Emergent Hamiltonians from boundary symmetries

We introduce a systematic framework for constructing emergent local commuting Hamiltonians in (d+1) dimensions from symmetries defined on a d-dimensional boundary. Starting from a generic d-dimensional symmetry, we gauge it by introducing layers of gauge fields carrying dual symmetries, which can in turn be gauged. The fixed points of this iterative gauging procedure yield (d+1)-dimensional stabilizer Hamiltonians built from local Gauss-law constraints. These Hamiltonians encode the emergent bulk order while realizing the original symmetry on the boundary.

I will show how, starting in one dimension, this construction reproduces familiar two-dimensional topological codes, such as the toric code, and naturally realizes all gapped boundary conditions as different one-dimensional symmetric phases at the boundary. The framework extends to higher-form and subsystem symmetries in arbitrary dimensions, yielding, for example, three-dimensional fracton models.

If time permits, I will also discuss how these symmetries can be classified within the framework of tensor networks.

Based on arXiv: 2403.07575, 2410.09044, and 2203.12563.