$$|\left<J_3 + i J_4\right>| = A \left(1-\frac{\tau}{\tau_c}\right)^{\frac 1 2}\;,$$ where $A=2.04$ and $\tau_c=0.37$. The critical exponent $1/2$ is typical for a mean field theory. Indeed, this can be expected from the absence of quadratic terms in the equations of motion and the fact that we consider the gravity system classically. The mean field behavior is also observed in the holographic description of superconductivity [@Hartnoll:2008vx; @Hartnoll:2008kx].

![The expectation value of the order parameter as a function of the temperature. The dotted curve is the numerical result and the solid curve is its fit with $(1-\tau/\tau_c)^{1/2}$.[]{data-label="fig:TvsO"}](tdep.jpg){width="7cm"}

Quantum Corrections {#sec:corrections}
===================

We found that the phase transition is of the second order in the classical supergravity approximation. In this section, we provide evidence that quantum corrections in the bulk change it to the first order. Such a phenomenon has been observed by Brazovskii [@Brazovskii] and elaborated by Swift and Hohenberg in [@SwiftHohenberg], in the context of a classical statistical model at finite temperature. We will extend this result to the gravity theory in $AdS_5$.

Let us review the Brazovskii model. It is a classical field theory in $d$ space dimensions at finite temperature with the following scalar field Hamiltonian in the momentum representation, 