  We report on a numerical study of real-time dynamics of electromagnetically
interacting chirally imbalanced lattice Dirac fermions within the classical
statistical field theory approach. Namely, we perform exact simulations of the
real-time quantum evolution of fermionic fields coupled to classical
electromagnetic fields, which are in turn coupled to the vacuum expectation
value of the fermionic electric current. We use Wilson-Dirac Hamiltonian for
fermions, and non-compact action for the gauge field. In general, we observe
that the backreaction of fermions on the electromagnetic field prevents the
system from acquiring chirality imbalance. In the case of chirality pumping in
parallel electric and magnetic fields, electric field is screened by the
produced on-shell fermions and the accumulation of chirality is hence stopped.
In the case of evolution with initially present chirality imbalance, axial
charge tends to transform to helicity of electromagnetic field. By performing
simulations on large lattices we show that in most cases this decay process is
accompanied by the inverse cascade phenomenon which transfers energy from
short-wavelength to long-wavelength electromagnetic fields. In some
simulations, however, we observe a very clear signature of inverse cascade for
the helical magnetic fields which is not accompanied by the axial charge decay.
This suggests that the relation between inverse cascade and axial charge decay
is not as straightforward as predicted by the simplest form of anomalous
Maxwell equations.
