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 TMF, 1974, Volume 21, Number 1, Pages 49–59 (Mi tmf3856)

Legendre transforms in the Ising model

A. N. Vasil'ev, R. A. Radzhabov

Abstract: The Legendre transforms of the logarithm of the partition function for the Ising model are considered. In the language of the first transform ($\Phi$) the magnetization is found by means of a variational principle: $\Phi$ plays the role of the varied functional whose stationarity points correspond to the desired values of the magnetization. Equations of motion are derived for $\Phi$, and their iterative solution is described (diagrams). The diagram expansion of $\Phi$ is equivalent to the high-temperature expansion of the logarithm of the partition function (free energy) in thetemperature – magnetizationvariables (instead of the usual temperature – external field variables). The possibility of using a diagram expansion for the first Legendre transform for the approximate calculation of the critical indices is discussed. The main advantage of the method is that it is equally applicable both above and below $T_c$.

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English version:
Theoretical and Mathematical Physics, 1974, 21:1, 963–970

Citation: A. N. Vasil'ev, R. A. Radzhabov, “Legendre transforms in the Ising model”, TMF, 21:1 (1974), 49–59; Theoret. and Math. Phys., 21:1 (1974), 963–970

Citation in format AMSBIB
\Bibitem{VasRad74} \by A.~N.~Vasil'ev, R.~A.~Radzhabov \paper Legendre transforms in the Ising model \jour TMF \yr 1974 \vol 21 \issue 1 \pages 49--59 \mathnet{http://mi.mathnet.ru/tmf3856} \transl \jour Theoret. and Math. Phys. \yr 1974 \vol 21 \issue 1 \pages 963--970 \crossref{https://doi.org/10.1007/BF01035593} 

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This publication is cited in the following articles:
1. A. N. Vasil'ev, R. A. Radzhabov, “Analysis of the nonstar graphs of the Legendre transform in the ising model”, Theoret. and Math. Phys., 23:3 (1975), 575–579
2. N. M. Bogolyubov, V. F. Brattsev, A. N. Vasil'ev, A. L. Korzhenevskii, R. A. Radzhabov, “High-temperature expansions at an arbitrary magnetization in the ising model”, Theoret. and Math. Phys., 26:3 (1976), 230–237
3. N. M. Bogolyubov, “Convergence of Feynman diagram expansions in the Ising model”, Theoret. and Math. Phys., 30:1 (1977), 88–90
4. M. I. Vladimir, L. A. Dogotar', V. A. Moskalenko, “Ising model for spin glasses”, Theoret. and Math. Phys., 50:2 (1982), 177–185
5. G. V. Matvienko, “Critical behavior of the ferromagnetic ising chain with interaction $J_{ij}=|i-j|^{-2}$”, Theoret. and Math. Phys., 63:3 (1985), 635–640
6. G. V. Matvienko, “Critical behavior of a plane Ising antiferromagnet with two-dimensional dipole interaction”, Theoret. and Math. Phys., 69:2 (1986), 1156–1163
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