512 (1996). 10.1016/0550-3213(96)00012-0
42. M. Krech, H. Karl, and H. W. Diehl, Physica A 297, 64 (2001). 10.1016/S0378-4371(01)00099-1
43. H. W. Diehl, M. Krech, and H. Karl, Phys. Rev. B 66, 024408 (2002). 10.1103/PhysRevB.66.024408
44. M. V. Manias, A. De Virgiliis, E. V. Albano, M. Müller, and K. Binder, Phys. Rev. E 75, 051603 (2007). 10.1103/PhysRevE.75.051603
45. H. Chamati and N. S. Tonchev, Phys. Rev. E 63, 026103 (2001). 10.1103/PhysRevE.63.026103
46. H. Chamati, E. Korutcheva, and N. S. Tonchev, Phys. Rev. E 65, 026129 (2002) 10.1103/PhysRevE.65.026129
47. H. Chamati and E. Korutcheva, Phys. Rev. B 77, 184416 (2008). 10.1103/PhysRevB.77.184416
48. A. Gambassi, Eur. Phys. J. B 64, 379 (2008). 10.1140/epjb/e2008-00043-y
49. A. Gambassi and S. Dietrich, J. Stat. Phys. 123, 929 (2006). 10.1007/s10955-006-9089-8
50. K. Binder, in Phase Transitions and Critical Phenomena, edited by, C. Domb, and, J. L. Lebowitz, (Academic, London, 1983), Vol. 8, pp. 1-144.
51. M. Krech and S. Dietrich, Phys. Rev. A 46, 1886 (1992). 10.1103/PhysRevA.46.1886
52. H. W. Diehl, D. Grüneberg, and M. A. Shpot, Europhys. Lett. 75, 241 (2006). 10.1209/epl/i2006-10090-0
53. D. Grüneberg and H. W. Diehl, Phys. Rev. B 77, 115409 (2008). 10.1103/PhysRevB.77.115409
54. S. Sachdev, Phys. Rev. B 55, 142 (1997). 10.1103/PhysRevB.55.142
55. H.-K. Janssen, Z. Phys. B 23, 377 (1976). 10.1007/BF01316547
56. C. de Dominicis, J. Phys. (Paris), Colloq. 37, C1-247 (1976). 10.1051/jphyscol:1976138
57. H. K. Janssen, in From Phase Transitions to Chaos, edited by, G. Györgyi, I. Kondor, L. Sasvári, and, T. Tel, (World Scientific, Singapore, 1992), pp. 68-91.
58. X. S. Chen and V. Dohm, Phys. Rev. E 66, 016102 (2002) 10.1103/PhysRevE.66.016102
59. X. S. Chen and V. Dohm, Phys. Rev. E 66, 059901 (E) (2002). 10.1103/PhysRevE.66.059901
60. V. Dohm, Phys. Rev. E 77, 061128 (2008). 10.1103/PhysRevE.77.061128
61. H. W. Diehl and E. Eisenriegler, Phys. Rev. Lett. 48, 1767 (1982). 10.1103/PhysRevLett.48.1767
62. H. W. Diehl and E. Eisenriegler, Phys. Rev. B 30, 300 (1984). 10.1103/PhysRevB.30.300
63. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, New York, 1992).
64. Form 2.4 of the action J involving is obtained directly if one keeps carefully track of the boundary contributions to δH= ∫V { δφ [- 2 + τ· + (u· /3!) φ2] φ } + ∫B δφ (c· - n) φ when going over from the Langevin equation to the Langrangian formulation. Rewriting the term ∫B φ in terms of ∫B (- 2) φ requires an integration by parts, which produces the boundary contribution ∫B (- n) φ. This contributes to the boundary terms of the classical equations of motion which yield the boundary conditions given in Eq. 2.12. The form of the action given in Eq. (10) of Ref. is consistent with our Eq. 2.4 provided contributions δ (z) and δ (z-L) are included in the functional derivative δH/δφ that yield the boundary terms ∫ Bj (c· j - n) φ. For details, see Refs..
