Tensor Large N and Melonic Dominance

Rank-three tensor models can have a controlled large-$N$ limit in which melonic graphs, rather than matrix-planar graphs, dominate. The result depends on the exact invariant and covariance. In the colored tetrahedral model below, a vertex carries $N^{-3/2}$, every closed strand face carries $N$, and a two-vertex melon adds three faces, preserving the leading power $N^3$.

Required background. Large-N limits, normalizations, and orders of limits supplies the declared tensor face count. Schwinger–Dyson identities supplies the functional identity used for the two-point equation.

Helpful background. Schwinger–Dyson hierarchies and inputs supplies the closure problem. Double-line counting supplies the matrix-genus argument that must not be imported into this model.

A colored tetrahedral model with fixed normalization

Use four distinct complex rank-three fields $\phi^A$, $A=0,1,2,3$. Every displayed lower-case index ranges from $1$ to $N$, and the Gaussian covariance is

$$\left\langle \phi^A_{abc}(x) \overline{\phi}^{\,B}_{a'b'c'}(y) \right\rangle_0 = \delta^{AB} \delta_{aa'}\delta_{bb'}\delta_{cc'} G_0(x-y),$$

so a propagator is $O(N^0)$. The slots carried by different colors correspond to the appropriate three of six independent $U(N)$ index groups. Write the tetrahedral interaction explicitly as

$$\begin{aligned} S_{\mathrm{int}} = \int\mathrm d^dx\, \biggl[ &\frac{g}{N^{3/2}}\, \phi^0_{abc} \phi^1_{ade} \phi^2_{fbe} \phi^3_{fdc}\\ &+ \frac{\bar g}{N^{3/2}}\, \overline\phi^{\,0}_{abc} \overline\phi^{\,1}_{ade} \overline\phi^{\,2}_{fbe} \overline\phi^{\,3}_{fdc} \biggr], \end{aligned}$$

with $g$ held fixed. Distinct colors remove an identical-field symmetry factor. Each propagator carries three strands and each interaction vertex contributes $N^{-3/2}$. If a connected graph has $V$ vertices and $F$ closed two-color strand faces, then

$$\mathcal A_{\mathcal G} \propto N^{F-\frac32V}.$$

This particular bosonic interaction is not bounded below on every real field direction. The equations below define its formal perturbative large-$N$ expansion, or a sector of a regulated/stabilized model whose additional invariants preserve the same leading two-point combinatorics. They are not a claim that the bare bosonic integral exists nonperturbatively without such data.

Melonic insertions preserve the leading power

Start from a propagator of color $0$. Replace it by two conjugate tetrahedral vertices joined by the three propagators of colors $1$, $2$, and $3$. The insertion adds

$$\Delta V=2$$

and one new closed face for each strand pair carried by the external color-$0$ line:

$$\Delta F=3.$$

Therefore

$$\Delta\!\left(F-\frac32V\right) = 3-\frac32(2) =0.$$

Iterating this operation generates melonic two-point graphs without changing their leading $N$ power. Closing a leading two-point graph gives an $O(N^3)$ vacuum graph, matching the $N^3$ tensor components.

Colored tensor graphs also admit a topological organization through several ribbon “jackets,” and the nonnegative Gurau degree measures suppression. Leading degree-zero graphs are melonic in this model Gurau 2011, §§2–4, pp. 831–842. This is not the matrix formula $N^{2-2h}$: a tensor graph has several jackets, and a graph can be melonic without being characterized by one surface genus.

Leading Schwinger–Dyson closure

Assume color symmetry,

$$\left\langle \phi^A_{abc}(x) \overline{\phi}^{\,B}_{a'b'c'}(y) \right\rangle = \delta^{AB}\delta_{aa'}\delta_{bb'}\delta_{cc'}G(x-y).$$

At leading order, cutting a color-$0$ propagator exposes the elementary melon. The three internal colors are distinct, so the self-energy in the displayed vertex convention is

$$\Sigma(x-y) = \lvert g\rvert^2G(x-y)^3.$$

Together with Dyson’s equation,

$$G(p)^{-1} = G_0(p)^{-1}-\Sigma(p),$$

this closes the leading two-point hierarchy. Vertex corrections and nonmelonic self-energies are suppressed by the tensor degree and powers of $1/N$.

