Summary
The phase diagram of the antiferromagnetic Ising model on the triangular lattice is studied in the presence of small ferromagnetic second-ncarest-neighbour andx-y transverse nearest-neighbour interactions. The free energy of the disordered phase is calculated either by using a thermodynamic perturbation expansion around Wannier’s i exact solution of the antiferromagnetic Ising model in zero magnetic fieldH or by using a high-field low-temperature expansion. For the ordered phase we use a self-consistently renormalized Bose-excitation picture. The resulting phase diagram has, for the first time, all the expected features. With the same approach we study also the lattice anisotropic triangular antiferromagnetic Ising model and our result compares favourably with the known exact solution atH = 0. We obtain also some, information on the phase diagram of the triangular antiferromagnetic Ising model. A number of two and four-spin correlation functions of the antiferromagnetic Ising model is explicitly calculated starting from the exact solution of the model. TheH-dependence of two-spin correlations is calculated for a cluster of nine spins.
Riassunto
Il diagramma di fase del modello di Ising antiferromagnetico relativo al reticolo triangolare è studiato in presenza di deboli interazioni ferromagnetiche a secondi vicini e trasversale di tipox-y a primi vicini. L’energia libera della fase disordinata si calcola usando lo sviluppo perturbativo termodinamico attorno all’esatta soluzione di Wannier del modello di Ising antiferromagnetico in campo magnetico nullo o usando uno sviluppo ad alti campi e bassa temperatura. Per la fase ordinata noi usiamo una descrizione ad eccitazioni bosoniche rinormalizzate autoconsistentemente. Il diagramma di fase ottenuto ha per la prima volta tutte le caratteristiche previste. Con questo stesso approccio si studia anche il modello di Ising antiferromagnetico trangolare con interazione direzionalinente anisotropa ed i nostri risultati si accordano soddisfacentemente con la soluzione esatta per H = 0. Si ottengono anche alcune informazioni sul diagramma di fase del modello di Ising antiferromagnetico triangolare. Si calcolano inoltre esplicitamente alcune fvnzioni di correlazione a due e a quattro spin del modello di Ising antiferromagnetico partendo dalla soluzione esatta ad H = 0. La dipendenza da H delle funzioni di correlazione a due spin è ottenuta sulla base di un cluster di nove spin.
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Literatur
G. H. Wannier:Phys. Rev., 79, 357 (1950);Phys. Rev. B,7, 5017 (1973); R. M. F. Houtappel:Physica,16, 425 (1950).
D. M. Burley: inPhase Transition and Critical Phenomena, Vol. 2, edited by C. Domb and M. S. Green (New York, N.Y., 1972), p. 329.
C. G. B. Garrett:Journ. Chem. Phys., 19, 1154 (1951).
P. W. Kasteleijn:Physica, 22, 387 (1956).
D. M. Burley:Proc. Phys. Soc., 85, 1163 (1965).
B. D. Metcalf:Phys. Lett., 45 A, 1 (1973).
M. Schick, J. S. Walker and M. Wortis:Phys. Lett., 58 A, 479 (1976).
C. E. Campbell and M. Schick:Phys. Rev. A, 5, 1919 (1972).
J. A. Tjon:Phys. Lett., 49A, 289 (1974), and private commuHication.
Preliminary results have been presented at theIUPAP International Conference on Statistical Physics (Budapest, 1975).
M. Schick and R. L. Siddon:Phys. Rev. A, 8, 339 (1973).
J. Stephenson:Can. Journ. Phys., 47, 2621 (1969).
M. F. Sykes, D. S. Gaunt, P. D. Roberts and J. A. Wyles:J. Phys. A, 5, 624 (1972).
J. Stephenson:Journ. Mat. Phys., 5, 1009 (1964).
M. Bloch:Phys. Rev. Lett., 9, 286 (1962);Journ. Appl. Phys.,34, 1151 (1963).
E. Rastelli, A. Tassi and L. Reatto:J. Phys. C, 7, 1735 (1974).
See, for instance, C. Domb: inPhase Transition and Critical Phenomena, Vol. 3, edited by C. Domb and M. S. Green (New York, N.Y., 1974), p. 357.
These expressions include also the effect of zero-point motion to second order in τ, effeet which is not included in (28), (29).
N. W. Dalton and D. W. Wood:Journ. Mat. Phys., 10, 1271 (1969).
S. Alexander:Phys. Lett., 54A, 353 (1975).
J. Stephenson:Journ. Math. Phys., 7, 1123 (1966).
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Work supported in part by Gruppo Nazionale Struttura della Materia del C.N.R.
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Rastelli, E., Tassi, A. & Reatto, L. Antiferromagnetic models on the triangular lattice. Nuov Cim B 42, 120–150 (1977). https://doi.org/10.1007/BF02906754
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DOI: https://doi.org/10.1007/BF02906754