Laurent Amour, Claudy Cancelier, Pierre L vy-Bruhl, Jean Nourrigat
Thermodynamic limits for a quantum crystal by Heat Kernel methods.
(187K, Plain Tex)

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.