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Ising model

  1. Sep 2, 2012 #1
    1. The problem statement, all variables and given/known data
    Calculate magnetisation for partition function
    ##Z=12+4\cosh (8\beta J)## for Ising model 2x2 lattice.


    2. Relevant equations
    [tex]F=-k_BTln Z[/tex]
    [tex]M(H,T)=-\frac{\partial}{\partial H}(\frac{F}{k_BT})[/tex]



    3. The attempt at a solution
    For me it looks that magnetisation is zero. By just doing the derivatives. But solution of problem is ##\frac{16+8e^{\beta J}}{Z}##. If I combine relation from relevant equation it looks like for me that solution is zero. Where I making the mistake.
     
  2. jcsd
  3. Sep 2, 2012 #2
    Where does the partition function come from?

    There is no H in the partition function, so the system does not react to any external field, just as you calculated.

    What are the units of J?
     
  4. Sep 2, 2012 #3
    ##J## is constant. Well you have 2x2 lattice. You calculate
    [tex]Z=\sum e^{-\beta H}[/tex]
     
  5. Sep 4, 2012 #4

    kreil

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    Gold Member

    You are confusing notation here. The derivative to find the magnetization should be with respect to h, the external magnetic field. H is the Hamiltonian used to find the partition function.


    If there is no term dependent on h in your partition function expression, then you have used an incorrect Hamiltonian. Assuming a constant interaction J and constant external field h, the 2D Ising Model has a Hamiltonian of the form

    [tex]H=-J \sum_{i,j} \sigma_i \sigma_j - h\sum_j \sigma_j[/tex]

    The first sum is over all nearest neighbors.


    As written your partition function trivially has no magnetization.
     
    Last edited: Sep 4, 2012
  6. Sep 5, 2012 #5
    I think there are a few sources of confusion here...

    It seems that the problem presents a certain partition function as a given. Presumably it's some kind of approximation for the 2-d Ising model Z.

    Also it looks like J is some parameter that's proportional to the applied magnetic field. So dZ/dH won't be zero.
     
  7. Sep 6, 2012 #6

    Mute

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    Homework Helper

    J is typically the spin-spin coupling in Ising models, and I suspect that is the case here. The partition function given simply doesn't depend on the magnetic field.

    To the OP: why should one expect the magnetization to be non-zero in the absence of a magnetic field? You have a 2x2 Ising model at finite temperature. Furthermore, I assume that the spin-spin interactions are nearest neighbor only, is that correct?

    If so, consider this question: what is the difference between your 2x2 Ising model and a 4-spin 1d Ising model with periodic boundary conditions (and no external field)?
     
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