|
|
| Line 1: |
Line 1: |
| I am looking at your page 6 of your lecture notes from [http://www.physics.fsu.edu/Users/Dobrosavljevic/Phase%20Transitions/ssb.pdf here]. | | 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. |
|
| |
|
| Looking at the Ising Model:
| | I found the two self-consistent equations for <math>m_A</math> and <math>m_B</math> to be: |
|
| |
|
| <math>H = -\frac{J}{2}\sum_{<i,j>}S_iS_j - h\sum_{}S_i</math> | | <math>m_A = tanh(-\beta(h + \frac{1}{2}Jzm_B))</math> |
|
| |
|
| Then going to the formula for <math>h_i</math>. Does the subscript i have anything to do with the index of summation i or does it stand for internal?
| | <math>m_B = tanh(-\beta(h + \frac{1}{2}Jzm_A))</math> |
|
| |
|
| '''Vlad: it is the index of the considered site i. It feels an internal field from all the z sites connected to site i.'''
| | Then I solved the relations for <math>m</math> and <math>m^\dagger</math> and found that: |
|
| |
|
| Also, why is the third formula:
| | <math>m_A = m + m^\dagger</math> |
|
| |
|
| <math>H = (h + Jzm)\sum_{i}S_i</math> | | <math>m_B = m - m^\dagger</math> |
|
| |
|
| I thought it would be: | | Then substituting this into the previous equations I get: |
|
| |
|
| <math>H = (-h + Jzm)\sum_{i}S_i</math> | | <math>m_A = tanh(-\beta(h + \frac{1}{2}Jz(m - m^\dagger))</math> |
|
| |
|
| since the external field is being subtracted in the initial equation above.
| | <math>m_B = tanh(-\beta(h + \frac{1}{2}Jz(m + m^\dagger))</math> |
| | |
| | |
| '''Vlad: there actually is a typo in these old notes. The correct expression should be:'''
| |
| | |
| '''<math>H = -(h + Jzm)\sum_{i}S_i</math>'''
| |
| | |
| '''since in the first expression <math>S_j\; \rightarrow m </math>, and the sum over the <math> j=1,...,z\; </math> neighbors of the site <math>i</math> just produce a factor <math>z\;</math>.'''
| |
| | |
| | |
| Also on the homework, I am not sure where the <math>m_A\;</math> and <math>m_B\;</math> come into play. Do we just use them for the self consistent equations like we did in class:
| |
| | |
| <math>tanh(\beta(\frac{1}{2}Jzm+h)) = m_0</math>
| |
| | |
| 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:'''
| |
| | |
| '''<math>m_A = tanh(\beta(\frac{1}{2}Jzm_B +h))</math>'''
| |
| | |
| and
| |
| | |
| '''<math>m_B = tanh(\beta(\frac{1}{2}Jzm_A +h))</math>, etc.'''
| |
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.
I found the two self-consistent equations for 
and 
to be:
Then I solved the relations for 
and 
and found that:
Then substituting this into the previous equations I get:
