2D Ising model entropy as a function of external field

I am trying to figure out what is the entropy expression as a function of the external field in a 2D Ising model with nearest neighbour interaction.

My Hamiltonian is the following:

$$\mathcal{H}=-\Lambda\sum_i c_i -\varepsilon\sum_{}c_i c_j$$

where $$\Lambda$$ is given by some function of the external field $$h$$ and $$\varepsilon$$ is constant. $$c_i \in \{1,-1\}$$.

I found a measure of the entropy with respect to the temperature, but not with respect to an external field.

If you have an answer please also direct me to a citeable source.

• You want an explicit expression for the entropy of the 2d Ising model as a function of $h$? None is known. The model has only been solved in the absence of a magnetic field. – Yvan Velenik Aug 4 at 11:30

Ising model in the external field has three thermodynamical degrees of freedom. I mean, all extensive quantities, such as entropy and internal energy, are functions of three independent variables, for example, $$\theta, \Lambda, N$$. What do you mean when asking about entropy as a function of the external field? The analytical solution for the 2D Ising model in the external field is not known. Hence, it is hardly possible to find $$S(\theta, \Lambda, N)$$ for the 2D model.
You can find $$S(\theta, \Lambda, N)$$, and probably even $$S(E, \Lambda, N)$$, in 1D model.