  We study Nielsen's circuit complexity for a charged thermofield double state
(cTFD) of free complex scalar quantum field theory in the presence of
background electric field. We show that the ratio of the complexity of
formation for cTFD state to the thermodynamic entropy is finite and it depends
just on the temperature and chemical potential. Moreover, this ratio smoothly
approaches the value for real scalar theory. We compare our field theory
calculations with holographic complexity of charged black holes and confirm
that the same cost function which is used for neutral case continues to work in
presence of $U(1)$ background field. For $t>0$, the complexity of cTFD state
evolves in time and contrasts with holographic results, it saturates after a
time of the order of inverse temperature. This discrepancy can be understood by
the fact that holographic QFTs are actually strong interacting theories, not
free ones.
