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 $\mathcal{O}(d{x}_{\mu }^2)$ 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 $Q={\mathrm{\Delta }}_{\mu }^{+}{K}^{\mu }$, reproducing to order $\mathcal{O}(d{x}_{\mu }^2)$ the continuum expression $Q = \partial\mu K^\mu \propto \vec E \cdot \vec B$\mathrm{.}Ifweconsiderahomogeneousfield$a(x) = a(t)$, 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 $\mathcal{O}(d{x}_{\mu }^2)$ accuracy). We discuss an iterative scheme allowing to overcome this difficulty.