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

The multicritical generalizations of the Lee-Yang universality class arise as renormalization-group fixed points of scalar field theories with complex $iφ^{2n+1}$ 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 $iφ^{2n+1}$ theories from their upper critical dimension to two dimensions. The $iφ^{2n+1}$ 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 $iφ^{2n+1}$ fixed points.

💡 Research Summary

The paper investigates a family of non‑unitary scalar field theories defined by a complex, (\mathcal{PT})‑symmetric interaction (i\phi^{2n+1}) with integer (n\ge1). These models generalize the Lee‑Yang edge singularity (the case (n=1)) and are expected to flow to non‑unitary minimal conformal field theories (\mathcal{M}(2,2n+3)) or (\mathcal{M}(2,4n+1)) in two dimensions. The authors employ the functional renormalization group (FRG) in the Wetterich formulation, using two truncations of the effective average action: the Local Potential Approximation (LPA) with a fixed wave‑function renormalization ((Z_k=1), hence (\eta=0)), and the improved LPA′ where the field‑strength renormalization (Z_k) is allowed to run, giving a non‑zero anomalous dimension (\eta).

First, they perform an (\epsilon)-expansion around the upper critical dimension (d_c= \frac{4n+2}{2n-1}). By expanding the fixed‑point potential in powers of the field and solving the FRG flow order by order in (\epsilon), they obtain analytic expressions for the fixed‑point couplings and for (\eta). Remarkably, (\eta) turns out to be real but negative for all (n), a direct consequence of the (\mathcal{PT}) symmetry. This negativity implies that the scaling dimension of the fundamental field, \