Abstract
We employ the "complexity equals action" conjecture to investigate the action growth rate for the charged and neutral AdS black branes of a holographic toy model consisting of Einstein-Maxwell theory in $d + 1$-dimensional bulk spacetime with $d - 1$ massless scalar fields which is called Einstein-Maxwell-Axion (EMA) theory. From the holographic point of view, the scalar fields source a spatially dependent field theory with momentum relaxation on the boundary, which is dual to the homogeneous and isotropic black branes. We find that the growth rate of the holographic complexity within the Wheeler-DeWitt (WDW) patch saturates the corresponding Lloyd's bound at the late time limit. Especially for the neutral AdS black branes, it will be shown that the complexity growth rate at late time vanishes for a particular value of relaxation parameter $\beta_{max}$ where the temperature of the black hole is minimal. Then, we investigate the transport properties of the holographic dual theory in the minimum temperature. A non-linear contribution of the axion field kinetic term in the context of k-essence model in the four-dimensional spacetime is considered as well. We also study the time evolution of the holographic complexity for the dyonic AdS black branes in this model.
Highlights
The AdS=CFT correspondence [1,2,3], as the most important realization of the holographic principle [4,5], relates a gravity theory in an asymptotically anti–de Sitter (AdS) spacetime in the bulk to a conformal field theory (CFT) without gravity living on the boundary of this spacetime
We find that there is a minimum temperature for which the complexity growth rate vanishes, just like what happens in the case of AdS black holes with hyperbolic geometry in Ref. [39]
We extended the study of holographic complexity via AdS black branes with momentum relaxation into (d þ 1)-dimensional EMA theory by using the complexity 1⁄4 action” (CA) conjecture
Summary
The AdS=CFT correspondence [1,2,3], as the most important realization of the holographic principle [4,5], relates a gravity theory in an asymptotically anti–de Sitter (AdS) spacetime in the bulk to a conformal field theory (CFT) without gravity living on the boundary of this spacetime. The AdS=CFT correspondence provides two proposals to compute the complexity of states in the boundary quantum field theory of two-sided AdS black holes. As the main purpose of this paper, we employ the CA conjecture to study holographic complexity and its time evolution in EMA theory with momentum relaxation by following the approach used in Ref. The structure of this paper is organized as follows: in Sec. II, we review the EMA theory with momentum relaxation and study the time evolution of the holographic complexity for charged/neutral AdSdþ black branes. In the context of the CA conjecture, we consider the WDW patch that includes null sheets bounding the bulk and joint terms, and investigate how the holographic complexity approaches the late-time limit on them. IV is dedicated to a brief summary and concluding remarks
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