Abstract
A study is made of a two-dimensional Ising model with staggered three-spin interactions in one direction and two-spin interactions in the other. The phase diagram of the model and its critical behavior are explored by conventional finite-size scaling and by exploiting relations between mass gap amplitudes and critical exponents predicted by conformal invariance. The model is found to exhibit a line of continuously varying critical exponents, which bifurcates into two Ising critical lines. This similarity of the model with the Ashkin-Teller model leads to a conjecture for the exact critical indices along the nonuniversal critical curve. Earlier contradictions about the universality class of the uniform (isotropic) case of the model are clarified.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. J. Baxter,Ann. Phys. (N.Y.)70:193–228 (1972).
J. Ashkin and E. Teller,Phys. Rev. 64:178–184 (1943).
R. J. Baxter,Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982).
R. J. Baxter and F. Y. Wu,Phys. Rev. Lett. 31:1294–1297 (1973).
M. N. Barber,Phys. Rep. 59:375–409 (1980).
R. V. Ditzian, J. R. Banavar, C. S. Grest, and L. P. Kadanoff,Phys. Rev. B22:2542–2553 (1980).
J. M. Debierre and L. Turban,J. Phys. A: Math. Gen. 16:3571–3584 (1983).
K. A. Penson, R. Jullien, and P. Pfeuty,Phys. Rev. B 26:6334–6337 (1982).
L. Turban,J. Phys. Lett. (Paris) 43:L259–265 (1982).
F. Iglói, D. V. Kapor, M. Škrinjar, and J. Sólyon,J. Phys. A: Math. Gen. 16:4067–4071 (1983).
F. C. Alcaraz and M. N. Barber,J. Phys. A: Math. Gen. 20, in press.
T. M. Schultz, E. Lieb, and D. Mathis,Rev. Mod. Phys. 36:856 (1964); J. B. Kogut,Rev. Mod. Phys. 51:659–713 (1979).
H. A. Kramers and G. H. Wannier,Phys. Rev. 60:252 (1941).
R. Savit,Rev. Mod. Phys. 52:453–487 (1980).
E. Fradkin and L. Susskind,Phys. Rev. D 17:2637–2658 (1978).
M. Kohmoto, M. den Nijs, and L. P. Kadanoff,Phys. Rev. B 24:5229–5241 (1981).
M. N. Barber, inPhase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1983), p. 143.
M. P. M. Nightingale,J. Appl. Phys. 53:7927 (1982).
J. M. van den Broeck and L. W. Schwartz,SIAM J. Math. Anal. 10:658 (1979).
C. J. Hamer and M. N. Barber,J. Phys. A: Math. Gen. 14:2009–2025 (1981).
F. C. Alcaraz and J. R. Drugowich de Felício,J. Phys. A: Math. Gen. 17:L651–655 (1984).
C. J. Hamer, J. Kogut, and L. Susskind,Phys. Rev. D 19:3091–3105 (1979).
M. N. Barber and C. J. Hamer,J. Aus. Math. Soc. B 23:229–240 (1982).
D. A. Smith and W. F. Ford,SIAM J. Num. Anal. 16:223–240 (1979).
H. W. J. Blöte and M. P. Nightingale,Physica 112A:405 (1982).
A. M. Polyakov,Sov. Phys. JETP Lett. 12:381 (1970).
A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov,J. Stat. Phys. 34:763 (1984);Nucl. Phys. B 241:333 (1984).
J. L. Cardy, inPhase Transitions and Critical Phenomena, C. Domb and J. L. Lebowitz, eds. (Academic Press, in press).
J. L. Cardy,J. Phys. A: Math. Gen. 17:L385–357 (1984).
J. L. Cardy,Nucl. Phys. B270 [FS16]:186 (1986).
G. V. Gehlen, V. Rittenberg, and H. J. Ruegg,J. Phys. A: Math. Gen. 19:107–119 (1985).
K. A. Penson and M. Kolb,Phys. Rev. B 29:2854–2856 (1984).
F. C. Alcaraz, J. R. Drugowich de Felício, R. Köberle, and F. Stick,Phys. Rev. B 32:7469–7475 (1985).
F. C. Alcaraz and J. R. Drugowich de Felício,Rev. Bras. Fís. 15:128–141 (1985).
H. W. J. Blöte, J. C. Cardy, and M. P. Nightingale,Phys. Rev. Let. 56:742–745 (1986).
I. Affleck,Phys. Rev. Lett. 56:746–748 (1986).
D. Friedan, Z. Qiu, and S. Shenker,Phys. Rev. Lett. 52:1575(1984).
G. v. Gehlen and V. Rittenberg,J. Phys. A: Math. Gen. 20, in press.
J. R. Drugowich de Felicio, private communication
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Alcaraz, F.C., Barber, M.N. On the critical behavior of the Ising model with mixed two- and three-spin interactions. J Stat Phys 46, 435–453 (1987). https://doi.org/10.1007/BF01013367
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01013367