Abstract
An accurate numerical calculation of the ground-state entropy associated to two-dimensional ±J Ising lattices is presented. The method is based on the use of the thermodynamic integration method. Total energy is calculated by means of the Monte Carlo method. Then the entropy (or degeneracy) of a state of interest is obtained by using thermodynamic integration starting at a known reference state. Results for small sizes are compared to exact values obtained by exhaustive scanning of the entire ground-state manifold, which serves as a test for the reliability of the simulation model developed here. The close agreement between simulated and exact results for energy and remnant entropy supports the validity of the technique used for describing the properties of ±J Ising lattices at the fundamental level. Finally, the results are extrapolated in order to estimate tendencies for larger systems.
Similar content being viewed by others
REFERENCES
K. Binder and A. P. Young, Rev. Mod. Phys. 58:801(1986).
M. Mezard, G. Parisi, and M. A. Virasoro, Spin glass theory and beyond (World Scientific, Singapur, 1987).
E. E. Vogel, J. Cartes, S. Contreras, W. Lebrecht, and J. Villegas, Phys. Rev. B 49:6018(1994).
A. J. Ramirez-Pastor, F. Nieto, and E. E. Vogel, Phys. Rev. B 55:14323(1997).
E. E. Vogel, S. Contreras, M. A. Osorio, J. Cartes, F. Nieto, and A. J. Ramirez-Pastor, Phys. Rev. B 58:8475(1998).
A. J. Ramirez-Pastor, F. Nieto, S. Contreras, and E. E. Vogel, Physica A 283:94(2000).
A. J. Ramirez-Pastor, F. Nieto, and E. E. Vogel, Physica A 310:384(2002).
A. K. Hartmann, Europhys. Lett. 40:429(1997).
M. A. Moore, H. Bokil, and B. Drossel, Phys. Rev. Lett. 81:4252(1998).
M. Palassini and A. P. Young, Phys. Rev. Lett. 83:5126(1999).
K. F. Pá;l, Physica A 223:283(1996).
A. K. Hartmann, Physica A 224:480(1996).
A. K. Hartmann, J. Phys. A: Math. Gen. 33:657(2000).
Z. F. Zhan, L. W. Lee, and J.-S. Wang, Physica A 285:239(2000).
G. Hed, A. K. Hartmann, D. Stauffer, and E. Domany, Phys. Rev. Lett. 86:3148(2001).
S. Kirkpatrick, Phys. Rev. B 16:4630(1977).
J. Vannimenus and G. Toulouse, J. Phys. C 10:L537(1977).
I. Morgenstern and K. Binder, Phys. Rev. B 22:288(1980).
H.-F. Cheung and W. L. McMillan, J. Phys. C 16:7027(1983).
A. J. Kolan and R. G. Palmer, J. Appl. Phys. 53:2198(1982).
A. K. Hartmann, Phys. Rev. E 63:016106(2001).
K. Binder, The Monte Carlo Method for the Study of Phase Transitions:A Review of Some Recent Progress, J. Computational Physics 59:1(1985).
S. F. Edwards and P. W. Anderson, J. Phys. F 5:965(1975)
G. Toulouse, Commun. Phys. 2:115(1977).
I. Morgenstern and K. Binder, Phys. Rev. Lett. 43:1615(1979).
K. H. Fisher and J. A. Hertz, Spin Glasses, Cambridge Studies in Magnetism I (Cambridge University Press, Cambridge, 1991).
Z. Salsburg, J. Jacobsen, W. Fickett, and W. Wood, J. Chem. Phys. 30:65(1959).
S. K. Ma, J. Stat. Phys. 26:221(1981).
Z. Alexandrowicz, J. Chem. Phys. 55:2765(1971).
H. Meirovitch, Chem. Phys. Lett. 45:389(1977).
C. H. Bennett, J. Comput. Phys. 22:245(1976).
J. P. Valleau and D. N. Card, J. Chem. Phys. 57:5457(1972).
G. Torrie and J. P. Valleau, Chem. Phys. Lett. 28:578(1974).
K. Binder, J. Stat. Phys. 24:69(1981)
K. BinderZ. Phys. B 45:61(1981).
T. L. Polgreen, Phys. Rev. B 29:1468(1984).
F. Romá;, A. J. Ramirez-Pastor, and J. L. Riccardo, Langmuir 16:9406(2000).
F. Romá;, A. J. Ramirez-Pastor, and J. L. Riccardo, J. of Chem. Phys. 114:10932(2001).
K. Kawasaki, in Phase Transitions and Critical Phenomena, Vol. 2, C. Domb and M. Green, eds. (Academic, London, 1972).
N. Metropolis, A. W Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21:1087(1953).
K. Binder, Applications of the Monte Carlo Method in Statistical Physics. Topics in current Physics, Vol. 36 (Springer, Berlin, 1984), p. 23.
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++, 2nd edn. (Cambridge University Press, London, 2002).
R. Brout, Phys. Rev. 115:824(1959).
R. G. Palmer and J. Adler, Int. J. Mod. Phys. C 10:667(1999).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Romá, F., Nieto, F., Vogel, E.E. et al. Ground-State Entropy of ±J Ising Lattices by Monte Carlo Simulations. Journal of Statistical Physics 114, 1325–1341 (2004). https://doi.org/10.1023/B:JOSS.0000013967.52237.6e
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000013967.52237.6e