Analyticity of critical exponents of the $O(N)$ models from nonperturbative renormalization

TL;DR

This study uses the nonperturbative renormalization group to analyze the critical exponents of $O(N)$ models across different dimensions and component numbers, finding no evidence of the predicted nonanalytic behavior except near the special point $(2,2)$.

Contribution

It provides a detailed numerical investigation of the critical exponents' analyticity in $O(N)$ models, challenging the Cardy-Hamber prediction of nonanalyticities across the $(d,N)$ plane.

Findings

No evidence of nonanalytic behavior of critical exponents for $d>2$.

Clear indications of a crossover line near $(2,2)$ in the $(d,N)$ plane.

Absence of the predicted vanishing eigenvalue in the RG transformation for $d>2$.

Abstract

We employ the functional renormalization group framework at the second order in the derivative expansion to study the $O(N)$ models continuously varying the number of field components $N$ and the spatial dimensionality $d$. We in particular address the Cardy-Hamber prediction concerning nonanalytical behavior of the critical exponents $\nu$ and $\eta$ across a line in the $(d,N)$ plane, which passes through the point $(2,2)$. By direct numerical evaluation of $\eta(d,N)$ and $\nu^{-1}(d,N)$ as well as analysis of the functional fixed-point profiles, we find clear indications of this line in the form of a crossover between two regimes in the $(d,N)$ plane, however no evidence of discontinuous or singular first and second derivatives of these functions for $d>2$. The computed derivatives of $\eta(d,N)$ and $\nu^{-1}(d,N)$ become increasingly large for $d\to 2$ and $N\to 2$ and it is only in this limit that $\eta(d,N)$ and $\nu^{-1}(d,N)$ as obtained by us are evidently nonanalytical. By scanning the dependence of the subleading eigenvalue of the RG transformation on $N$ for $d>2$ we find no indication of its vanishing as anticipated by the Cardy-Hamber scenario. For dimensionality $d$ approaching 3 there are no signatures of the Cardy-Hamber line even as a crossover and its existence in the form of a nonanalyticity of the anticipated form is excluded.