Towards a phase diagram for spin foams

TL;DR

This paper introduces a systematic coarse-graining scheme for 3D spin foams using decorated tensor networks, enabling the study of continuum limits and phase diagrams in non-Abelian lattice gauge models.

Contribution

It develops the first tensor network coarse-graining algorithm for proper spin foams with non-Abelian gauge groups, handling gauge redundancy efficiently.

Findings

Phase diagrams for finite group models were obtained.

Identified phase transitions in models with non-trivial simplicity constraints.

The scheme mimics 4D gravity spin foam constructions.

Abstract

One of the most pressing issues for loop quantum gravity and spin foams is the construction of the continuum limit. In this paper, we propose a systematic coarse-graining scheme for three-dimensional lattice gauge models including spin foams. This scheme is based on the concept of decorated tensor networks, which have been introduced recently. Here we develop an algorithm applicable to gauge theories with non-Abelian groups, which for the first time allows for the application of tensor network coarse-graining techniques to proper spin foams. The procedure deals efficiently with the large redundancy of degrees of freedom resulting from gauge symmetry. The algorithm is applied to 3D spin foams defined on a cubical lattice which, in contrast to a proper triangulation, allows for non--trivial simplicity constraints. This mimics the construction of spin foams for 4D gravity. For lattice gauge models based on a finite group we use the algorithm to obtain phase diagrams, encoding the continuum limit of a wide range of these models. We find phase transitions for various families of models carrying non--trivial simplicity constraints.