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

A bstract The multicritical generalizations of the Lee-Yang universality class arise as renormalization-group fixed points of scalar field theories with complex iφ 2 n +1 interaction, n ∈ 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 ℳ(2 , 2 n + 3). Motivated by that, we revisit the functional renormalization group approach to complex 𝒫𝒯-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 2 n +1 theories from their upper critical dimension to two dimensions. The iφ 2 n +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 𝒫𝒯-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 ∆ = ( 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 2 n +1 fixed points.