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 thermo- dynamic 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 temper- ature. This discrepancy can be understood by the fact that holographic QFTs are actually strong interacting theories, not free ones.

中文翻译：

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带电热场双态的复杂性

我们研究了在存在背景电场的情况下，自由复标量量子场理论的带电热场双态 (cTFD) 的 Nielsen 电路复杂性。我们表明 cTFD 状态的形成复杂性与热力学熵的比率是有限的，它仅取决于温度和化学势。此外，这个比率平稳地接近实标量理论的值。我们将我们的场论计算与带电黑洞的全息复杂性进行比较，并确认用于中性情况的相同成本函数在存在 U(1) 背景场的情况下继续工作。对于 t > 0，cTFD 状态的复杂性随时间演变并与全息结果形成对比，它在逆温量级的时间后饱和。