Complete phase diagrams with respect to external fields at low temperatures for models with nearest-neighbor interaction in the case of a finite or countable number of ground states

A. G. Basuev

Teoretičeskaâ i matematičeskaâ fizika, Tome 58 (1984) no. 2, pp. 261-278

Résumé

It is shown that at low temperatures and for arbitrary external fields (activities $z_k$, $\hat z=\{z_k\}$) the ensemble with the Hamiltonian (1) and particles in the set $\Phi$ is equivalent to $|\Phi|$ Ising models with activities $b_k(\hat z), \hat b(\hat z) = \{b_k(\hat z)\}$. The mapping $\hat b(\hat z)$ is a homeomorphism on the positive octant $l_\infty (\Phi)$ if $\sup\limits_k \sum\limits_{l \neq k} \exp\{-\beta\varepsilon(k,l)\}\leq \bar\psi_1$, where $\bar\psi_1$ is a small number. The pressure in the ensemble is $p(\hat z)=\sup\limits_{k \in \Phi}b_k(\hat z) = | \hat b(\hat z) |$. The limit Gibbs states corresponding to the vector $\hat z$ are small perturbations of the ground states $\alpha(x)= q \in G_1(\hat z)$ and are labeled by elements of the set $G_1(\hat z) = \{ \hat q: \ln b_q(\hat z) = p(\hat z)\}$, where the function $G_1(\hat z)$ defines the phase diagram of the ensemble. In the regions of constancy of $G_1(\hat z)$ the pressure can be continued to a holomorphie function, and the particle densities $z_l \partial p/\partial z_l$ are continuous in the closure of a region of constancy of $G_1(\hat z)$.