Tensorial Quartic Yukawa Model

• The tensorial quartic Yukawa model is a field theory where symmetric traceless tensor fields couple to fermions via Yukawa, quartic, and sextic interactions, defining distinct universality classes.
• Renormalization group flows reveal critical exponents, mass-gap ratios, and stability regions that differentiate chiral XY and novel SO(3) tensorial universality classes.
• Large-N and melonic extensions inform low-energy theories for spin-orbital liquids and Dirac systems, guiding precision computations in critical phenomena.

The tensorial quartic Yukawa model is a class of field theories featuring real, symmetric, traceless tensor fields of rank two or higher, coupled via Yukawa interactions to multiplets of Dirac or Majorana fermions, supplemented by a single quartic tensor self-interaction and, where symmetry allows, higher-order invariants such as sextic terms. These models generalize vector and matrix Gross–Neveu–Yukawa systems by utilizing tensor fields as order parameters, admitting a rich structure of universality classes, renormalization group (RG) flows, and critical phenomena. They have applications as low-energy effective theories for fractionalized spin-orbital liquids and Dirac electronic systems, and exhibit distinctive behavior for low rank (notably SO(2) and SO(3)) as well as in large-$N$ tensor limits (Han et al., 2024, Han et al., 25 Jun 2025, Fraser-Taliente et al., 2024).

1. Lagrangian Structure and Symmetry

The prototypical tensorial quartic Yukawa model is characterized by a real, symmetric, traceless rank-two tensor field $T_{ij}(x)$, transforming as the adjoint under SO$(N)$, coupled to $N_f$ flavors of Dirac or Majorana fermions: $$\mathcal{L} = \bar\psi_{aI}\,\gamma^\mu\partial_\mu\,\psi_{aI} + \tfrac{1}{2}\,\partial_\mu T_{ij}\,\partial_\mu T_{ij} + \tfrac{1}{2}\,r\,T_{ij}T_{ij} + g\,\bar\psi_{aI}\,(T_{ij}\,\Lambda_{ij})_{ab}\,\psi_{bI} + \frac{\lambda}{4}\,(T_{ij}T_{ij})^{2} + V_{\rm sextic}$$ where $a=1,\ldots,N$, $I=1,\ldots,N_f$, and the $\Lambda_{ij}$ generate the symmetric traceless tensor representation. For $N=3$, two independent sextic invariants appear: $$V_{\rm sextic}(T) = \frac{\kappa_1}{6}\,(T_{ij}T_{ij})^3 + \frac{\kappa_2}{6}\,(T_{ij}T_{jk}T_{ki})^2$$ Analogous constructions hold for models with higher-rank tensors or multi-index symmetries, such as $O(N)^3$-invariant rank-3 tensors in the melonic sector (Han et al., 2024, Han et al., 25 Jun 2025, Fraser-Taliente et al., 2024).

For $N=2$, the theory reduces to the chiral XY model, while $N=3$ defines a new SO(3) tensorial universality class. For $N\geq 4$, the structure of quartic invariants and the stability of the theory are more intricate, leading to distinct RG behavior (Han et al., 2024, Han et al., 25 Jun 2025).

2. Renormalization Group Flows and Fixed Points

The key RG equations are formulated in terms of dimensionless couplings: $$\alpha_g = \frac{g^2}{2\pi^2}, \quad \lambda \to \frac{\lambda}{2\pi^2}$$ The two-loop $\beta$-functions for general $N$ are: $$\begin{aligned} \frac{d\alpha_g}{d\ell} &= \epsilon \alpha_g - \frac{N^2 + (2N_f + 3)N - 6}{8}\alpha_g^2 - \frac{N^2 + N + 2}{64}\,\alpha_g\,\lambda^2 + \frac{2N^2 + 3N - 6}{16}\,\alpha_g^2\,\lambda + \mathcal{O}(\alpha_g^3) \ \frac{d\lambda}{d\ell} &= \epsilon\,\lambda - \frac{NN_f}{2}\,\alpha_g\lambda + \frac{N^2N_f}{4}\alpha_g^2 - \frac{N^2 + N + 14}{8}\lambda^2 + \mathcal{O}(\alpha_g^3, \lambda^3) \end{aligned}$$ For $N=3$ and $N=2$, the quartic sector simplifies to a single invariant, leading to a unique RG structure. At higher $N$, two quartic invariants exist, and RG flows yield two critical values $N_{c1}$ and $N_{c2}$ below and above which real and stable fixed points exist or vanish. In the presence of fermions, for $N \geq 4$ there are critical flavor numbers $N_{f,c1}$ and $N_{f,c2}$ partitioning the $(N,N_f)$ plane into regions with second-order, weakly first-order, or no critical fixed points:

Region	$N_f$ domain	Behavior
No fixed point	$N_f < N_{f,c1}$	No real stable FP
Unstable fixed point	$N_{f,c1} < N_f < N_{f,c2}$	Real FP, stability violated
Stable fixed point	$N_f > N_{f,c2}$	Real, stable continuous PT

Two-loop corrections reduce $N_{f,c2}$, bringing the physically stable critical window closer to the Gross–Neveu limit $N_f=1$ for large $N$ (Han et al., 25 Jun 2025).

