Abstract
We study the holographic “complexity = action” (CA) and “complexity = volume” (CV) proposals in Einstein-dilaton gravity in all spacetime dimensions. We analytically construct an infinite family of black hole solutions and use CA and CV proposals to investigate the time evolution of the complexity. Using the CA proposal, we find dimensional dependent violation of the Lloyd bound in early as well as in late times. Moreover, depending on the parameters of the theory, the bound violation relative to the conformal field theory result can be tailored in the early times as well. In contrast to the CA proposal, the CV proposal in our model yields results similar to those obtained in the literature.
Highlights
Complexity of a state (B) with respect to a given initial state (A) is defined as the least possible number of unitary transformations required to construct the state B from A
We study the holographic “complexity = action” (CA) and “complexity = volume” (CV) proposals in Einstein-dilaton gravity in all spacetime dimensions
We analytically construct an infinite family of black hole solutions and use CA and CV proposals to investigate the time evolution of the complexity
Summary
We will briefly discuss the gravity solution corresponding to the Einsteindilaton system. We have an infinite family of analytic black hole solutions for the gravity system of eq (2.1) These different solutions correspond to different dilaton potentials, as different forms of A(r) will give different V (r). We will take a more general approach and consider different values of n and a ( different dilaton potential) to see the effects of different running dilaton profiles on the time evolution of the holographic complexity in different dimensions. It is important to mention that there exists another admissible solution to the Einstein-dilaton equations of motion which corresponds to thermal AdS. This solution is obtained by taking the rh → 0 limit, which translates to g(r) = 1. An interesting question, which we leave for future study, is to investigate how the time dependence of holographic complexity varies as we pass through the confinement/deconfinement critical point
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