Motivated by ideas of fractionalization and topological order in bosonic models with short-range
interactions, we consider similar phenomena in formal lattice gauge theory models, which are
models where basic constituents are quantum lines. In the first example, we show that a compact
quantum electrodynamics (CQED) can have, besides familiar Coulomb and confined phases,
additional unusual confined phases where excitations are quantum lines carrying fractions of the
elementary unit of electric field strength; specifically, we construct a model that has $N$-tupled
monopole condensation and realizes 1/N fractionalization of the quantum Faraday lines. In the
second example, we consider a system consisting of two copies of CQED in (4+1)D and engineer
condensation of bound states of monopoles (which are quantum lines in four spatial dimensions)
and U(1) electric field lines. When the bound states contain a single monopole, we find lattice
gauge theory analogs of Symmetry Protected Topological phases, while when the bound states
contain multiple monopoles, we find analogs of Symmetry-Enriched Topological phases, where
we also have fractionalization of Faraday lines. The distinct character of these “topological”
phases of quantum lines is revealed by unusual response properties and physics at a boundary of a
spatial region in such a phase.
Fractionalization of Faraday lines in generalized compact quantum electrodynamics and SPT- and SET-like phases of quantum lines and particles.
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