Perfectly hidden order and Z2 confinement transition in a fully packed monopole liquid

We investigate a variant of spin ice whose degenerate ground states are densely packed monopole configurations. An applied field drives this model through a Z2 confinement transition. This phase change is a variant of the U(1) Kasteleyn transition, but instead of saturated order the system has fluctuations in the confined phase, and shows critical scaling on both sides of the transition. Remarkably, the magnetic response scales with the critical exponent of the specific heat in the 3D Ising universality class. We prove this universality using a Kramers--Wannier duality to map to an Ising model. The dual order parameter maps back to a non-local string order parameter in the original model, invisible to any local probe. Further, we describe the transition in terms of a bosonic field theory including a pairing term.