Spontaneous symmetry breaking and optimization of functional renormalization group

The requirement for the absence of spontaneous symmetry breaking in the $d=1$ dimension has been used to optimize the regulator dependence of functional renormalization group equations in the framework of the sine-Gordon scalar field theory. Results obtained by the optimization of this kind were compared to those of the Litim-Pawlowski and the principle of minimal sensitivity optimization scenarios. The optimal parameters of the compactly supported smooth (CSS) regulator, which recovers all major types of regulators in appropriate limits, have been determined beyond the local potential approximation, and the Litim limit of the CSS was found to be the optimal choice.

Figure 1

Phase diagram of the SG model for $d=1$ dimensions obtained by the numerical solution of Eqs. (8) and (9) using the power-law regulator (6b) with $b=3$. Arrows indicate the direction of the flow. The distance $D$ is defined by (10).

Figure 2

Positions of the saddle point of the SG model for dimensions $1\le d\le 2$ obtained by the power-law regulator (6b). The inset shows the dependence of the distance $D$ defined by Eq. (10) on the parameter $b$ for dimension $d=1$.

Figure 3

Dependence of $D$ (10) on the parameter $b$ of the CSS regulator (5) represented in various limits. Best values are obtained in terms of the parameters $c$ and $h$, for “fixed” $b$.