# Susceptibility with a complex order parameter

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?