---
title: Tensorial Quartic Yukawa Model
url: https://www.emergentmind.com/topics/tensorial-quartic-yukawa-model
type: topic
---

# Tensorial Quartic Yukawa Model

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 [2411.16842][2506.20710][2410.09152].

## 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 [2411.16842][2506.20710][2410.09152].

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 [2411.16842][2506.20710].

## 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 $N$ [2506.20710].

## 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 [2411.16842].

## 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 [2410.09152].

## 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 [2506.20710].

## 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 [2110.05496].

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.

Source: https://www.emergentmind.com/topics/tensorial-quartic-yukawa-model