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I want to compute mean-field exponents in a theory that has a complex order parameter. So, let's say I have

$$ F=\int d\vec x \left[ a|\psi|^2 - \frac{b}{2}|\psi|^4\right] \equiv \int d\vec x A[\psi,\psi^*] $$

I know that I can find the extrema of the free energy by deriving with respect to $\psi^*$ (as pointed out here: Variation over complex function in Ginzburg-Landau theory).

When I do it, I find something like

$$ \dfrac{\partial A(\psi, \psi^*)}{\partial \psi^*}=a\psi - b \psi|\psi|^2 = 0 $$

Then I can find the extrema of the function, to find some $\bar\psi=0$ and some $|\bar\psi|^2 \neq 0$, which allows me to identify the exponent $\beta$.

But, how should I proceed for the susceptibility? When I add some external fields to my free energy,

$$ F=\int d\vec x \left[ a|\psi|^2 - \frac{b}{2}|\psi|^4- h_1\psi - h_2\psi^* \right] \equiv \int d\vec x A[\psi,\psi^*] $$

I compute the extrema of the function inside,

$$ \dfrac{\partial A(\psi, \psi^*)}{\partial \psi^*}=a\psi - b \psi|\psi|^2 - h_2 = 0 $$

and I obtain the solutions $\bar\psi$. The next step would be to take derivatives with respect to the field $h_2$ to identify and take the susceptibility. If the parameter were real, I know that I can use the definition of susceptibility

$$ \chi = \dfrac{\partial m}{\partial h} $$

to substitute any derivative $\partial_h f(m) = \chi\partial_m f(m)$. Then, I would take the derivative $\partial_h [\partial_m A]_{m=m_c}$, which will allow me to find the susceptibility $\chi = 1 / f(m)$, and use the scaling $m\sim |T-T_c|^\beta$ to get the scaling of $\chi$.

With the complex order parameter, I do not know how to proceed. In principle, I would be interested in the susceptibility of $|\psi|$, so I'm letting $\chi = \partial_{h_2} |\psi|$, and then

$$ \dfrac{\partial}{\partial h_2} \left [a\psi - b \psi|\psi|^2 - h_2 \right ] = e^{i\theta} \chi \left( a - 3b|\psi|^2 \right) - 1 = 0 $$

I solve for $\chi$ and evaluate the order parameter near the critical point to have

$$ \chi \sim \dfrac{e^{-i\theta}}{a - 3b|\bar\psi|^2}. $$

In principle I should be able to identify the exponent $\gamma$ using the scaling of $|\bar\psi|$, but I suspect that my expression is not correct: the susceptibility should be real and there is a complex exponential in my numerator. But if I define it the susceptibility to be computed without the absolute value, then I cannot compute the derivative of the $\psi|\psi|^2$ term... If I let $m=|\psi|$ in the Hamiltonian and I work with it as a real parameter I know how to do it, but I need the general procedure for more complicated cases.

Which is the correct procedure to obtain the susceptibility with a complex order parameter?

Thank you in advance!

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