Is there a reasonable lower bound for free energy per site of the 2D Ising model in the presence of an external field?

Question

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

Comments

Your definition of the free energy differs from the standard one by a minus sign. The answer can't be that simple (or is too weak to be of any use). Set $h=0$ and the exact solution (due to Onsager) is known and exhibits different behaviour for $\theta<\theta_c$ and $\theta>\theta_c$ for a critical value that I denote by $\theta_c$ (given by $(\sinh 2\theta_c)^2=1$ for the square lattice). A similar exact solution is not known for non-zero $h$. Now one expects a phase transition across a real codimension one "wall" in the $(\theta,h)$ plane that can be written as $\theta_c(h)$. I expect two different bounds on either side of the wall.

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@Suresh1: the sharp phase-transition doesn't happen at any nonzero h, and at large h, you might as well use a single-site lower bound.