Entanglement of Gauge Theories: from the Toric Code to the $\mathbb{Z}_2$ Lattice Gauge Higgs Model

The toric code (TC) model subjected to a magnetic field can be mapped to the $\mathbb{Z}_2$ lattice gauge Higgs ($\mathbb{Z}_2$ GH) model. Although this isometric mapping preserves the bulk energy spectrum, here, we show that it has a non-trivial effect on the entanglement structure. We derive a quantum channel that allows us to obtain the reduced density matrix of the $\mathbb{Z}_2$ GH model from the one of the TC model. We then contrast the ground state entanglement spectra (ES) of the two models. Analyzing the role of the electric-magnetic duality, we show that while the ES of the TC model is enriched by the duality, the ES of the $\mathbb{Z}_2$ GH model is in fact not. This thus represents an example where the bulk-boundary correspondence fails. Moreover, the quantum channel allows us to investigate the entanglement distillation of the $\mathbb{Z}_2$ GH model from the TC model.