Critical and multicritical Lee-Yang fixed points in the local potential approximation

TL;DR

This paper investigates the behavior of multicritical Lee-Yang fixed points in complex scalar field theories across dimensions using the functional renormalization group, revealing their connection to non-unitary conformal models and the challenges in lower dimensions.

Contribution

It provides a detailed analysis of the continuation of $i\,\varphi^{2n+1}$ fixed points from upper critical dimensions to two dimensions, highlighting the limitations of the Local Potential Approximation.

Findings

The Lee-Yang universality class can be followed down to two dimensions.

Multicritical fixed points with $n>1$ cannot be continued to $d=2$ within LPA'.

Unexpected non-perturbative fixed points interfere with the continuation of higher multicritical fixed points.

Abstract

The multicritical generalizations of the Lee-Yang universality class arise as renormalization-group fixed points of scalar field theories with complex $i\varphi^{2n+1}$ interaction, $n\in\mathbb{N}$, just below their upper critical dimension. It has been recently conjectured that their continuation to two dimensions corresponds to the non-unitary conformal minimal models $\mathcal{M}(2,2n+3)$. Motivated by that, we revisit the functional renormalization group approach to complex $\mathcal{P}\mathcal{T}$-symmetric scalar field theories in the Local Potential Approximation, without or with wavefunction renormalization (LPA and LPA' respectively), aiming to explore the fate of the $i\varphi^{2n+1}$ theories from their upper critical dimension to two dimensions. The $i\varphi^{2n+1}$ fixed points are identified using a perturbative expansion of the functional fixed-point equation near their upper critical dimensions, and they are followed to lower dimensions by numerical integration of the full equation. A peculiar feature of the complex $\mathcal{P}\mathcal{T}$-symmetric potentials is that the fixed points are characterized by real but negative anomalous dimensions $\eta$, and in low dimension $d$, this can lead to a change of sign of the scaling dimensions $\Delta=(d-2+\eta)/2$, thus requiring a novel analysis of the analytical properties of the functional fixed-point equations. We are able to follow the Lee-Yang universality class ($n=1$) down to two dimensions, and numerically determine the scaling dimension of the fundamental field as a function of $d$. On the other hand, within the LPA', multicritical Lee-Yang fixed points with $n>1$ cannot be continued to $d=2$ due to the existence of unexpected non-perturbative fixed points that annihilate with the $i\varphi^{2n+1}$ fixed points.