Critical dynamics of stochastic models with energy conservation (model <i>C</i>)

R. Folk; G. Moser

doi:10.1103/PhysRevE.69.036101

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Critical dynamics of stochastic models with energy conservation (model C)

R. Folk

G. Moser

Institute for Theoretical Physics, University of Linz, Linz, Austria

Institute for Physics and Biophysics, University of Salzburg, Salzburg, Austria

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Phys. Rev. E 69, 036101 – Published 4 March, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.036101

Abstract

We calculate the field-theoretic functions of the generalized dynamical model $C^{*},$ in which a conserved secondary density is coupled to a nonconserved complex order parameter, in two-loop order. We show that the fixed points in this extended model are equal to the fixed points obtained in model C with a real order parameter, which has been introduced by Halperin, Hohenberg, and Ma. Our results correct long-standing errors in the field-theoretic functions in model C published by several authors leading also to different fixed point values $w^{*}$ for the ratio of the two time scales involved. The stability regions of the fixed points, which remained partially unclear, considered in a “phase diagram”—whose axes are the spatial dimension d and number of order parameter components n—are now clarified. Especially an anomalous region found by previous authors, in which the scaling properties remained unsolved, does not exist. There are only two regions: one with a finite fixed point $w^{*}$ where the dynamical exponent z of the order parameter is $z = 2 + α / ν$ and another region where $w^{*} = 0$ and z is equal to the model A value.

References (38)

B. I. Halperin, P. C. Hohenberg, and Shang-keng Ma, Phys. Rev. Lett. 29, 1548 (1972) (model A).
B. I. Halperin, P. C. Hohenberg, and Shang-keng Ma, Phys. Rev. B 10, 139 (1974) (model C).
K. Oerding and H. K. Janssen, J. Phys. A 26, 3369 (1993).
W. Koch and V. Dohm, Phys. Rev. E 58, R1179 (1998).
G. Grinstein, Shang-keng Ma, and G. F. Mazenko, Phys. Rev. B 15, 258 (1977).
P. Calabrese and A. Gambassi, Phys. Rev. E 67, 036111 (2003).
V. I. Gorentsveig, P. Fratzl, and J. L. Lebowitz, Phys. Rev. B 55, 2912 (1997).
K. Binder, W. Kinzel, and D. P. Landau, Surf. Sci. 112, 232 (1982).
H. Tanaka, J. Phys.: Condens. Matter 11, L159 (1999).
E. D. Siggia and D. R. Nelson, Phys. Rev. B 15, 1427 (1977).
B. Berdnikov and K. Rajagopal, Phys. Rev. D 61, 105017 (2000).
S. Borsányi, A. Patkós, D. Sexty, and Zs. Szép, Phys. Rev. D 64, 125011 (2001).
E. Brezin and C. De Dominicis, Phys. Rev. B 12, 4954 (1975).
K. K. Murata, Phys. Rev. B 13, 2028 (1976).
B. I. Halperin, P. C. Hohenberg, and Shang-keng Ma, Phys. Rev. B 13, 4119 (1976).
R. Folk and G. Moser, Phys. Rev. Lett. 89, 125301 (2002).
R. Folk and G. Moser, Acta Phys. Slov. 52, 285 (2002).
R. Folk and G. Moser, Phys. Rev. Lett. 91, 030601 (2003).
V. Dohm, Phys. Rev. B 44, 2697 (1991).
G. Parisi, J. Stat. Phys. 23, 49 (1980).
D. J. Amit, Field Theory the Renormalization Group and Critical Phenomena, 2nd ed. (World Scientific, Singapore, 1984).
R. Schloms and V. Dohm, Nucl. Phys. B 328, 639 (1989).
Although the definitions of the ζ functions (11) for the dynamical parameters and the renormalization factors (B12) are different, the explicit expressions of the ζ functions are in agreement with [19].
After finishing our calculations we have been informed by V. Dohm of unpublished calculations by K. Oerding. His results for the dynamical ζ functions agree with ours.
We use, however, also in this case the ε-expanded fixed-point values of the static couplings. The two-loop β function for u has no real solution and one had to use the Borel-summed β function, which is not known for general n.
A. Pelissetto and E. Vicari, Phys. Rep. 368, 549 (2002).
Yu. Holovatch, M. Dudka, and T. Yavors’kii, J. Phys. Stud. 5, 233 (2001).
The parameter c has been calculated in three-loop order in [13] and corrected by Antonov and Vasil’ev [29]. The reason why the result of [29] is more reliable is the fact that c would be negative for $d < 3.70$ (ε>0.296) in [13], whereas this happens only for $d < 1.35$ (ε>2.65) in [29].
N. V. Antonov and A. N. Vasil’ev, Theor. Math. Phys. 60, 671 (1984).
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 1997).
Note that in [18] the dynamical critical exponent z for model A has been missprinted due to an erroneous minus sign before $c η .$
In [13] a renormalization with normalization conditions has been used. Therefore, in the comparison with our renormalization scheme using the minimal subtraction scheme in the ζ functions of [13] ε has to be set to zero.
R. Bausch, H. K. Janssen, and H. Wagner, Z. Phys. B 24, 113 (1976).
E. Brezin, J. C. Le Guillou, and J. Zinn-Justin, in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, New York, 1976), Vol. 4.
I. D. Lawrie, J. Phys. A 9, 961 (1976).
V. Dohm, Z. Phys. B: Condens. Matter 60, 61 (1985).
E. Eisenriegler and B. Schaub, Z. Phys. B: Condens. Matter 39, 65 (1980).
D. Stauffer, Int. J. Mod. Phys. C 8, 1263 (1997); P. Sen, S. Dasgupta, and D. Stauffer, Eur. Phys. J. B 1, 107 (1998); B. Zheng and H. J. Luo, Phys. Rev. E 63, 066130 (2001).

