03-541 Laurent Amour, Claudy Cancelier, Pierre L vy-Bruhl, Jean Nourrigat

Thermodynamic limits for a quantum crystal by Heat Kernel methods. (187K, Plain Tex) Dec 12, 03

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Abstract. We consider a d-dimensional quantum anharmonic crystal, where the interaction between the ions satisfies hypotheses, based on the idea that the ions are not too far from the points $\Z ^d$, and that the interaction between them decreases exponentially with their distance. Under these conditions, we study carefully the heat kernel of the Hamiltonian related to each finite set of $\Z^d$, with constants in the inqualities that are independent of this set, (thus, improving an earlier result of Sj\"ostrand). Then, taking the limit when the finit set `tends to $\Z^d$', we define as usual a Gibbs state on the algebra of quasilocal observables, proving the convergence in norm, with an exponential rate, for the usual limit defining the state. Then, proving first an exponential decay of the correlations for finite sets and passing to the limit, we prove some properties, (mixing, triviality at infinity when restricted to a suitable subalgebra), of our Gibbs state. The decay of correlation relies on the study of the heat kernel, and is itself used for estimating the rate of convergence in the thermodynamic limit. Another consequence of this decay of correlations is the continuity of the mean energy per site with respect of the temperature, but all the results in this paper are valid under some inequalities between the temperature, the coupling constant, and the Planck's constant, making impossible to let the temperature tend to 0 when the other parameters are fixed.

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