Question

I am looking at your page 6 of your lecture notes from here.

Looking at the Ising Model:

$H=-{\frac {J}{2}}\sum _{<i,j>}S_{i}S_{j}-h\sum _{}S_{i}$

Then going to the formula for $h_{i}$. Does the subscript i have anything to do with the index of summation i or does it stand for internal?

Also, why is the third formula:

$H=(h+Jzm)\sum _{i}S_{i}$

I thought it would be:

$H=(-h+Jzm)\sum _{i}S_{i}$

since the external field is being subtracted in the initial equation above.

Also on the homework, I am not sure where the $m_{A}$ and $m_{B}$ come into play. Do we just use them for the self consistent equations like we did in class:

$tanh(\beta ({\frac {1}{2}}Jzm+h))=m_{0}$

or are we supposed to substitute one of them into the initial equation of the Ising Model?