Conformal invariance constraints in the $O(N)$ models: a first study within the nonperturbative renormalization group

TL;DR

The paper investigates whether full conformal invariance constrains critical behavior in $O(N)$ models within the FRG framework, using a derivative expansion at order $O(\partial^2)$ and a source for composite operators. It derives modified Ward identities for dilatation and special conformal transformations and studies their compatibility with the dilatation identity, applying the PMS and PMC criteria to fix regulator-induced breaking. Across $N=1,2,3,4$ and larger $N$, PMS and PMC yield essentially identical predictions for the critical exponents $\nu$ and $\omega$, agreeing with other high-precision approaches and showing the DE converges towards large-$N$ results; a notable pathology appears at $N=5$ where the conformal constraint becomes ill-defined due to $\Upsilon$-related non-invertibility. The work also demonstrates agreement between the full and strict DE implementations at this order, and extends the analysis to non-unitary cases $N=0$ and $N=-2$, suggesting conformal symmetry remains a useful organizing principle in FRG for a broad range of models.

Abstract

The behavior of many critical phenomena at large distances is expected to be invariant under the full conformal group, rather than only isometries and scale transformations. When studying critical phenomena, approximations are often required, and the framework of the nonperturbative, or functional renormalization group is no exception. The derivative expansion is one of the most popular approximation schemes within this framework, due to its great performance on multiple systems, as evidenced in the last decades. Nevertheless, it has the downside of breaking conformal symmetry at a finite order. This breaking is not observed at the leading order of the expansion, denoted LPA approximation, and only appears once one considers, at least, the next-to-leading order of the derivative expansion ($\mathcal{O}(\partial^2)$) when including composite operators. In this work, we study the constraints arising from conformal symmetry for the $O(N)$ models using the derivative expansion at order $\mathcal{O}(\partial^2)$. We explore various values of $N$ and minimize the breaking of conformal symmetry to fix the non-physical parameters of the approximation procedure. We compare our prediction for the critical exponents with those coming from a more usual procedure, known as the principle of minimal sensitivity.

Figures (9)

• Figure 1: Function $f(\rho,\alpha)$ corresponding to the relevant perturbation $\nu$ for the $O(2)$ model. The continuous line indicates the $\alpha_{\text{PMC}}$ value, while the dashed one indicates the $\alpha_{\text{PMS}}$ one. This figure corresponds to calculations performed with the exponential regulator given in \ref{['regprofiles']}.
• Figure 2: Function $f(\rho,\alpha)$ corresponding to the irrelevant perturbation $\omega$ for the $O(2)$ model. The continuous line indicates the $\alpha_{\text{PMC}}$ value, while the dashed one indicates the $\alpha_{\text{PMS}}$ one. This figure corresponds to calculations performed with the exponential regulator given in \ref{['regprofiles']}.
• Figure 3: Function $f(\rho,\alpha)$ corresponding to the relevant perturbation $\nu$ for the $O(4)$ model. The continuous line indicates the $\alpha_{\text{PMC}}$ value, while the dashed one indicates the $\alpha_{\text{PMS}}$ one. This figure corresponds to calculations performed with the exponential regulator given in \ref{['regprofiles']}.
• Figure 4: Function $f(\rho,\alpha)$ corresponding to the irrelevant perturbation $\omega$ for the $O(4)$ model. The continuous line indicates the $\alpha_{\text{PMC}}$ value, while the dashed one indicates the $\alpha_{\text{PMS}}$ one. This figure corresponds to calculations performed with the exponential regulator given in \ref{['regprofiles']}.
• Figure 5: Function $f(\rho,\alpha)$ corresponding to the irrelevant perturbation $\omega$ for the $O(5)$ model. The dashed line indicates the $\alpha_{\text{PMS}}$ value. Notice that the function $|f(\rho,\alpha)|$ presents a maximum, rather than a minimum, thus the absence of the $\alpha_{\text{PMC}}$ value. This figure corresponds to calculations performed with the exponential regulator given in \ref{['regprofiles']}.