Coupled intertwiner dynamics: A toy model for coupling matter to spin foam models

TL;DR

This paper introduces a simple 2D toy model coupling matter (Ising model) to spin foam quantum gravity, revealing mutual influences and phase transition sensitivities that shed light on matter-gravity interactions in quantum gravity.

Contribution

It presents a novel coupled model of matter and spin foam quantum gravity, exploring their mutual dynamics and phase transitions through tensor network renormalization.

Findings

Ising phase transition temperature depends on background configurations

Matter can induce phase transitions in the gravitational background

Strong coupling occurs near both phase transitions

Abstract

The universal coupling of matter and gravity is one of the most important features of general relativity. In quantum gravity, in particular spin foams, matter couplings have been defined in the past, yet the mutual dynamics, in particular if matter and gravity are strongly coupled, are hardly explored, which is related to the definition of both matter and gravitational degrees of freedom on the discretisation. However extracting this mutual dynamics is crucial in testing the viability of the spin foam approach and also establishing connections to other discrete approaches such as lattice gauge theories. Therefore, we introduce a simple 2D toy model for Yang--Mills coupled to spin foams, namely an Ising model coupled to so--called intertwiner models defined for $\text{SU}(2)_k$. The two systems are coupled by choosing the Ising coupling constant to depend on spin labels of the background, as these are interpreted as the edge lengths of the discretisation. We coarse grain this toy model via tensor network renormalization and uncover an interesting dynamics: the Ising phase transition temperature turns out to be sensitive to the background configurations and conversely, the Ising model can induce phase transitions in the background. Moreover, we observe a strong coupling of both systems if close to both phase transitions.