I am considering the following ferromagnetic Hamiltonian for the 2-d Ising Model, say with periodic boundary condition in the torus $\Lambda_n=\mathbb{T}^2_n := (\mathbb{Z}/ \mathbb{Z}_n)^2$:
$$ H_n(\sigma)= - \sum_{x\sim y}J \sigma_x \sigma_y- a_n\sum_{x}h \sigma_x $$ where $x\sim y$ denotes that $d_{\mathbb{T}^2_n}(x,y)=1$. Notice that $h_n$ is also changing according to $n$. We can also define the pressure $$ p(\beta,h)= \lim_{n \to \infty} \frac{1}{|\Lambda_n|} \log Z_{\Lambda_n,\beta,h\cdot a_n}. $$ where $Z_{\Lambda_n,\beta,h\cdot a_n}$ is the partition function for the volume $\Lambda_n$.
The Lee-Yang Circle Theorem guarantees that if I choose $a_n \equiv 1 \neq 0$, and $h \neq 0$, I have that my the pressure is analytic in $h$, and therefore I have no phase transition.
However, if one has a non-homogeneous magnetic field, one can still have phase transition (see BCCP14).
I would like to know if there there exists a "right order" for $a_n$ at which we could observe a phase in $h$. That is, can I choose ${a_n}$ such that there exists $h^* >0$ so that for all $h>h^*$, $p(\beta,h)$ is analytic in $h$, and for $0<h<h^*$, the pressure ceases to be differentiable?
If perhaps looking at the pressure is not the right type of phase transition, one could also think in terms of Gibbs measures.
Are there any articles in this direction? Comments and suggestions are appreciated.
EDIT: If this is this too difficult to explicit find in the Ising model, could one do that in the Curie-Weiss?