Ginzburg-Landau description of a class of non-unitary minimal models

Abstract: It has been proposed that the Ginzburg-Landau description of the non-unitary conformal minimal model $M(3,8)$ is provided by the Euclidean theory of two real scalar fields with third-order interactions that have imaginary coefficients. The same lagrangian describes the non-unitary model $M(3,10)$, which is a product of two Yang-Lee theories $M(2,5)$, and the Renormalization Group flow from it to $M(3,8)$. This proposal has recently passed an important consistency check, due to Y. Nakayama and T. Tanaka, based on the anomaly matching for non-invertible topological lines. In this paper, we elaborate the earlier proposal and argue that the two-field theory describes the $D$ series modular invariants of both $M(3,8)$ and $M(3,10)$. We further propose the Ginzburg-Landau descriptions of the entire class of $D$ series minimal models $M(q, 3q-1)$ and $M(q, 3q+1)$, with odd integer $q$. They involve $PT$ symmetric theories of two scalar fields with interactions of order $q$ multiplied by imaginary coupling constants.

• The paper presents a Ginzburg-Landau framework for non-unitary minimal models using a two-field system with imaginary coupling terms.
• It employs loop expansion methods in transitional dimensions to verify anomaly matching and predicted RG flows between models like M(3,8) and M(3,10).
• The study highlights potential applications in turbulence and non-equilibrium systems through extended GL theories for complex interactions.

Ginzburg-Landau Description of Non-Unitary Minimal Models

Introduction to Non-Unitary Minimal Models

Non-unitary minimal models $M(p, q)$, characterized by relatively prime positive integers $p$ and $q$, have long been central in the study of critical phenomena and string theory. Their central charges and operator dimensions are well-documented, yet a comprehensive field theory representation, particularly for non-unitary cases where $|p - q| > 1$, poses intriguing challenges. While the Ginzburg-Landau (GL) descriptions have been well-established for unitary models through even-powered scalar fields, non-unitary models, often differentiated by $\mathcal{PT}$ symmetry, demand more unconventional approaches, including imaginary couplings.

Theoretical Background and Previous Work

Recent advancements have proposed that the GL framework for certain non-unitary minimal models such as $M(3,8)$ and $M(3,10)$ can be effectively described by a two-field system with imaginary cubic interactions. This system models the $D$ series modular invariants for both $M(3,8)$ and $M(3,10)$, contributing to the broader classification of models $M(q, 3q-1)$ and $M(q, 3q+1)$ as GL theories of two scalar fields. The novelty here lies in handling odd integer $q$ with $\mathcal{PT}$-symmetric theories where scalar fields interact through higher-order, imaginary-coupled terms (2410.11714).

Key Developments and Methodology

The work elaborates on earlier findings by Nakayama and Tanaka, who provided anomaly mismatch arguments, advocating for two-field GL representations. These theories capture the $D$ series modular invariants and facilitate the understanding of RG flows between models like $M(3,10)$ and $M(3,8)$. Here, the main strategy involves extrapolating from unitary model GL actions to include interactions of the form $\phi^q$ multiplied by imaginary factors which create $\mathcal{PT}$-symmetry, thus stabilizing otherwise ill-defined path integrals.

Numerical and Analytical Insights

Analyses reveal that both $M(3,8)$ and $M(3,10)$, when probed within a scalar field framework in 6-$\epsilon$ dimensions, affirm these anomaly matching conditions. Employing loop expansion methods in these transitional dimensions, the research identifies these models' scaling and RG flow characteristics, managing to align predictions with expected minimal model dimensions: $h_{2,2}$ becomes particularly significant. For instance, two fixed points yielding accurate IR behaviors align with imaginary coupling analyses, which previously lacked analytical backing in higher dimensions (2410.11714).

Implications and Future Directions

Successfully extending GL models to account for non-unitary behaviors suggests rich applications in the physics of turbulence and beyond. These findings endorse the broader class of $M(q, 3q\pm1)$ models as potential GL candidates, strengthened by RG flows evidencing systematic anomaly correlations through larger unit spaces. Future work could build on these foundations, investigating more complex interactions and potential applications in understanding scale invariance in non-equilibrium systems.

Conclusion

This research establishes a robust framework for understanding non-unitary minimal models through innovative GL descriptions. The bridging of anomaly matching principles with speculative yet substantive GL frameworks sets a substantive base for exploring complex theories in minimal model physical landscapes. This novel approach paves the way for further explorations into higher-order interactions and their implications in diverse physical phenomena.