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

TL;DR

This work investigates PT-symmetric scalar theories with iφ^{2n+1} interactions using the functional renormalization group in the Local Potential Approximation (LPA) and LPA'. By perturbatively locating fixed points near their upper critical dimensions d_{2n+1} and numerically solving the fixed-point equations down to d=2, the authors track the Lee-Yang fixed point (n=1) to two dimensions with Δφ matching the exact M(2,5) value within a few percent. They also study higher multicritical points, finding that in LPA' these fixed points annihilate with nonperturbative partners below d≈2.72, preventing a clean d=2 continuation, which has implications for the proposed CFT interpretations M(2,2n+3) or M(2,4n+1). Analytic investigations of moveable singularities and large-field behavior underpin the numerical results, and a dynamical-systems view clarifies the global structure of fixed-point solutions. Overall, the work clarifies the strengths and limitations of the LPA/LPA' FRG framework for nonunitary multicritical Lee-Yang universality and highlights the need for extended truncations to resolve the fate of higher multicritical fixed points in two dimensions.

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 $η$, and in low dimension $d$, this can lead to a change of sign of the scaling dimensions $Δ=(d-2+η)/2$, 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.

Figures (13)

• Figure 1: Snapshots of the dynamical system \ref{['eq:dyn-syst']} with $\gamma=0.3$, for (from left to right) ${\varphi}=0.3$, ${\varphi}=1$, and ${\varphi}=7$. The horizontal orange lines represent the locus of singularities at ${h}"=\pm1$. The green line represents the nullcline \ref{['eq:nullcline']}. In the first panel we have plotted in red a separatrix that we claim corresponds to a solution passing smoothly across the locus of singularities.
• Figure 2: Snapshots of the dynamical system \ref{['eq:dyn-syst']} with $\gamma=-0.3$, for (from left to right) ${\varphi}=0.3$, ${\varphi}=1$, and ${\varphi}=7$. The horizontal orange lines represent the locus of singularities at ${h}"=\pm1$. The green line represents the nullcline \ref{['eq:nullcline']}.
• Figure 3: The Newtonian potential of equation \ref{['eq:NewtonU']}. Outside of the plotted range, $U( {h})$ becomes complex.
• Figure 4: Stream plot of the dynamical system \ref{['eq:dyn-syst']} with $\gamma=0$. The green line is the nullcline $y=0$, on which we choose our initial conditions $x(0)=x_0$. The horizontal orange lines represent the locus of singularities at ${h}"=\pm1$. The red lines are the separatrices: starting from a point on the green line, the separatrix corresponds to the $E=0$ solution.
• Figure 5: Left: Integration endpoint for \ref{['eq:FP-pureIm']} with different initial conditions $h"'(0)=g$ at $d=5.8$, in the LPA and the LPA'. Right: Potential solution $h(\varphi)$ at the "spike" in the LPA' in $d=5.8$, approached from the "left" (dashed) and from the "right" (up to dots) of the spike.