[c1] B. I. Halperin, P. C. Hohenberg, and Shang-keng Ma, Phys. Rev. Lett. 29, 1548 (1972) (model A).

[c2] B. I. Halperin, P. C. Hohenberg, and Shang-keng Ma, Phys. Rev. B 10, 139 (1974) (model C).

[c3] K. Oerding and H. K. Janssen, J. Phys. A 26, 3369 (1993).

[c4] W. Koch and V. Dohm, Phys. Rev. E 58, R1179 (1998).

[c5] G. Grinstein, Shang-keng Ma, and G. F. Mazenko, Phys. Rev. B 15, 258 (1977).

[c6] P. Calabrese and A. Gambassi, Phys. Rev. E 67, 036111 (2003).

[c7] V. I. Gorentsveig, P. Fratzl, and J. L. Lebowitz, Phys. Rev. B 55, 2912 (1997).

[c8] K. Binder, W. Kinzel, and D. P. Landau, Surf. Sci. 112, 232 (1982).

[c9] H. Tanaka, J. Phys.: Condens. Matter 11, L159 (1999).

[c10] E. D. Siggia and D. R. Nelson, Phys. Rev. B 15, 1427 (1977).

[c11] B. Berdnikov and K. Rajagopal, Phys. Rev. D 61, 105017 (2000).

[c12] S. Borsányi, A. Patkós, D. Sexty, and Zs. Szép, Phys. Rev. D 64, 125011 (2001).

[c13] E. Brezin and C. De Dominicis, Phys. Rev. B 12, 4954 (1975).

[c14] K. K. Murata, Phys. Rev. B 13, 2028 (1976).

[c15] B. I. Halperin, P. C. Hohenberg, and Shang-keng Ma, Phys. Rev. B 13, 4119 (1976).

[c16] R. Folk and G. Moser, Phys. Rev. Lett. 89, 125301 (2002).

[c17] R. Folk and G. Moser, Acta Phys. Slov. 52, 285 (2002).

[c18] R. Folk and G. Moser, Phys. Rev. Lett. 91, 030601 (2003).

[c19] V. Dohm, Phys. Rev. B 44, 2697 (1991).

[c20] G. Parisi, J. Stat. Phys. 23, 49 (1980).

[c21] D. J. Amit, Field Theory the Renormalization Group and Critical Phenomena, 2nd ed. (World Scientific, Singapore, 1984).

[c22] R. Schloms and V. Dohm, Nucl. Phys. B 328, 639 (1989).

[c23] Although the definitions of the ζ functions (11) for the dynamical parameters and the renormalization factors (B12) are different, the explicit expressions of the ζ functions are in agreement with [19].

[c24] After finishing our calculations we have been informed by V. Dohm of unpublished calculations by K. Oerding. His results for the dynamical ζ functions agree with ours.

[c25] We use, however, also in this case the ε-expanded fixed-point values of the static couplings. The two-loop β function for u has no real solution and one had to use the Borel-summed β function, which is not known for general n.

[c26] A. Pelissetto and E. Vicari, Phys. Rep. 368, 549 (2002).

[c27] Yu. Holovatch, M. Dudka, and T. Yavors’kii, J. Phys. Stud. 5, 233 (2001).

[c28] The parameter c has been calculated in three-loop order in [13] and corrected by Antonov and Vasil’ev [29]. The reason why the result of [29] is more reliable is the fact that c would be negative for $d < 3.70$ (ε>0.296) in [13], whereas this happens only for $d < 1.35$ (ε>2.65) in [29].

[c29] N. V. Antonov and A. N. Vasil’ev, Theor. Math. Phys. 60, 671 (1984).

[c30] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 1997).

[c31] Note that in [18] the dynamical critical exponent z for model A has been missprinted due to an erroneous minus sign before $c η .$

[c32] In [13] a renormalization with normalization conditions has been used. Therefore, in the comparison with our renormalization scheme using the minimal subtraction scheme in the ζ functions of [13] ε has to be set to zero.

[c33] R. Bausch, H. K. Janssen, and H. Wagner, Z. Phys. B 24, 113 (1976).

[c34] E. Brezin, J. C. Le Guillou, and J. Zinn-Justin, in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, New York, 1976), Vol. 4.

[c35] I. D. Lawrie, J. Phys. A 9, 961 (1976).

[c36] V. Dohm, Z. Phys. B: Condens. Matter 60, 61 (1985).

[c37] E. Eisenriegler and B. Schaub, Z. Phys. B: Condens. Matter 39, 65 (1980).

[c38] D. Stauffer, Int. J. Mod. Phys. C 8, 1263 (1997); P. Sen, S. Dasgupta, and D. Stauffer, Eur. Phys. J. B 1, 107 (1998); B. Zheng and H. J. Luo, Phys. Rev. E 63, 066130 (2001).

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Critical dynamics of stochastic models with energy conservation (model C)

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