Global solutions of functional fixed point equations via pseudospectral methods

We apply pseudospectral methods to construct global solutions of functional renormalization group equations in field space to high accuracy. For this, we introduce a basis to resolve both finite as well as asymptotic regions of effective potentials. Our approach is benchmarked using the critical behavior of the scalar $O(1)$ model, providing results for the global fixed point potential as well as leading critical exponents and their respective global eigenfunctions. We provide new results for (1) multicritical $O(1)$ models in fractional dimensions, (2) the three-dimensional Gross-Neveu model at both small and large $N$, and (3) the scalar-tensor model, also in three dimensions.

Figure 1

Derivative of the effective potential of the Wilson-Fisher fixed point in LPA and LPA’.

Figure 2

Decay of coefficients of Chebyshev and rational Chebyshev expansion of the derivative of the Wilson-Fisher fixed point potential, LPA on the left and LPA’ on the right. Notice the algebraic decay in the latter case in the rational Chebyshev region, which needs a factor of 10 as many coefficients as in the case of exponential convergence in LPA to achieve the same order of accuracy.

Figure 3

Eigenperturbations of the Wilson-Fisher fixed point, normalized to one at $\rho =0$.

Figure 4

First derivative of multicritical fixed point potentials exhibiting two minima regarded as function of the dimensionless scalar field (blue, Wilson-Fisher potential), three minima (yellow), four minima (red), five minima (green). The small insets depict the global behavior of the solutions.

Figure 5

Decay of coefficients of Chebyshev and rational Chebyshev expansion in the large-$N$ limit of the Gross-Neveu model.

Figure 6

Decay of rational Chebyshev coefficients in dependence on $L$ in the large-$N$ limit of the Gross-Neveu model.

Figure 7

Fixed point potential (upper left panel), relevant exponent (upper right panel), Yukawa-coupling (lower left panel) and anomalous dimensions (lower right panel) for the Gross-Neveu model with $0.3\le N\le 12$. For increasing flavor number $u_{*}^{\prime }(0)$ increases as well. The dashed line depicts the flavor number $N_t$ of the transition.

Figure 8

Global fixed point solution for the scalar-tensor model in three dimensions. The scalar potential closely resembles the Wilson-Fisher fixed point potential on flat background. The nonminimal coupling is strictly positive, admitting a positive Newton constant.

Figure 9

Eigenfunctions of the fixed point solution to the three-dimensional scalar-tensor model. They are normalized such that $\delta v(0)=1$. From relevant to irrelevant critical exponents: thin blue, thick orange, dotted green, dashed red, dot-dashed violet.