Given the standard Ising partition function:
$$Z(\theta ,h) = \sum\limits_{\bf{x}} {\exp \left\{ {\theta \sum\limits_{(i,j) \in E} {{x_i}{x_j}} + h\sum\limits_{i \in V} {{x_i}} } \right\}}, $$

is there a closed form expression for the lower bound of the free-energy (Pressure) per-site, defined as,

$$\psi (\theta ,h) = \mathop {\lim }\limits_{n \to \infty } \frac{{\log \left( {Z(\theta ,h)} \right)}}{n}.$$

My hunch is that is $2\theta + h$ based on taking limits of an approximation suggested in this article

Any ideas on its suitability as a lower bound?

This post imported from StackExchange Physics at 2014-06-21 09:02 (UCT), posted by SE-user user48476