$SO(3)$-invariant phase of the $O(N)^3$ tensor model

TL;DR

This paper investigates an $SO(3)$-invariant phase in large-$N$ bosonic tensor models with $O(N)^3$ symmetry, revealing its significance in understanding the melonic large-$N$ limit and spontaneous symmetry breaking.

Contribution

It introduces an $SO(3)$-invariant solution expressed via Wigner $3jm$ symbols and demonstrates its role in justifying the melonic large-$N$ limit in a broken phase.

Findings

Identification of an $SO(3)$-invariant tensor solution

Connection between the solution's scaling and melonic large-$N$ limit

Insights into spontaneous symmetry breaking patterns

Abstract

We study classical and quantum (at large-$N$) field equations of bosonic tensor models with quartic interactions and $O(N)^3$ symmetry. Among various possible patterns of spontaneous symmetry breaking we highlight an $SO(3)$ invariant solution, with the tensor field expressed in terms of the Wigner $3jm$ symbol. We argue that such solution has a special role in the large-$N$ limit, as in particular its scaling in $N$ can provide an on-shell justification for the melonic large-$N$ limit of the two-particle irreducible effective action in a broken phase.