Symmetry breaking in tensor models

TL;DR

This paper investigates a quartic tensor model with a focus on symmetry breaking at a critical point, revealing phase transitions that lead to different types of refined random geometries and surfaces.

Contribution

It introduces an analysis of symmetry breaking in tensor models, connecting phase transitions to the emergence of different geometric structures in the double scaling limit.

Findings

Critical point corresponds to a phase transition with symmetry breaking.

Symmetric phase relates to infinitely refined random surfaces.

Broken phase relates to infinitely refined random nodal surfaces.

Abstract

In this paper we analyze a quartic tensor model with one interaction for a tensor of arbitrary rank. This model has a critical point where a continuous limit of infinitely refined random geometries is reached. We show that the critical point corresponds to a phase transition in the tensor model associated to a breaking of the unitary symmetry. We analyze the model in the two phases and prove that, in a double scaling limit, the symmetric phase corresponds to a theory of infinitely refined random surfaces, while the broken phase corresponds to a theory of infinitely refined random nodal surfaces. At leading order in the double scaling limit planar surfaces dominate in the symmetric phase, and planar nodal surfaces dominate in the broken phase.