On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems |
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Authors: | Jürg Fröhlich Charles Pfister |
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Institution: | (1) Institut des Hautes Etudes Scientifiques, 35, route de Chartres, F-91440 Bures-sur-Yvette, France;(2) Département de Mathématiques, Ecole Polytechnique Fédérale, 61, av. de Cour, CH-1007 Lausanne, Switzerland |
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Abstract: | We develop a unified approach, based on Araki's relative entropy concept, to proving absence of spontaneous breaking of continuous, internal symmetries and translation invariance in two-dimensional statistical-mechanical systems. More precisely, we show that, under rather general assumptions on the interactions, all equilibrium states of a two-dimensional system have all the symmetries, compact internal and spatial, of the dynamics, except possibly rotation invariance. (Rotation invariance remains unbroken if connected correlations decay more rapidly than the inverse square distance.) We also prove that two-dimensional systems with a non-compact internal symmetry group, like anharmonic crystals, typically do not have Gibbs states. |
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