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?

Vlad: it is the index of the considered site i. It feels an internal field from all the z sites connected to site i.

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.

Vlad: there actually is a typo in these old notes. The correct expression should be:

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

since in the first expression $S_{j}\;\rightarrow m$, and the sum over the $j=1,...,z\;$ neighbors of the site $i$ just produce a factor $z\;$.

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?

Vlad: you see, here the same IDEA applies. However, the magnetization on sublattice A is expressed in terms of the magnetization on sublattice B. Thus we get a similar self-consistency conditions:

$m_{A}=tanh(\beta ({\frac {1}{2}}Jzm_{B}+h))$

and

$m_{B}=tanh(\beta ({\frac {1}{2}}Jzm_{A}+h))$, etc.