The closure is combinatorial, not exact at finite $N$. It also does not determine a physical solution until one specifies the contour or stable completion, boundary conditions, spectral or reflection-positivity requirements, and which solution of the nonlinear equation is selected.

First application: the infrared scaling solution

Suppose the model has a translation-invariant critical regime in $d<4$ where the self-energy dominates a standard $p^2$ kinetic term. Then

$$G(p)\Sigma(p)\simeq-1.$$

Use the scaling ansatz

$$G(x) \sim \frac{b}{\lvert x\rvert^{2\Delta}}.$$

Fourier transformation gives

$$G(p)\sim p^{2\Delta-d}, \qquad \Sigma(p)\sim p^{6\Delta-d}.$$

For their product to be momentum independent,

$$8\Delta-2d=0, \qquad \Delta=\frac d4.$$

The self-energy then behaves as $p^{d/2}$, which dominates $p^2$ as $p\to0$ when $d<4$. This power-counting check is necessary. The coefficient $b$ and even the existence of a physical conformal solution depend on statistics, signs, stability, and boundary conditions; they cannot be inferred from melonic counting alone.

For an uncolored $O(N)^3$ tetrahedral tensor with a suitably normalized interaction, the same melonic two-point structure occurs, but additional pillow and double-trace invariants can be generated and must be treated in the renormalization problem. The solvable uncolored construction and its relation to SYK-like equations are developed in Klebanov and Tarnopolsky 2017, §§2–4.

Shared calculation. The large-N counting and topology map compares the melonic face increment with matrix Euler counting. The large-N scaling comparison records the model dependence and normalization warning.

Control and failure boundaries

The leading melonic equation is reliable only under the following checks:

• the interaction invariant and covariance are exactly those used in the face count;
• $g$ is held fixed with the displayed $N^{-3/2}$ vertex;
• the selected contour or added stabilizing interactions define the model;
• no alternative tensor invariant changes the leading degree;
• the $N\to\infty$ limit precedes any infrared limit whose enhancement has not been bounded;
• the nonlinear Schwinger–Dyson solution satisfies the model’s positivity, analyticity, and boundary conditions.

Changing the rank, symmetry group, covariance, or invariant can change the correct power of $N$ and the leading graph family. “Tensor large $N$” is a class of limits, not one universal expansion.

Common pitfalls

Drawing a tensor graph as a ribbon graph and assigning one genus. Three-strand colored graphs require their own face and degree count. Matrix Euler counting does not select melons.

Deriving the infrared exponent before checking dominance. The ansatz $\Delta=d/4$ is self-consistent against a $p^2$ term only for $d<4$ and only if a physical solution exists.

Ignoring bosonic stability. A formal melonic series is not a convergent functional integral. State the contour or stable completion before making nonperturbative claims.

Exercises

1. Verify the $N$ power of an elementary melonic insertion.

Solution

Two vertices supply $N^{-3}$ and three new closed faces supply $N^3$. Their product is $N^0$, equivalently

$$\Delta\!\left(F-\frac32V\right)=3-3=0.$$

2. Derive the infrared dimension $\Delta=d/4$.

Solution

If $G(x)\sim\lvert x\rvert^{-2\Delta}$, then $G(p)\sim p^{2\Delta-d}$. Since $\Sigma(x)\sim G(x)^3$, $\Sigma(p)\sim p^{6\Delta-d}$. Requiring $G(p)\Sigma(p)$ to be constant gives $8\Delta-2d=0$.

3. Why is a planar drawing not evidence of tensor leading order?

Solution

The tensor power is $N^{F-3V/2}$ and depends on colored strand faces. A two-dimensional projection can be crossing-free while having a subleading face count, and there is no single ribbon surface whose genus determines the answer.

References

• Gurau, R. (2011). “The 1/N Expansion of Colored Tensor Models.” Annales Henri Poincaré 12, 829–847. doi:10.1007/s00023-011-0118-z. Open PDF.
• Klebanov, I. R., and Tarnopolsky, G. (2017). “Uncolored Random Tensors, Melon Diagrams, and the Sachdev–Ye–Kitaev Models.” Physical Review D 95, 046004. doi:10.1103/PhysRevD.95.046004. Open PDF.