65. A. Romeo and A. A. Saharian, J. Phys. A 35, 1297 (2002). 10.1088/0305-4470/35/5/312
66. F. M. Schmidt, Diplomarbeit (Universität Duisburg, Essen, 2008).
67. F. M. Schmidt and H. W. Diehl, Phys. Rev. Lett. 101, 100601 (2008). 10.1103/PhysRevLett.101.100601
68. R. Courant and D. Hilbert, Methoden der Mathematischen Physik I (Heidelberger Taschenbücher), 3rd ed. (Springer-Verlag, Berlin, 1968), Vol. 30.
69. H. W. Diehl and S. Dietrich, Phys. Rev. B 24, 2878 (1981). 10.1103/PhysRevB.24.2878
70. H. W. Diehl and S. Dietrich, Z. Phys. B: Condens. Matter 50, 117 (1983). 10.1007/BF01304094
71. M. E. Fisher and A. Aharony, Phys. Rev. B 10, 2818 (1974). 10.1103/PhysRevB.10.2818
72. D. J. Amit and V. Martin-Mayor, Field Theory, The Renormalization Group, and Critical Phenomena, 3rd ed. (World Scientific, Singapore, 2005).
73. L. Schäfer and H. Horner, Z. Phys. B: Condens Matter 29, 251 (1978) 10.1007/BF01321190.
74. D. J. Amit and Y. Y. Goldschmidt, Ann. Phys. (N.Y.) 114, 356 (1978). 10.1016/0003-4916(78)90274-9
75. I. D. Lawrie, J. Phys. A 14, 2489 (1981). 10.1088/0305-4470/14/9/041
76. R. B. Griffiths, Phys. Rev. 176, 655 (1968). 10.1103/PhysRev.176.655
77. D. G. Kelly and S. Sherman, J. Math. Phys. 9, 466 (1968). 10.1063/1.1664600
78. M. Krech and S. Dietrich, Phys. Rev. Lett. 66, 345 (1991) 10.1103/PhysRevLett.66.345
79. M. Krech and S. Dietrich, Phys. Rev. Lett. 67, 1055 (E) (1991). 10.1103/PhysRevLett.67.1055.2
80. M. Krech, E. Eisenriegler, and S. Dietrich, Phys. Rev. E 52, 1345 (1995). 10.1103/PhysRevE.52.1345
81. For background on the thermodynamic Casimir effect and further references, see Ref..
82. M. Krech, Casimir Effect in Critical Systems (World Scientific, Singapore, 1994).
83. R. Zandi, A. Shackell, J. Rudnick, M. Kardar, and L. P. Chayes, Phys. Rev. E 76, 030601 (2007). 10.1103/PhysRevE.76.030601
84. A. Maciołek, A. Gambassi, and S. Dietrich, Phys. Rev. E 76, 031124 (2007). 10.1103/PhysRevE.76.031124
85. RG-improved Landau theory was used in Ref. to explain the dip in the measured thickness dependence of He wetting layers (Ref.). Unfortunately, Landau theory predicts for this n=2 component case that a transition to a phase with long-range order occurs for finite thickness L at a shifted temperature Tc,L < Tc,∞. However, for finite L and n=2, one rather expects a transition to a low-temperature phase with quasi-long-range order. Moreover, the minimum of the (much too deep) dip that Landau theory yields is located precisely in the temperature regime where a transition to a phase with long-range order is erroneously predicted. Evidently, Landau theory can hardly be trusted in this regime. It fails both quantitatively by predicting a much too deep dip as well as qualitatively, giving the usual power-law singularities for the finite-size susceptibility and a jump singularity for the second temperature derivative of the excess free energy per area. Recent Monte Carlo work (Refs.) does not suffer from such problems.
