From doubled Chern–Simons–Maxwell lattice gauge theory to extensions of the toric code

We regularize compact and non-compact Abelian Chern–Simons–Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group RR, each local Hilbert space is analogous to the one of a charged “particle” moving in the link-pair group space R2R2 in a constant “magnetic” background field. In the compact case, the link-pair group space is a torus U(1)2U(1)2 threaded by kk units of quantized “magnetic” flux, with kk being the level of the Chern–Simons theory. The holonomies of the torus U(1)2U(1)2 give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry from U(1)U(1) to Z(k)Z(k). The local Hilbert space of a link-pair then decomposes into representations of a magnetic translation group. In the pure Chern–Simons limit of a large “photon” mass, this results in a Z(k)Z(k)-symmetric variant of Kitaev’s toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields. Electric charges on the original lattice and on the dual lattice obey mutually anyonic statistics with the statistics angle 2πk. Non-Abelian U(k)U(k) Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing.