Question

Can you tell me if this is right? Ive spent some time now working on parts 2,3, and 4 and I am worried that I made an error in part 1 that is propagating to the later parts.

Vlad: I think you have some sign errors I corrected below. Note that in the case of antiferromagnetism, the interaction J has the opposite sign.

I found the two self-consistent equations for $m_{A}$ and $m_{B}$ to be:

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

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

Then I solved the relations for $m$ and $m^{\dagger }$ and found that:

$m_{A}=m+m^{\dagger }$

$m_{B}=m-m^{\dagger }$

Then substituting this into the previous equations I get:

$m_{A}=tanh(-\beta (h+{\frac {1}{2}}Jz(m-m^{\dagger }))$

$m_{B}=tanh(-\beta (h+{\frac {1}{2}}Jz(m+m^{\dagger }))$