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Homework Help: Quick question: Ginzberg-Landau theory.

  1. Nov 11, 2011 #1
    1. The problem statement, all variables and given/known data

    I have an effective Hamiltonian, in the mean field approximation of the form:

    [tex] \beta H = tx^{2}+(a+\frac{b}{2})^{4} +c^{6} [/tex]

    Where x is a minimized order parameter, t is related to temperature and a,b,c are some system parameters. The system is a Bose-Einstein condensate such a a superconductor/superfluid etc.

    I've been asked to consider the phase behavior near the critical point (t=0) and would like to check my reasoning as I'm not 100% confident in it.

    3. The attempt at a solution

    For t>0 (above the critical temperature) we can say it is okay to have an order parameter that is potentially zero (non-finite). Also, in the region near the critical point we can expect x to be small. In this case we can ignore the higher order powers in x and take only the quadratic term for our approximation. Taking the derivative of the hamiltonian and setting it to zero we get

    [tex] 2tx=0 [/tex] which if t is positive and small but non-zero means that x must be zero, hence we have no form of BEC formation at t>0.

    For t<0, we would expect some sort of ordering so x must be finite. This means we need to keep, at least the quartic term, in our consideration. S ignoring the sixth power and setting the derivative to zero, gives us a quadratic equation with two, one positive and one negative. This would cause the function to turn and be finite at t=0 displaying a finite order parameter (and BEC formation) at the critical point.

    My question is: does this make sense. Specifically, is it okay to make the assumption of expected order for t<0 and (potentially) x=0 for t>0, in order to discard some higher powers. Or would that be introducing a bias and fixing things to get the result I want out?

    I hope that was clear but let me know if I need to clarify something.
  2. jcsd
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