Abelian Topological Order on Lattice Enriched with Electromagnetic Background

In topological phases of matter, the interplay between intrinsic order and global symmetry is an interesting task. study orders with discrete symmetry, important systematic approach construction exactly soluble lattice models. However, for continuous in particular electromagnetic U(1), has been less systematically developed. this paper, we introduce a effective theories large class abelian on three-dimensional spacetime background. We discuss associated properties, including Hall conductivity spin-c nature Some these can be readily mapped to microscopic Hamiltonians spatial lattice; others may also shed light their possible Hamiltonian realizations. Our based gauging 1-form $$\mathbb {Z}$$ symmetries. naturally related continuum path integral (doubled) U(1) Chern–Simons theory, through latter’s formal description terms Deligne–Beilinson cohomology; when dropped, our reduced Dijkgraaf–Witten model orders, as expected.