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One-Dimensional Ising Model in Bethe Approximation

  1. Nov 2, 2017 #1

    MathematicalPhysicist

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    1. The problem statement, all variables and given/known data
    The following question and its solution is from Bergersen's and Plischke's:
    Equation (3.38) is:
    $$m = \frac{\sinh (\beta h)}{\sqrt{\sinh^2(\beta h) + e^{-4\beta J}}}$$

    2. Relevant equations


    3. The attempt at a solution
    They provide the solution in their solution manual which I don't understand how did they come to it.

    I hope I don't have a typo.

    After that they are equating between ##\sigma_0## and ##\sigma_1## and solve for ##z##,

    My problem is how to derive the above three equations, (3.2) and (3.3), I'm lost.[/quote]
     
    Last edited: Nov 2, 2017
  2. jcsd
  3. Nov 6, 2017 #2

    MathematicalPhysicist

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  4. Nov 8, 2017 #3
    I don't have a copy of your text but the Bethe approximation considers a cluster of a central spin ##\sigma_0## in an external field h, and two surrounding spins (in the 1-s Ising model), in an effective field h', which mimics the remaining crystalline lattice. The spin Hamiltonian of this cluster is:$$\mathcal H_c=\mathcal J\sigma_0 \left (\sigma_1 + \sigma_2 \right ) - h\sigma_0 -h'\left (\sigma_1 + \sigma_2 \right )$$
    and the partition function of this cluster is:$$\mathcal Z_c=\sum_{ \sigma_i } exp\left (-\beta \mathcal H_c \right )$$
    The spins in the sum can take the values +1 or -1. Using the notation from your book construct the following table:
    ##\sigma_0## ##\sigma_1## ##\sigma_2## ##\mathcal Z_c##
    ## 1## ## 1## ## 1## ## xyz##
    ## 1## ## -1## ## 1## ## x##
    ## 1## ## 1## ## -1## ## x##
    ## 1## ## -1## ## -1## ## xy^{-1}z^{-1}##
    ## -1## ## 1## ## 1## ## x^{-1}y^{-1}z##
    ## -1## ## -1## ## 1## ## x^{-1}##
    ## -1## ## 1## ## -1## ## x^{-1}##
    ## -1## ## -1## ## -1## ## x^{-1}yz^{-1}##
    Having found ##\mathcal Z_c## I assume you can calculate:
    $$ < \sigma_0 >=\frac {1} {\beta}\frac {\partial ln \mathcal Z_c} {\partial h}$$
    and
    $$ < \sigma_1 >=\frac {1} {\beta}\frac {\partial ln \mathcal Z_c} {\partial h'}$$
     
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