Abstract
Using a method for composing self-consistent equations, we construct a class of approximate solutions of the Ising problem that are a generalization of the Bethe approximation. We show that some of the approximations in this class can be interpreted as exact solutions of the Ising model on recursive lattices. For these recursive lattices, we find exact values of the thresholds of percolation through sites and couplings and show that for the Ising model of a diluted magnet, our method leads to exact values for these thresholds.
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References
R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Acad. Press, London (1982).
S. V. Sjomkin, V. P. Smagin, and E. G. Gusev, “Potts model on the Bethe lattice with nonmagnetic impurities in an external magnetic field,” Theor. Math. Phys., 197, 1645–1649 (2018).
S. V. Semkin, V. P. Smagin, and E. G. Gusev, “Magnetic susceptibility of a diluted Ising magnet,” Theor. Math. Phys., 201, 1655–1663 (2019).
S. V. Semkin and V. P. Smagin, “Cluster method of constructing Bethe approximation for the Ising model of a dilute magnet,” Russian Phys. J., 60, 1803–1810.
S. V. Semkin and V. P. Smagin, “The method of cyclic clusters in the Ising model of a dilute magnet [in Russian],” Vestn. VOUÉS, 10, 116–123 (2018).
A. A. Zykov, Foundations of Graph Theory [in Russian], Vuzovskaya Kniga, Moscow (2004).
L. N. Ananikian, “Magnetic properties of 3He on recursive lattices,” Izv. NAN Armenii. Fizika, 42, 17–23 (2007).
N. S. Ananikian, L. N. Ananikian, and L. A. Chakhmakhchyan, “Cyclic period-3 window in antiferromagnetic potts and Ising models on recursive lattices,” JETP Lett., 94, 39–43 (2011).
S. V. Semkin and V. P. Smagin, “The Potts model on a Bethe lattice with nonmagnetic impurities,” JETP, 121, 636–639.
S. V. Semkin and V. P. Smagin, “Bethe approximation in the Ising model with mobile impurities,” Phys. Solid State, 57, 943–948 (2015).
J. M. Ziman, Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems, Cambridge Univ. Press, Cambridge (1979).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 202, No. 2, pp. 304–311, February, 2020.
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Semkin, S.V., Smagin, V.P. & Gusev, E.G. Ising Model with Nonmagnetic Dilution on Recursive Lattices. Theor Math Phys 202, 265–271 (2020). https://doi.org/10.1134/S0040577920020099
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DOI: https://doi.org/10.1134/S0040577920020099