Holographic complexity growth rate in Horndeski theory
- 86 Downloads
Based on the context of complexity = action (CA) conjecture, we calculate the holographic complexity of AdS black holes with planar and spherical topologies in Horndeski theory. We find that the rate of change of holographic complexity for neutral AdS black holes saturates the Lloyd’s bound. For charged black holes, we find that there exists only one horizon and thus the corresponding holographic complexity can’t be expressed as the difference of some thermodynamical potential between two horizons as that of Reissner–Nordstrom AdS black hole in Einstein–Maxwell theory. However, the Lloyd’s bound is not violated for charged AdS black hole in Horndeski theory.
However, it was pointed out that the action growth expression is different for charged black hole with a single horizon in the Einstein–Maxwell–Dilaton and Born–Infeld theories . It is thus worthwhile taking a further step to explore the pattern of the action growth of charged AdS black holes with only one horizon in higher derivative gravity theory. In this paper, we shall study the action growth of AdS black holes in Horndeski gravity theory.
Horndeski theory is a kind of higher derivative scalar-tensor theory which has the similar property of Lovelock gravity, that the Largrangian involves terms which are more than two derivatives, but the equations of motion are consisted of terms which have at most two derivatives acting on each field . AdS black holes have been constructed in Horndeski gravities in [38, 39] and their thermodynamics were studied in [40, 41]. The stability and causality of these black holes were carried out in [42, 43, 44]. Holographic application of Horndeski theory were investigated in [45, 46, 47, 48, 49, 50, 51, 52], especially, it was shown in  that although there is no holographic a-theorem for general Horndeski gravity, there does exist a critical point in parameter space where the holographic a-theorem can be achieved, which suggests the Horndeski theory should have a holographic field theory dual. The AdS black holes we studied in this paper are in this critical point.
We shall study the action growth of unchanged black holes in \(D=4\) Horndeksi gravity in Sect. 2, and in Sect. 3, we compute the action growth of the charged black holes in the four dimensional Einstein–Maxwell–Horndeski gravity. We conclude our results in Sect. 4.
2 Neutral black holes in Horndeski theory
2.1 Planar black hole
2.2 Spherical black hole
3 Charged black holes in Horndeski theory
3.1 Planar black hole
3.2 Spherical black hole
In this paper, we studied the holographic complexity in Horndeski gravity theories through the “Complexity = Action” conjecture. In particular, we calculated the gravitational action growth of neutral and charged AdS black holes in Horndeski gravities. We found that the rate of change of action for neutral black holes with planar and spherical topologies is 2M , which is the same as the universal result [11, 12] and saturates the Lloyd’s bound .
The charged black holes are more complicated. We analysed the metric profiles carefully and found that the charged black holes with planar and spherical topology both have only one event horizon which are quite different from that of the RN black holes. We computed the gravitational action growth for the charged black holes with planar and spherical topologies. It turns out that the action growth for planar topology is less then 2 M thus satisfies the Lloyd’s bound. Whilst, for spherical case, we showed that the action growth is less than 2M when q is small. Though we did not prove analytically that the result holds for the whole range of parameters, we did numerically studies a substantial parameter choices and found that the action growth is less than 2M, which leads us to believe that the action growth is always less than 2M and satisfies the Lloyd’s bound.
Here, we just studied the late time rate of change of holographic complexity, it is worthwhile going a step further to see the effect of the higher-derivative non-minimally coupled Horndeski term to the complexity of formation , subregion complexity [16, 55, 56, 57] and also to explore full time dependence of the holographic complexity [58, 59, 60].
We are grateful to Luis Lehner and Hong Lu for useful discussion. H.-S.L. is grateful for hospitality at Yangzhou University, during the course of this work. X.-H.F. is supported in part by NSFC Grants No. 11475024 and No. 11875200. H.-S.L. is supported in part by NSFC Grants No. 11475148 and No. 11675144.
- 6.S. Sachdev, What can gauge-gravity duality teach us about condensed matter physics? Ann. Rev. Condens. Matter Phys. 3, 9 (2012). arXiv:1108.1197 [cond-mat.str-el]
- 7.J. McGreevy, TASI lectures on quantum matter (with a view toward holographic duality). arXiv:1606.08953 [hep-th]
- 19.Y. Zhao, Complexity and boost symmetry. arXiv:1702.03957 [hep-th]
- 22.Z.Y. Fan, M. Guo, Holographic complexity under a global quantum quench. arXiv:1811.01473 [hep-th]
- 24.S.A. Hosseini Mansoori, V. Jahnke, M.M. Qaemmaqami, Y.D. Olivas, Holographic complexity of anisotropic black branes. arXiv:1808.00067 [hep-th]
- 25.H. Ghaffarnejad, M. Farsam, E. Yaraie, Effects of quintessence dark energy on the action growth and butterfly velocity. arXiv:1806.05735 [hep-th]
- 26.E. Yaraie, H. Ghaffarnejad, M. Farsam, Complexity growth and shock wave geometry in AdS-Maxwell-power-Yang-Mills theory. arXiv:1806.07242 [gr-qc]
- 35.P.A. Cano, R.A. Hennigar, H. Marrochio, Complexity growth rate in lovelock gravity. arXiv:1803.02795 [hep-th]
- 56.C.A. Agn, M. Headrick, B. Swingle, Subsystem complexity and holography. arXiv:1804.01561 [hep-th]
- 57.S.J. Zhang, Subregion complexity and confinement-deconfinement transition in a holographic QCD model. arXiv:1808.08719 [hep-th]
- 59.Y.S. An, R.G. Cai, Y. Peng, Time dependence of holographic complexity in Gauss–Bonnet gravity. Phys. Rev. D. 98, 106013 (2018). arXiv:1805.07775 [hep-th]
- 60.S. Mahapatra, P. Roy, On the time dependence of holographic complexity in a dynamical Einstein–Dilaton model. arXiv:1808.09917 [hep-th]
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP3