  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.
