Numerical and analytical studies of a matrix model with non-pairwise contracted indices

TL;DR

This paper investigates a matrix model with non-pairwise contracted indices related to the canonical tensor model, analyzing its phase transitions and configurations both analytically and numerically, and discussing implications for the tensor model.

Contribution

It provides the first analytical and numerical analysis of a matrix model with non-pairwise index contractions linked to the canonical tensor model, revealing phase and dimensional transitions.

Findings

Evidence for a continuous phase transition at a key consistency point.

Identification of dimensional transitions in configurations near the transition.

Analytical and numerical confirmation of the model's critical behavior.

Abstract

The canonical tensor model, which is a tensor model in the Hamilton formalism, can be straightforwardly quantized and has an exactly solved physical state. The state is expressed by a wave function with a generalized form of the Airy function. The simplest observable on the state can be expressed by a matrix model which contains non-pairwise index contractions. This matrix model has the same form as the one that appears when the replica trick is applied to the spherical $p$-spin model for spin glasses, but our case has different ranges of variables and parameters from the spin glass case. We analyze the matrix model analytically and numerically. We show some evidences for the presence of a continuous phase transition at the location required by a consistency condition of the canonical tensor model. We also show that there are dimensional transitions of configurations around the transition region. Implications of the results to the canonical tensor model are discussed.