# Investigating the holographic complexity in Einsteinian cubic gravity

## Abstract

In this paper, we investigate the holographic complexity of a small mass AdS black hole in Einsteinian cubic gravity by using the “complexity equals action” (CA) and “complexity equals volume” (CV) conjectures. In the CA context, the late-time growth rate satisfies the Lloyd bound for the \(k=0\) and \(k=1\) cases but it violates it for the \(k=-1\) case in the first-order approximation of the small mass parameter. However, by a full-time analysis, we find that this late-time limit is approached from above, which implies that this bound in all of these cases will be violated. In the CV context, we considered both the original and the generalized CV conjectures. Differing from the CA conjecture, the late-time rate here is non-vanishing in the zeroth-order approximation, and this shows that the Lloyd bound is exactly violated even in the late-time limit. These results show numerous differences from the neutral case of the Einstein gravity in both the CA and the CV holographic contexts where all of their late-time results saturate the Lloyd bound. These differences illustrate the influence of the higher curvature correction in Einstein gravity.

## 1 Introduction

*e.g.*, [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. As argued in [4], there is a bound on the complexity growth rate at late times,

While the properties of the complexity in Einstein gravity have been investigated in many situations for both CA and CV conjectures, relatively little is known about its behavior in higher curvature gravitational theories, especially its time-dependent behaviors. The holographic principle suggests that the higher-order corrections of the bulk action are dual to finite *N* and finite coupling effects in the boundary CFT. To study the holographic complexity for a higher curvature gravity could help us to investigate the hidden structures caused by the higher-order correction. Therefore, it is necessary for us to study the features of the holographic complexity in higher curvature gravity and show the influences of the higher curvature correction.

One of the most suitable higher curvature gravitational theories for holographic applications is the 4-dimensional Einsteinian cubic gravity, which provides a holographic toy model of a non-supersymmetric CFT in three dimensions. In [16], the authors have given a general formula for the full action for general \(F(R_{abcd})\) gravity and applied it to the calculation of the CA complexity in Einsteinian cubic gravity for a massless black hole solution. Their result shows that the late-time CA complexity growth rate, in this case, vanishes because the mass of the black hole also vanishes. To gain a better understanding of the holographic complexity in Einsteinian cubic gravity, in this paper, we consider a small mass AdS black hole solution and then study the time-dependent behaviors of the holographic complexity growth rate, and we take the Lloyd bound as the criterion to present the influence of the higher curvature term.

The structure of this paper is as follows. In Sect. 2, we review the critical Einsteinian cubic gravity and its small mass AdS solution. In Sect. 3, we apply CA conjecture to calculate the complexity growth rate of the small mass AdS black hole. In Sect. 3, we also study the holographic complexity by using the original CV conjecture and the generalized CV conjecture which is modified by the Wald entropy density. Finally, we conclude the paper in Sect. 5.

## 2 Critical Einsteinian cubic gravity

*m*. Here \(k={1, 0, -1}\) denotes the 2-dimensional spherical, planar, and hyperbolic geometry, respectively,

*L*is the AdS radius, and \(\mu \) is a dimensionless parameter. This solution describes an asymptotic AdS black hole with a horizon at \(r=r_h\). According to the blackening factor, we can see that this geometry has a curvature singular at the horizon under the first-order approximation of

*m*. However, this black hole is regular at the zeroth-order approximation. Therefore, we hope that the small mass correction does not affect the singularity at the horizon. This can be understood if the blackening factor is a nonanalytic function of the mass parameter, which means that when all corrections are concerned, this singularity will disappear.

## 3 CA conjecture

*m*. Then, by using these coordinates, the typical point \(r_1\) can be obtained by

*r*as the affine parameter, i.e., \(k_1^{a}=\left( \frac{\partial }{\partial r}\right) ^a\), which gives rise to the expansion \(\Theta =2/{r}\). Then the counter term of the past right null segment can be written

## 4 CV conjecture

*u*–

*r*plane, i.e.,

*u*, we can find the conserved quantity

*t*and the radius \(r_\text {min}\) of the turning point. From (38), we can further obtain

*t*can be regarded as a function of \(r_\text {min}\). Together with (40) and (42), the CA complexity growth rate can be further obtained,

*W*.

Finally, we show the time dependence of the holographic complexity for these two CV conjectures in Fig. 4 for the planar case \((k=0)\). From this figure, we can see that both of the CV holographic complexities increase with time and finally saturate the late-time value, which shows similar behaviors to the CV conjecture in the neutral case for Einstein gravity.

## 5 Conclusion

According to the calculation for Einsteinian cubic gravity in [16], the authors found that the late-time CA complexity growth rate of the massless black hole vanishes. Their result is in agreement with the Lloyd bound since the mass of this black hole also vanishes. In this paper, in order to gain a better understanding of the holographic complexity in Einsteinian cubic gravity, we study the holographic complexity growth rate of a small mass AdS black hole with both CA and CV conjectures. In the CA context, we find that the late-time growth rate only contains the first-order contributions of \(\delta m\). Moreover, different from the neutral case of the Einstein gravity, where the late-time growth rate of complexity saturates the bound, here the late-time result shows other features. For the planar case \((k=0)\), it saturates the Lloyd bound, for the spherical case \((k=1)\), the late-time value is lower than the saturated value of this bound, while for the hyperbolic case \((k=-1)\), it will violate the bound. By showing the time dependence of the growth rate, we find that the late-time value is approached, i.e., the complexity has a non-zero growth rate even for the massless case, which might imply that the Lloyd bound will also be violated after the time evolution of the complexity is taken into account. In the CV context, we considered both the original and the generalized CV conjectures. Different from the CA results, its late-time limit growth rate has non-vanishing zeroth-order contributions of *m*, which means that the Lloyd bound will be violated even though we only consider the late-time limit. The above results show numerous differences from the neutral case of the Einstein gravity in both the CA and CA contexts where all of their late-time results saturate the Lloyd bound. These illustrate the influence of the higher curvature correction in the Einstein gravity.

