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

TL;DR

The authors propose that a PT-symmetric massless scalar theory with the $i\phi^5$ interaction realizes the non-unitary minimal model $M(2,7)$ at $d=2$. They establish operator identifications $\phi \leftrightarrow \phi_{1,2}$ and $\phi^2 \leftrightarrow \phi_{1,3}$ with $i\phi^3$ as a Virasoro descendant, and validate this via $d=\frac{10}{3}-\epsilon$ expansions and constrained Padé extrapolations to estimate dimensions across $2\le d\le \frac{10}{3}$, including rough $d=3$ values. They show that deforming $M(2,7)$ by $\phi_{1,3}$ yields an integrable two-breather theory with a known S-matrix, and they explore RG flows between minimal models (notably $M(4,5)\to M(2,7)$ and $M(2,7)\to M(2,5)$) using semiclassical arguments and Truncated Conformal Space Approach, respectively. The work suggests a broader Ginzburg–Landau description for the family $M(2,2n+1)$ realized by $i\phi^{2n-1}$ interactions and outlines potential lattice realizations and future numerical tests.

Abstract

We discuss dimensional continuation of the massless scalar field theory with the $iφ^5$ interaction term. It preserves the so-called $\mathcal{PT}$ symmetry, which acts by $φ\rightarrow -φ$ accompanied by $i\rightarrow -i$. Below its upper critical dimension $10/3$, 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 $φ$ and $φ^2$ with the Virasoro primaries $φ_{1,2}$ and $φ_{1,3}$, respectively, and $iφ^3$ with a quasi-primary operator, which is a Virasoro descendant of $φ_{1,3}$. 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 $iφ^{2n-1}$ interaction terms.

Figures (2)

• Figure 2.1: Two-sided Padé extrapolations for the dimensions of operators $\phi, \phi^2, \phi^3$ and $\phi^5$. We use the [1,1] Padé approximant, except for $\phi^3$ where we use the [0,2] Padé approximant because [1,1] has a pole.
• Figure 4.1: Operator flow (a) and effective central charge (b) for the $M(2,7)\rightarrow M(2,5)$ flow.