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(New page: I am looking at your page 6 of your lecture notes from [http://www.physics.fsu.edu/Users/Dobrosavljevic/Phase%20Transitions/ssb.pdf here]. Looking at the Ising Model: <math>H = -\frac{J}...)
 
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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:
<font color=red>'''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.'''</font>


<math>H = -\frac{J}{2}\sum_{<i,j>}S_iS_j - h\sum_{}S_i</math>
I found the two self-consistent equations for <math>m_A</math> and <math>m_B</math> to be:


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_A = tanh (+\beta(h-\frac{1}{2}Jzm_B))</math>


Also, why is the third formula:
<math>m_B = tanh (+\beta(h-\frac{1}{2}Jzm_A))</math>


<math>H = (h + Jzm)\sum_{i}S_i</math>
Then I solved the relations for <math>m</math> and <math>m^\dagger</math> and found that:


I thought it would be:
<math>m_A = m + m^\dagger</math>


<math>H = (-h + Jzm)\sum_{i}S_i</math>
<math>m_B = m - m^\dagger</math>


since the external field is being subtracted in the initial equation above.
Then substituting this into the previous equations I get:


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>m_A = tanh(+\beta(h - \frac{1}{2}Jz(m - m^\dagger))</math>


<math>tanh(\beta(\frac{1}{2}Jzm+h)) = m_0</math>
<math>m_B = tanh(+\beta(h - \frac{1}{2}Jz(m + m^\dagger))</math>
 
or are we supposed to substitute one of them into the initial equation of the Ising Model?

Latest revision as of 15:04, 5 February 2009

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 and to be:

Then I solved the relations for and and found that:

Then substituting this into the previous equations I get:

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