Criticality of the $O(N)$ universality via global solutions to nonperturbative fixed-point equations

TL;DR

This paper develops a numerical method to solve nonperturbative fixed-point equations in the functional renormalization group approach, accurately capturing the global potential and critical exponents across dimensions and universality classes.

Contribution

It introduces a high-precision numerical integration method for fixed-point equations that incorporates correct asymptotic behavior and computes critical exponents for the $O(N)$ universality class.

Findings

Global fixed-point potentials match small and large field expansions.

Critical exponents are computed for various dimensions and $N$ values.

Large-field Laurent expansion of the potential is derived for $2 \,\leq d \leq 4$.

Abstract

Fixed-point equations in the functional renormalization group approach are integrated from large to vanishing field, where an asymptotic potential in the limit of large field is implemented as initial conditions. This approach allows us to obtain a global fixed-point potential with high numerical accuracy, that incorporates the correct asymptotic behavior in the limit of large field. Our calculated global potential is in good agreement with the Taylor expansion in the region of small field, and it also coincides with the Laurent expansion in the regime of large field. Laurent expansion of the potential in the limit of large field for general case, that the spatial dimension $d$ is a continuous variable in the range $2\leq d \leq 4$, is obtained. Eigenfunctions and eigenvalues of perturbations near the Wilson-Fisher fixed point are computed with the method of eigenperturbations. Critical exponents for different values of $d$ and $N$ of the $O(N)$ universality class are calculated.