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

Abstract

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

50 pages, 13 figures, 2 tables, 7 raw data files for figures 9,11,12; references added, corrections to eq. (3.20) and (3.21), minor improvements

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