Skip to Main Content

Access Provided by Xinjiang University

Physical Review E

Access by Xinjiang University

Partition function zeros of the one-dimensional Blume-Capel model in transfer matrix formalism

R. G. Ghulghazaryan, K. G. Sargsyan, and N. S. Ananikian

Department of Theoretical Physics, Yerevan Physics Institute, Alikhanian Brothers 2, 375036, Yerevan, Armenia

Phys. Rev. E 76, 021104 – Published 2 August, 2007

DOI: https://doi.org/10.1103/PhysRevE.76.021104

Abstract

Zeros of the partition function of the one-dimensional ferromagnetic and antiferromagnetic Blume-Capel models have been studied by using the transfer matrix method in the thermodynamic limit and for finite size chains. The equation for the distribution of zeros of the partition function in the thermodynamic limit is derived. The distribution of the Yang-Lee and Fisher zeros are studied for a variety of values of the parameters of the model. Densities of the Yang-Lee and Fisher zeros are investigated and a singular behavior of the corresponding densities of zeros at the edge points is shown. The edge singularity exponents are calculated analytically for the densities of the Yang-Lee and Fisher zeros. It is found that for both cases edge singularity exponents are universal and equal to $σ = - \frac{1}{2}$ .

Article Text

References (27)

C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952); T. D. Lee and C. N. Yang, ibid. 87, 410 (1952).
M. E. Fisher, in Lectures in Theoretical Physics, edited by W. E. Brittin (University of Colorado Press, Boulder, CO, 1965), Vol. 7C, p. 1.
I. Bena, M. Droz, and A. Lipowski, Int. J. Mod. Phys. B 19, 4269 (2005).
R. G. Ghulghazaryan, N. S. Ananikian, and P. M. A. Sloot, Phys. Rev. E 66, 046110 (2002); N. S. Ananikian and R. G. Ghulghazaryan, J. Comput. Methods Sci. Eng. 2, 75 (2002); S.-Y. Kim, Phys. Rev. E 74, 011119 (2006); 71, 017102 (2005).
M. Biskup, C. Borgs, J. T. Chayes, L. J. Kleinwaks, and R. Kotecký, Phys. Rev. Lett. 84, 4794 (2000).
R. G. Ghulghazaryan and N. S. Ananikian, J. Phys. A 36, 6297 (2003); S.-Y. Kim, J. Korean Phys. Soc. 44, 495 (2004); Z. Glumac and K. Uzelac, J. Phys. A 24, 501 (1991).
D. Ruelle, Phys. Rev. Lett. 26, 303 (1971); C. M. Newman, J. Stat. Phys. 15, 399 (1976); E. H. Lieb and A. D. Sokal, Commun. Math. Phys. 80, 153 (1981).
C. M. Newman, Commun. Math. Phys. 41, 1 (1975); A. D. Sokal, J. Stat. Phys. 25, 25 (1981).
P. J. Kortman and R. B. Griffiths, Phys. Rev. Lett. 27, 1439 (1971); M. E. Fisher, ibid. 40, 1610 (1978).
W. Janke, D. A. Johnston, and R. Kenna, Nucl. Phys. B 682, 618 (2004); M. Suzuki, J. Math. Phys. 9, 2064 (1968); R. Abe, Prog. Theor. Phys. 38, 586 (1967); Y. Park and M. E. Fisher, Phys. Rev. E 60, 6323 (1999).
V. Matveev and R. S. Shrock, Phys. Rev. E 53, 254 (1996); W. T. Lu and F. Y. Wu, J. Stat. Phys. 102, 953 (2001); M. L. Glasser, V. Privman, and L. S. Schulman, Phys. Rev. B 35, 1841 (1987); J. Stephenson, J. Phys. A 20, 4513 (1987).
P. Tong and X. Liu, Phys. Rev. Lett. 97, 017201 (2006); J. L. Jacobsen and J. Salas, J. Stat. Phys. 122, 705 (2006); Sh.-Ch. Chang and R. Shrock, J. Phys. Math. Gen. 39, 10277 (2006).
M. E. Fisher, Suppl. Prog. Theor. Phys. 69, 14 (1980).
X.-Z. Wang and J. S. Kim, Phys. Rev. E 58, 4174 (1998).
D. A. Kurtze, J. Stat. Phys. 30, 15 (1983); W. van Saarloos and D. A. Kurtze, J. Phys. A 17, 1301 (1984).
S. Katsura and M. Ohminami, J. Phys. A 5, 95 (1972).
L. A. F. Almeida and D. Dalmazi, J. Phys. A 38, 6863 (2005).
M. Blume, Phys. Rev. 141, 517 (1966); H. W. Capel, Physica (Amsterdam) 32, 966 (1966).
M. Blume, V. J. Emery, and R. B. Griffiths, Phys. Rev. A 4, 1071 (1971).
E. R. Dobbs, Helium Three (Oxford University Press, Oxford, UK, 2002); C. La Pair et al., Physica (Amsterdam) 29, 755 (1963).
A. Blatter and M. von Allmen, Phys. Rev. Lett. 54, 2103 (1985); W. Sinkler, C. Michaelsen, R. Bormann, D. Spilsbury, and N. Cowlam, Phys. Rev. B 55, 2874 (1997).
G. P. Johari, Physiol. Chem. Phys. 3, 2483 (2001); P. E. Cladis, R. K. Bogardus, W. B. Daniels, and G. N. Taylor, Phys. Rev. Lett. 39, 720 (1977).
O. Mishima and H. E. Stanley, Nature (London) 396, 329 (1998).
D. Ertas and D. R. Nelson, Physica C 272, 79 (1996); N. Avraham et al., Nature (London) 411, 451 (2001).
J. Zhang, X. Peng, A. Jonas, and J. Jonas, Biochemistry 34, 8631 (1995); M. I. Marqués, J. M. Borreguero, H. E. Stanley, and N. V. Dokholyan, Phys. Rev. Lett. 91, 138103 (2003).
S. Rastogi, G. W. H. Hohne, and A. Keller, Macromolecules 32, 8897 (1999); A. L. Greer, Nature (London) 404, 134 (2000).
MPSolve 2.2 multiprecision polynomial solver by D. A. Bini was used for solving polynomial equations (http://fibonacci.dm.unipi.it/~bini/ric.html).