Ising Spin Network States for Loop Quantum Gravity: a Toy Model for Phase Transitions

TL;DR

This paper introduces Ising spin network states in loop quantum gravity, linking quantum geometry to statistical physics models, and explores their phase structure and emergent geometry from correlations.

Contribution

It develops a new class of spin network states modeled on the Ising system, enabling the study of geometry emergence and phase transitions in quantum gravity.

Findings

Constructed 2D and 3D Ising spin network states from loop gravity operators.

Derived local Hamiltonian constraints characterizing these states.

Analyzed phase diagram and showed how distance emerges from correlations.

Abstract

Non-perturbative approaches to quantum gravity call for a deep understanding of the emergence of geometry and locality from the quantum state of the gravitational field. Without background geometry, the notion of distance should entirely emerge from the correlations between the gravity fluctuations. In the context of loop quantum gravity, quantum states of geometry are defined as spin networks. These are graphs decorated with spin and intertwiners, which represent quantized excitations of areas and volumes of the space geometry. Here, we develop the condensed matter point of view on extracting the physical and geometrical information out of spin network states: we introduce new Ising spin network states, both in 2d on a square lattice and in 3d on a hexagonal lattice, whose correlations map onto the usual Ising model in statistical physics. We construct these states from the basic holonomy operators of loop gravity and derive a set of local Hamiltonian constraints which entirely characterize our states. We discuss their phase diagram and show how the distance can be reconstructed from the correlations in the various phases. Finally, we propose generalizations of these Ising states, which open the perspective to study the coarse graining and dynamics of spin network states using well-known condensed matter techniques and results.