3. Universality Classes and Critical Exponents

Distinct universality classes emerge as a consequence of the tensor symmetry and coupling structure:

• For $N=2$, the equivalence to the chiral XY universality class is manifest in the RG flows and critical quantities.
• For $N=3$, a new SO(3) tensorial universality class is established, with universal critical exponents that diverge from both the standard chiral XY and SO($N$)-vector classes.

At the infrared-stable fixed point, anomalous dimensions and the inverse correlation length exponent are: $$\begin{aligned} \eta_\psi &= \frac{N^2+N-2}{16}\,\alpha_g^* - \frac{(N^2+N-2)[N^2+(12N_f+1)N-2]}{1024}(\alpha_g^*)^2 \ \eta_T &= \frac{NN_f}{4}\,\alpha_g^* - \frac{(3N^2+5N-10)NN_f}{128}(\alpha_g^*)^2 + \frac{N^2+N+2}{64}(\lambda^*)^2 \ \nu^{-1} &= 2 - \frac{NN_f}{4}\alpha_g^* - \frac{N^2+N+2}{8}\lambda^* + \frac{(3N^2+N-2)NN_f}{128}(\alpha_g^*)^2 \end{aligned}$$ For example, with $N=3$, $N_f=1$, and $\epsilon=1$, numerical estimates yield $\eta_\psi \approx 0.312$, $\eta_T \approx 0.373$, $\nu^{-1} \approx 1.042$, distinguishing the SO(3) tensorial universality class (Han et al., 2024).

4. Mass-Gap Ratios and Symmetry-Broken Phases

In the ordered (symmetry-broken) phase for $N=3$, a unique bosonic mass $m_T$ and two fermion masses $m_\psi^{(1)}$, $m_\psi^{(2)} = 2\,m_\psi^{(1)}$ emerge. The universal mass-gap ratio is: $$\mathcal{R}_G = \frac{m_T^2}{m_\psi^2} = \frac{2\lambda^*}{\alpha_g^*}$$ For $N=3, N_f=1, \epsilon=1$, $\mathcal{R}_G \approx 1.874$, marking a distinctly tensorial feature of this universality class.

Minimizing the sextic invariant $\kappa_2 (T^3)^2$ selects the uniaxial nematic condensate $$\langle T_{ij} \rangle \propto \mathrm{diag}(1,1,-2)$$, breaking SO(3) $\to$ SO(2) and fully gapping the fermion spectrum.

5. Higher-Rank Generalizations and Melonic Large-$N$ Limit

For rank-3 and higher tensorial extensions, such as $O(N)^3$-invariant models, the dominant RG and CFT structure are dictated by “melonic” diagrams in the large-$N$ limit. The action involves both quartic Yukawa ($\phi^2 \bar\psi\psi$) and sextic bosonic ($\phi^6$) interactions, with the melonic limit taken via specific $N$-scaling of couplings. The resulting Schwinger–Dyson equations yield IR scaling solutions:

• Melonic-Yukawa fixed point: $h=0, \lambda\neq 0$, fundamental fields have $\Delta_\phi = D/4$, $\Delta_\psi = (D-2)/4$.
• Bosonic-melonic fixed point: $\lambda=0, h\neq 0$, $\Delta_\phi = D/6$.
• Prismatic and supersymmetric fixed points: additional structures with intersecting bosonic and Yukawa sectors, manifest supersymmetry at special dimensionality (e.g., $D=3$).

Stable CFTs exist only within certain dimensional windows where all local singlet scaling dimensions are real, with instability indicated by complexification of operator spectra (Fraser-Taliente et al., 2024).

6. Stability, Loop Corrections, and Universality Constraints

For $N\geq 4$, the RG fixed-point structure is highly sensitive to loop corrections:

• One-loop analysis predicts $N_{f,c2}\approx N$, but two-loop and partial three-loop corrections lower $N_{f,c2}$, sharpening the domain of second-order transitions and narrowing the window for fluctuation-induced first-order behavior.
• The stability of quartic couplings and positivity of the effective scalar potential partition the theory space into regions with physically admissible, weakly first-order, and non-existent phase transitions.
• The slow convergence and large, alternating coefficients in the $\epsilon$-expansion for critical values ($N_{c1}, N_{c2}, N_{f,c2}$) necessitate advanced resummation techniques such as Borel–Padé procedures for quantitative precision (Han et al., 25 Jun 2025).

7. Generalizations, Extensions, and Applications

The tensorial quartic Yukawa framework extends naturally to general scalar-fermion models with arbitrary tensor structures, higher loops, and gauge interactions. The full set of RG $\beta$-functions for general marginal interactions has been tabulated up to 4-loop (gauge), 3-loop (Yukawa), and 2-loop (quartic scalar) order, supporting high-precision model building and universality class searches, including asymptotic safety scenarios and BSM scalar sector explorations (Davies et al., 2021).

Tensorial quartic Yukawa models unify phenomena observed in spin-orbital liquid transitions, Dirac materials, and emergent large-$N$ CFT behavior. The identification of distinct universality classes, such as the SO(3) tensorial type, and the role of higher-order invariants in ground state selection, mark these models as a central framework for contemporary studies of criticality in tensor field theories.