86. R. Garcia and M. H. W. Chan, Phys. Rev. Lett. 83, 1187 (1999). 10.1103/PhysRevLett.83.1187
87. A. Hucht, Phys. Rev. Lett. 99, 185301 (2007). 10.1103/PhysRevLett.99. 185301
88. O. Vasilyev, A. Gambassi, A. Maciołek, and S. Dietrich, EPL 80, 60009 (2007). 10.1209/0295-5075/80/60009
89. The functions Qd,2 are encountered also in the analysis of the quantum spherical model; see Ref..
90. H. Chamati, E. S. Pisanova, and N. S. Tonchev, Phys. Rev. B 57, 5798 (1998). 10.1103/PhysRevB.57.5798
91. D. Dantchev, H. W. Diehl, and D. Grüneberg, Phys. Rev. E 73, 016131 (2006). 10.1103/PhysRevE.73.016131
92. H. J. F. Knops, J. Math. Phys. 14, 1918 (1973). 10.1063/1.1666269
93. For a recent analysis of conventional mean spherical models under various free boundary conditions, see H. Chamati, J. Phys. A: Math. Theor. 41, 375002 (2008) and its references. 10.1088/1751-8113/41/37/375002
94. Note that there is a misprint in the second part of Eq. (4.72) of Ref.: the argument of the exponential function under the square root should be ř∞ rather than ř∞ /2.
95. M. Le Bellac, Thermal Field Theory, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 2004).
96. I. T. Drummond, R. R. Horgan, P. V. Landshoff, and A. Rebhan, Nucl. Phys. B 524, 579 (1998). 10.1016/S0550-3213(98)00210-7
97. See, e.g., G. Baym, J.-P. Blaizot, and J. Zinn-Justin, Europhys. Lett. 49, 150 (2000), and Ref.. 10.1209/epl/i2000-00130-3
98. D. Dantchev and J. Rudnick, Eur. Phys. J. B 21, 251 (2001). 10.1007/s100510170201
99. H. Chamati and D. M. Dantchev, Eur. Phys. J. B 26, 89 (2002) 10.1140/epjb/e20020070.
100. D. Dantchev, M. Krech, and S. Dietrich, Phys. Rev. E 67, 066120 (2003). 10.1103/PhysRevE.67.066120
101. As discussed in Ref., the linear scaling field associated with the subleading long-range interaction is coupled beyond linear order to other linear scaling fields such as the one ∼ (u- u) associated with the usual Wegner-type corrections to scaling. This coupling implies a mixing of these irrelevant scaling fields whereby the nonlinear scaling field associated with the latter corrections to scaling acquires a dependence on the strength of the subleading long-range interaction.
102. P. G. de Gennes, Macromolecules 14, 1637 (1981). 10.1021/ma50007a007
103. L. Peliti and S. Leibler, J. Phys. C 16, 2635 (1983). 10.1088/0022-3719/16/13/027
104. I. M. Gel'fand and G. E. Shilov, Generalized Functions (Academic, New York, 1964), Vol. 1, pp. 1-423.
105. A. Drewitz, R. Leidl, T. W. Burkhardt, and H. W. Diehl, Phys. Rev. Lett. 78, 1090 (1997) 10.1103/PhysRevLett.78.1090
106. R. Leidl and H. W. Diehl, Phys. Rev. B 57, 1908 (1998). 10.1103/PhysRevB.57.1908
107. S. Krimmel, W. Donner, B. Nickel, H. Dosch, C. Sutter, and G. Grübel, Phys. Rev. Lett. 78, 3880 (1997). 10.1103/PhysRevLett.78.3880
108. D. Dantchev and D. Grüneberg, Casimir force in O (n) lattice models with a diffuse interface (2008), http://www.citebase.org/abstract?id=oai:arXiv. org:0806.3718.
109. For the definition of the Hurwitz zeta function see, e.g., W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, Berlin, 1966).
110. MATHEMATICA, Version 6, Wolfram Research.