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

Complexity for Charged Thermofield Double States