Lattice implementation of Abelian gauge theories with Chern-Simons
  number and an axion field
  Real time evolution of classical gauge fields is relevant for a number of
applications in particle physics and cosmology, ranging from the early Universe
to dynamics of quark-gluon plasma. We present a lattice formulation of the
interaction between a $shift$-symmetric field and some $U(1)$ gauge sector,
$a(x)\tilde{F}_{\mu\nu}F^{\mu\nu}$, reproducing the continuum limit to order
$\mathcal{O}(dx_\mu^2)$ and obeying the following properties: (i) the system is
gauge invariant and (ii) shift symmetry is exact on the lattice. For this end
we construct a definition of the {\it topological number density} $Q =
\tilde{F}_{\mu\nu}F^{\mu\nu}$ that admits a lattice total derivative
representation $Q = \Delta_\mu^+ K^\mu$, reproducing to order
$\mathcal{O}(dx_\mu^2)$ the continuum expression $Q = \partial_\mu K^\mu
\propto \vec E \cdot \vec B$. If we consider a homogeneous field $a(x) = a(t)$,
the system can be mapped into an Abelian gauge theory with Hamiltonian
containing a Chern-Simons term for the gauge fields. This allow us to study in
an accompanying paper the real time dynamics of fermion number non-conservation
(or chirality breaking) in Abelian gauge theories at finite temperature. When
$a(x) = a(\vec x,t)$ is inhomogeneous, the set of lattice equations of motion
do not admit however a simple explicit local solution (while preserving an
$\mathcal{O}(dx_\mu^2)$ accuracy). We discuss an iterative scheme allowing to
overcome this difficulty.
