Multi-critical behaviour of 4-dimensional tensor models up to order 6

TL;DR

This paper investigates the critical behavior of 4-dimensional tensor models with various interactions, revealing a universal two-phase structure and emphasizing the need for new interactions to achieve higher-dimensional geometries.

Contribution

It systematically analyzes the enhancement scaling, diagram counting, and multi-critical behavior of tensor models with complex interactions, including $K_{3,3}$-type, at order 6.

Findings

Two-phase structure of branched-polymer and 2d gravity regimes is common in these models.

Both necklace and $K_{3,3}$ interactions generate planar regimes.

Universality of mixed phases at criticality suggests the need for higher-order interactions.

Abstract

Tensor models generalize the matrix-model approach to 2-dimensional quantum gravity to higher dimensions. Some models allowing a $1/N$ expansion have been explored, most of them generating branched-polymer geometries. Recently, enhancements yielding an additional 2d quantum-gravity (planar) phase and an intermediate regime of proliferating baby-universes have been found. It remains an open issue to find models escaping these lower dimensionality universality classes. Here we analyse the dominant regime and critical behaviour of a range of new models which are candidates for such effective geometries, in particular interactions based on the utility graph $K_{3,3}$. We find that, upon proper enhancement, the two-phase structure of a branched-polymer and a 2d gravity regime is the common case in $U(N)$-invariant rank $D=4$ tensor models of small orders. Not only the well known so-called necklace interactions but also $K_{3,3}$-type interactions turn out as the source for the planar regime. We give a systematic account of the enhancement scaling, the counting of leading-order diagrams and the multi-critical behaviour of a wide range of interactions, in particular for all order-6 interactions of rank 3 and 4. These findings support the claim of universality of such mixtures of branched-polymer and planar diagrams at criticality. In particular, this hints at the necessity to consider new ingredients, or interactions of higher order and rank, in order to obtain higher dimensional continuum geometry from tensor models.