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 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.
Highlights
According to complexity action (CA) proposal of Susskind [5,6,7], the complexity of a boundary state is dual to a value of gravitational action in a special part of spacetime, known as WDW patch, which interestingly contains the black hole interior
We study Nielsen’s circuit complexity for a charged thermofield double state of free complex scalar quantum field theory in the presence of background electric field
It is shown by Lloyd [8] that the complexity growth rate for a given state is smaller than twice the average energy of the system at that state and intriguingly the CA proposal gives the same value for neutral black holes at late times [6, 9, 10]
Summary
Since each mode k is decoupled from the other modes in (2.11), the respective cTFD state will be the product of TFD states for each of the oscillators. The complexity of a target state in a basis independent way can be achieved by defining the relative covariance matrix. One way to estimate the length of shortest path between the two states is to consider orthonormal symplectic group generators (with respect to Frobenius inner product) as our gates and calculate the length of the path that minimizes F2 cost function, i.e. the linear path. Since this path is not necessary the minimal geodesic of F1, .
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