Lattice systems with a continuous symmetry |
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Authors: | Jean Bricmont Jean-Raymond Fontaine Joel L Lebowitz Thomas Spencer |
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Institution: | 1. Department of Mathematics, Princeton University, 08544, Princeton, NJ, USA 2. Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA
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Abstract: | We investigate a continuous Ising system on a lattice, equivalently an anharmonic crystal, with interactions: $$\sum\limits_{\left\langle {x,y} \right\rangle } {\left( {\phi _x - \phi _y } \right)} ^2 + \lambda \left( {\phi _x - \phi _y } \right)^4 , \phi _x \in \mathbb{R}, x \in \mathbb{Z}^d .$$ We prove that the perturbation expansion for the free energy and for the correlation functions is asymptotic about λ=0, despite the fact that the reference system (λ=0) does not cluster exponentially. The results can be extended to more general systems of this type, e.g. an even polynomial semibounded from below instead of a quartic interaction. By a suitable scaling, λ corresponds to the temperature. |
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