Boundary criticality in the Gross-Neveu-Yukawa model at higher orders

Abstract: We extend the study of boundary criticality in the Gross-Neveu-Yukawa universality class beyond leading order. Using the hyperbolic space formulation of boundary conformal field theories, we compute the first subleading corrections at large $N$ to the free energies of the normal",ordinary" and ``special" boundary universality classes. We also determine the order $1/N$ correction to the dimension of the boundary fermion at the normal fixed point. In the Gross-Neveu-Yukawa theory in $d=4-ε$, we perform a higher-order analysis of the boundary free energy, and use it to extract estimates for the boundary central charge in $d=3$. The large $N$ and $ε$-expansion results are shown to be precisely consistent in overlapping regimes, providing nontrivial consistency checks for the identification of the boundary universality classes. Our calculations rely on a combination of AdS harmonic analysis and boundary conformal field theory techniques.

• The paper presents higher-order corrections in the Gross-Neveu-Yukawa model’s boundary criticality using large N and 4-ε expansions.
• The paper employs AdS/CFT methods and spectral analysis to derive explicit integral representations for free energies and anomalous dimensions at the boundary.
• The paper’s findings bridge theoretical predictions with practical applications, informing lattice simulations and conformal bootstrap studies in boundary phenomena.

Boundary Criticality in the Gross-Neveu-Yukawa Model at Higher Orders

Introduction

This work extends the theoretical analysis of boundary universality classes in the Gross-Neveu-Yukawa (GNY) model, emphasizing subleading corrections at large $N$ and higher-order perturbative results in the $4-\epsilon$ expansion. The GNY model, an interacting fermionic theory providing a UV-complete realization of the Gross-Neveu (GN) system, exhibits nontrivial boundary critical behavior. Multiple boundary universality classes—denoted as “normal,” “ordinary,” and “special” boundaries, in analogy with the O$(N)$ scalar model—are realized by distinct conformal boundary conditions and support a rich structure of RG flows.

This analysis relies on placing the GNY and GN models in hyperbolic space, thereby translating boundary CFT (BCFT) questions into calculable quantities via AdS/CFT techniques and conformal harmonic analysis. The work synthesizes large $N$ and $\epsilon$-expansion methods and demonstrates nontrivial agreement in their overlapping regimes, providing rigorous checks on both the phase structure and physical observables at the boundaries.

Large $N$ Expansion of Boundary Free Energies and Scaling Dimensions

Universality Classes and Boundary Phases

Each boundary universality class is constructed explicitly: the normal phase corresponds to standard boundary conditions for fermions, while the ordinary and special phases arise from the alternative quantization. The normal boundary saddle is realized at constant $\sigma_* = \frac{d}{2}-1$, with boundary fermion dimension $\hat\Delta_{(1/2)} = d-\frac{3}{2}$, and the fluctuation determinants are encoded in the AdS free energy.

AdS Harmonic Analysis and Spectral Methods

The evaluation of free energies up to order $N^{0}$ proceeds via spectral decomposition of the quadratic action for the $\sigma$-field. The authors present explicit calculations of the corresponding spectral densities for the fluctuation operators around each boundary saddle, leading to integral representations and efficient expansion schemes (e.g., by introducing auxiliary spectral functions and compactly supported integrals).

For the normal phase, the one-loop determinant of the $4-\epsilon$0 fluctuations is reduced to an explicit integral over the spectral parameter $4-\epsilon$1, and closed-form expressions for the relevant quantities are derived using hypergeometric summation and analytic continuation—yielding e.g., the order $4-\epsilon$2 correction to the boundary free energy and corresponding boundary central charge in $4-\epsilon$3.

Higher Loop Corrections and Consistency with $4-\epsilon$4 Expansion

In parallel, the authors analyze the $4-\epsilon$5 expansion of the boundary free energies and bulk one-point functions at higher orders, systematically resumming the series (Padé approximants) to estimate observable quantities in $4-\epsilon$6. The interplay between large $4-\epsilon$7 and $4-\epsilon$8 expansion is highlighted: the consistency across the two approaches demonstrates robustness of the universality class assignments and confirms the correctness of the analytic methods in the BCFT context.

Particularly, the explicit treatment of curvature counterterms is crucial to recover agreement between the two expansion schemes; these terms correct the free energy at $4-\epsilon$9 and their omission would yield spurious mismatches in, e.g., the central charge estimate.

Anomalous Dimensions at the Boundary

The calculation of the $(N)$0 correction to the scaling dimension of the leading boundary fermionic operator uses a refined equation-of-motion method, adapted to the fermion sector. Via boundary conformal block decompositions and the use of orthogonality relations, explicit integral and closed-form expressions for the anomalous dimensions are obtained. The analysis establishes that

$(N)$1

where $(N)$2 is the Dirac spinor multiplicity. This result is cross-checked against known $(N)$3-expansion data and provides a nonperturbative benchmark for the scaling dimension across dimensions for various $(N)$4.

Padé Resummation, Central Charges, and Connection to Boundary CFT

A detailed Padé resummation program is implemented, utilizing constraints from exact results in two dimensions (e.g., the minimal model realization of the super-Ising CFT for $(N)$5), to provide estimates of physically significant BCFT quantities in $(N)$6. This includes the boundary central charge $(N)$7 and one-point function coefficients for the scalar field, e.g., for the $(N)$8 super-Ising universality class. Numerical results demonstrate quantitative agreement between field-theoretic expansions and minimal-model CFT data in $(N)$9, and interpolated values in $N$0 are provided with controlled uncertainty.

Implications

The synthesis of large $N$1 expansion, perturbative $N$2 expansion, and BCFT harmonic analysis yields a robust framework for the study of interacting fermionic BCFTs.

• Theoretical implications: The explicit higher-order checks on matching between different expansion approaches reinforce the universality class structure and the predictive power of analytic methods for strongly correlated boundary phenomena, including supersymmetric and multicritical cases.
• Practical implications: Quantitative predictions for boundary scaling dimensions and central charges are directly relevant for lattice simulations, conformal bootstrap approaches, and surface criticality phenomena in quantum magnets and condensed matter platforms modeled by GNY-type fixed points.
• Future prospects: The harmonic techniques and boundary block decompositions developed here are likely extendable to nontrivial surface transitions and topological phases, as well as to the systematic study of RG defect flows, generalized free field theories, and AdS/CFT duals of boundary phenomena in higher dimensions.

Conclusion

This work provides a comprehensive and technically rigorous extension of boundary criticality analysis in the Gross-Neveu-Yukawa model, pushing large $N$3 and $N$4 expansion methodologies to higher orders. The construction and evaluation of spectral densities, careful treatment of counterterms, and matching to conformal data enable precise computations of free energies, central charges, and anomalous dimensions for all major BCFT boundary universality classes in this context. These results enhance understanding of boundary criticality in strongly correlated fermionic systems and serve as a foundation for future investigations into interacting BCFTs and their holographic duals.