The Ising model in a random magnetic field |
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Authors: | Daniel S. Fisher Jürg Fröhlich Thomas Spencer |
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Affiliation: | (1) Bell Laboratories, 07974 Murray Hill, New Jersey;(2) Theoretical Physics, ETH-Hönggerberg, CH 8093 Zurich, Switzerland;(3) Courant Institute, New York University, 251 Mercer Street, 10012 New York, New York |
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Abstract: | The existence of a spontaneous magnetization in the three-dimensional Ising model in a weak random magnetic field (RFIM) is investgated. Following Imry and Ma, we consider the energy change, E , from the fully aligned ferromagnetic state caused by flipping all the spins inside a connected surface, . It is proved rigorously that with high probability, E is positive forall enclosing the origin. Under the unproven assumption that the expectation value of the spin at one site is weakly correlated with the random fields at far away sites (which is true if surfaces within surfaces can be ignored) it follows that the three-dimensional RFIM has a spontaneous magnetization at low temperatures. The proof works for all dimensions greater than two, providing support for the conjecture that two is the lower critical dimension.Work supported in part by NSF grant No. DMR 8100417. |
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Keywords: | Domain wall ferromagnetic order random field |
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