Critical and multicritical points in the three-dimensional Z2 gauge theory with matter -- an information bottleneck perspective

Lattice gauge theories are a subject of great importance to many domains of physics, from particle physics to quantum computing. Despite extensive research, many open problems still exist. In particular, the recent interest in self-dual gauge theories has brought to the forefront the old question of RG treatment of such systems, or, more concretely put, how to locally coarse-grain discrete gauge fields. Here we apply the recently proposed Real Space Mutual Information (RSMI) approach to the 3D Ising gauge theory with matter. This technique, employing neural-network-based estimation and information bottlenecks, allows us to peer into the structure of the critical and multi-critical points in the theory. It provides direct access to leading and sub-leading operators, along with their symmetries and degeneracies. While this type of data is accessible for 2D critical systems via direct diagonalization techniques, for 3D critical systems we are not aware of a competing method capable of extracting such information in practice. We will briefly discuss the bearing of our numerical findings on the conjecture that the multi-critical point in this 3D system is an emergent XY theory.