Cluster Approximation for the Two-Dimensional Ising Model

[MathematicalPhysicist]

Homework Statement

In the attachments there is the question and its solution, it's problem 3.5.

Homework Equations

The Attempt at a Solution

My question is how did they get the dimensionless Hamiltonian in both cases, and how did they explicitly calculated $m$ in both cases?

I assume it's with $m = \frac{\sum_{\sigma_i}\sigma_i e^{-\beta H}}{\sum e^{-\beta H}}$, but I really don't see how to get to the same terms as in the solution.

Your help is appreciated!
https://www.physicsforums.com/attachments/214464 https://www.physicsforums.com/attachments/214465

 

[MathematicalPhysicist]

I can see that the pictures didn't get uploaded, so I am sending it again.