Complexity for Charged Thermofield Double States

TL;DR

This paper investigates the complexity of charged thermofield double states in free scalar quantum field theory, comparing field theory results with holographic predictions and analyzing their time evolution and dependence on physical parameters.

Contribution

It introduces the calculation of Nielsen's circuit complexity for charged thermofield double states in free scalar fields with background electric fields, extending previous neutral case analyses.

Findings

The complexity of formation ratio to entropy is finite and depends only on temperature and chemical potential.

The complexity evolution saturates after a time scale inversely proportional to temperature.

Holographic and field theory results agree for static properties but differ in time evolution due to interaction effects.

Abstract

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.