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Phase transition in ferromagnetic Ising models with non-uniform external magnetic fields. (English) Zbl 1197.82038
The authors study the phase transition phenomenon for the Ising model under the action of a non-uniform external magnetic field \(\mathbf{h}=(h_i)_{i\in \mathbb Z^d}\) and prove following theorems.
Theorem 1. If the magnetic field \(\mathbf{h}\) belongs to \(l_1(\mathbb Z^2)\), then the model presents a phase transition when \(J>3\|\mathbf{h}\|_1\).
Theorem 2. If \(\mathbf{h}\in l_{\infty}(\mathbb Z^d)\) is such that \(\liminf_{i\in\mathbb Z^d} h_i>0\), then the Ising model with external field \(\mathbf{h}\) has no phase transition.

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
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[1] Basuev, A.G.: Ising model in half-space: A series of phase transitions in low magnetic fields. Theor. Math. Phys. 153, 1539–1574 (2007) · Zbl 1139.82309 · doi:10.1007/s11232-007-0132-y
[2] Ellis, R.S.: Entropy, Large Deviations, and Statistical Mechanics. Grundlehren der Mathematischen Wissenschaften, vol. 271. Springer, New York (1985) · Zbl 0566.60097
[3] Fontes, L.R.G., Jordão Neves, E.: Phase uniqueness and correlation length in diluted-field Ising models. J. Stat. Phys. 80, 1327–1339 (1995) · Zbl 1081.82574 · doi:10.1007/BF02179873
[4] Fröhlich, J., Pfister, C.E.: Semi-infinite Ising model II. The wetting and layering transitions. Commun. Math. Phys. 112, 51–74 (1987) · Zbl 1108.82302 · doi:10.1007/BF01217679
[5] Georgii, H.-O.: Spontaneous magnetization of randomly dilute ferromagnets. J. Stat. Phys. 25, 369–396 (1981) · doi:10.1007/BF01010795
[6] Georgii, H.-O.: Gibbs Measures and Phase Transitions. de Gruyter, Berlin (1988)
[7] Jonasson, J., Steif, J.E.: Amenability and phase transition in the Ising model. J. Theor. Probab. 12, 549–559 (1999) · Zbl 0940.60093 · doi:10.1023/A:1021690414168
[8] Lebowitz, J.: Coexistence of phases in Ising ferromagnetics. J. Stat. Phys. 16, 462–476 (1977)
[9] Lebowitz, J., Martin-Löf, A.: On the uniqueness of the equilibrium state for Ising spin systems. Commun. Math. Phys. 25, 276–282 (1972) · doi:10.1007/BF01877686
[10] Lee, T.D., Yang, C.N.: Statistical theory of equations of state and phase transitions II. Lattice gas and Ising model. Phys. Rev. 87, 404–409 (1952) · Zbl 0048.43401 · doi:10.1103/PhysRev.87.410
[11] Lieb, E.H., Ruelle, D.: A property of zeros of the partition function for Ising spin systems. J. Math. Phys. 13(5), 781–784 (1972) · doi:10.1063/1.1666051
[12] Lieb, E.H., Sokal, A.D.: A general Lee-Yang theorem for one-component and multicomponent ferromagnets. Commun. Math. Phys. 80, 153–179 (1981) · doi:10.1007/BF01213009
[13] Newman, C.M.: Inequalities for Ising models and field theories which obey the Lee-Yang theorem. Commun. Math. Phys. 41, 1–9 (1975) · doi:10.1007/BF01608542
[14] Olivieri, E., Vares, M.E.: Large Deviations and Metastability. Encyclopedia of Mathematics and its Applications, vol. 100. Cambridge University Press, Cambridge (2005)
[15] Pfister, C.E.: Large deviations and phase separation in the two-dimensional Ising model. Helv. Phys. Acta 64, 953–1054 (1991)
[16] Ruelle, D.: On the use of ”small external fields” in the problem of symmetry breakdown in statistical mechanics. Ann. Phys. 69, 364–374 (1972) · doi:10.1016/0003-4916(72)90181-9
[17] Velenik, Y.: Phase separation as a large deviations problem. PhD Thesis, Lausanne (2003)
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