Towards a Quintic Ginzburg-Landau Description of the $(2,7)$ Minimal Model

We discuss dimensional continuation of the massless scalar field theory with the $i{\phi }^5$ interaction term. It preserves the so-called $\mathcal{P}\mathcal{T}$ symmetry, which acts by $\phi \to -\phi$ accompanied by $i\to -i$. Below its upper critical dimension $10\mathrm{/}3$, this theory has interacting infrared fixed points. We argue that the fixed point in $d=2$ describes the non-unitary minimal conformal model $M(2,7)$. We identify the operators $\phi$ and ${\phi }^2$ with the Virasoro primaries ${\phi }_{1,2}$ and ${\phi }_{1,3}$, respectively, and $i{\phi }^3$ with a quasi-primary operator, which is a Virasoro descendant of ${\phi }_{1,3}$. Our identifications appear to be consistent with the operator product expansions and with considerations based on integrability. Using constrained Padé extrapolations, we provide estimates of the critical exponents in $d=3$. We also comment on possible lattice descriptions of $M(2,7)$ and discuss RG flows to and from this CFT. Finally, we conjecture that the minimal models $M(2,2n+1)$ are described by the massless scalar field theories with the $i{\phi }^{2n-1}$ interaction terms.