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
R, each local Hilbert space is analogous to the one of a charged "particle" moving in the link-pair group space
R2 in a constant "magnetic" background field. In the compact case, the link-pair group space is a torus
U(1)2 threaded by
k units of quantized "magnetic" flux, with
k being the level of the Chern-Simons theory. The holonomies of the torus
U(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) to
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)-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
k2π. Non-Abelian
U(k) Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing.
We investigate how symmetry and topological order are coupled in the (
2+1)--dimensional
ZN rank-2 toric code for general
N, which is an exactly solvable point in the Higgs phase of a symmetric rank-2
U(1) gauge theory. The symmetry-enriched topological order present has a nontrivial realization of square-lattice translation (and rotation and reflection) symmetry, where anyons on different lattice sites have different types and belong to different superselection sectors. We call such particles ``position-dependent excitations.'' As a result, in the rank-2 toric code anyons can hop by one lattice site in some directions while only by
N lattice sites in others, reminiscent of fracton topological order in
3+1 dimensions. We find that while there are
N2 flavors of
e charges and
2N flavors of
m fluxes, there are not
NN2+2N anyon types. Instead, there are
N6 anyon types, and we can use Chern-Simons theory with six
U(1) gauge fields to describe all of them. While the lattice translations permute anyon types, we find that such permutations cannot be expressed as transformations on the six
U(1) gauge fields. Thus, the realization of translation symmetry in the
U6(1) Chern-Simons theory is not known. Despite this, we find a way to calculate the translation-dependent properties of the theory. In particular, we find that the ground-state degeneracy on an ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ torus is
N3gcd(Lx,N)gcd(Ly,N)gcd(Lx,Ly,N), where
gcd stands for ``greatest common divisor.'' We argue that this is a manifestation of UV/IR mixing which arises from the interplay between lattice symmetries and topological order.
Maxwell-Chern-Simons (MCS) theory in
2+1 dimensions provides a paradigmatic example of a topological gauge theory with both dynamical and topological degrees of freedom. Its Euclidean formulation suffers from a sign problem, making Hamiltonian numerical approaches particularly attractive. As a first step toward the non-perturbative Hamiltonian study of MCS theory, we investigate the constant mode sector on a spatial torus. Being analytically solvable in the continuum, it provides an ideal benchmark for understanding how the topological properties of the theory are encoded in a finite-dimensional lattice Hilbert space. We construct a finite-dimensional discretization of the torus of flat connections and show that the resulting lattice problem maps onto a generalized Harper-Hofstadter model with twisted boundary conditions. We identify the commensurability conditions under which the finite lattice exactly reproduces the magnetic translation algebra and the topological degeneracy of the continuum theory. A systematic analysis of gauge field truncation and its convergence toward the continuum limit is then presented.