Harmonic expansion of the effective potential in a functional renormalization group at finite chemical potential

In this paper we propose a method to study the functional renormalization group (FRG) at finite chemical potential. The method consists of mapping the FRG equations within the Fermi surface into a differential equation defined on a rectangle with zero boundary conditions. To solve this equation we use an expansion of the potential in a harmonic basis. With this method we determined the phase diagram of a simple Yukawa-type model; as expected, the bosonic fluctuations decrease the strength of the transition.

Figure 1

The structure of the differential equation. The $S_F$ Fermi surface is a circle in this coordinate system.

Figure 2

Reproduction of the exact mean field solution in a 10-element harmonic basis (left). The error of the reproduction (right).

Figure 3

The running of the couplings in the ${\mathcal{D}}_{>}$ regime. In the plot the couplings are rescaled in order to show in a single plot.

Figure 4

The boundary condition at $\mu ={\mu }_{MF}$ (solid blue line). To compare, we also plotted the mean field case (dashed orange line).

Figure 5

The effective potential resulting from the FRG calculation. The different curves correspond to different orders of the expansion of the radiative correction in (45). The orange dotted line corresponds to the mean field, the blue, magenta, brown and green lines (from up to down) correspond to $p=0$, 1, 2 and 3, respectively. In the left panel $\mu ={\mu }_{MF}$ is the mean field critical chemical potential; in the right panel $\mu ={\mu }_c=1.053{\mu }_{MF}$ is the true critical chemical potential. In the right panel the $\mathcal{P}=0$ case is not shown, and the mean field curve goes down to $U_{MF}(\phi =0)=-2.92$.

Figure 6

Identification of the order of the phase transition: if the $|m_0|$ vs $v$ curve crosses zero at $v=0$, then we have a second order, if it crosses zero at positive $v$, a first order transition.

Figure 7

Phase diagram in the coupling constant plane for the scalar Yukawa model. In the darker regimes we have second order, in the lighter regimes first order phase transition. The curves from bottom to top correspond to the mean field, one loop and FRG calculations, respectively. There are coupling pairs, where mean field predicts first order transition, but according to the FRG calculation it is still second order.

Figure 8

Relative difference of the borderline using second or third order expansion of the inverse square root. The difference is below 1.5%, which hints for a good convergence.