## Notes

### Acknowledgements

This research was supported by National Natural Science Foundation of China (NSFC) with Grants No. 11675015, the Cultivating Program of Excellent Innovation Team of Chengdu University of Technology (Grant No. KYTD201704), the Cultivating Program of Middle-aged Backbone Teachers of Chengdu University of Technology (Grant No. 10912-2019KYGG01511), the Open Research Fund of Computational Physics Key Laboratory of Sichuan Province, Yibin University (Grant No. JSWL2018KFZ01).

## References

- 1.L. Susskind, Fortsch. Phys.
**64**, 24 (2016)ADSCrossRefGoogle Scholar - 2.S. Aaronson, arXiv:1607.05256
- 3.D. Stanford, L. Susskind, Phys. Rev. D
**90**, 126007 (2014)ADSCrossRefGoogle Scholar - 4.A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle, Y. Zhao, Phys. Rev. Lett.
**116**, 191301 (2016)ADSCrossRefGoogle Scholar - 5.A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle, Y. Zhao, Phys. Rev. D
**93**, 086006 (2016)ADSMathSciNetCrossRefGoogle Scholar - 6.D.A. Roberts, D. Stanford, L. Susskind, Localized shocks. JHEP
**1503**, 051 (2015). https://doi.org/10.1007/JHEP0(2015)051 - 7.L. Susskind, Y. Zhao, arXiv:1408.2823
- 8.J. Jiang, Phys. Rev. D
**98**, 086018 (2018)ADSMathSciNetCrossRefGoogle Scholar - 9.R.G. Cai, S.M. Ruan, S.J. Wang, R.Q. Yang, R.H. Peng, Action growth for AdS black holes. JHEP
**1609**, 161 (2016). https://doi.org/10.1007/JHEP09(2016)161 - 10.L. Lehner, R.C. Myers, E. Poisson, R.D. Sorkin, Phys. Rev. D
**94**, 084046 (2016)ADSMathSciNetCrossRefGoogle Scholar - 11.R.A. Jefferson, R.C. Myers, JHEP
**1710**, 107 (2017)ADSCrossRefGoogle Scholar - 12.K. Hashimoto, N. Iizuka, S. Sugishita, Phys. Rev. D
**96**, 126001 (2017)ADSMathSciNetCrossRefGoogle Scholar - 13.D. Carmi, S. Chapman, H. Marrochio, R.C. Myers, S. Sugishita, JHEP
**1711**, 188 (2017)ADSCrossRefGoogle Scholar - 14.J. Jiang, J. Shan, J. Yang, arXiv:1810.00537
- 15.W. Cottrell, M. Montero, Complexity is simple!. JHEP
**1802**, 039 (2018)ADSMathSciNetCrossRefGoogle Scholar - 16.J. Jiang, H. Zhang, Phys. Rev. D
**99**, 086005 (2019)ADSMathSciNetCrossRefGoogle Scholar - 17.J. Jiang, M. Zhang, arXiv:1905.07576
- 18.M. Guo, J. Hernandez, R.C. Myers, S.M. Ruan, arXiv:1807.07677
- 19.Z.Y. Fan, M. Guo, arXiv:1811.01473
- 20.Z.Y. Fan, M. Guo, JHEP
**1808**, 031 (2018)ADSCrossRefGoogle Scholar - 21.R.Q. Yang, Y.S. An, C. Niu, C.Y. Zhang, K.Y. Kim, arXiv:1809.06678
- 22.Y.S. An, R.G. Cai, Y. Peng, arXiv:1805.07775
- 23.R.Q. Yang, Y.S. An, C. Niu, C.Y. Zhang, K.Y. Kim, arXiv:1803.01797
- 24.Y.S. An, R.H. Peng, Phys. Rev. D
**97**, 066022 (2018)ADSMathSciNetCrossRefGoogle Scholar - 25.A. Reynolds, S.F. Ross, Class. Quant. Grav.
**34**, 175013 (2017)ADSCrossRefGoogle Scholar - 26.S. Chapman, H. Marrochio, R.C. Myers, JHEP
**1701**, 062 (2017)ADSCrossRefGoogle Scholar - 27.J. Jiang, X. Liu, Phys. Rev. D
**99**, 026011 (2019)ADSMathSciNetCrossRefGoogle Scholar - 28.J. Jiang, Eur. Phys. J. C
**79**, 130 (2019)ADSCrossRefGoogle Scholar - 29.J. Jiang, X.W. Li, arXiv:1903.05476
- 30.J. Jiang, B.X. Ge, Phys. Rev. D
**99**, 126006 (2019)ADSCrossRefGoogle Scholar - 31.X.H. Feng, H.S. Liu, arXiv:1811.03303
- 32.J. Jiang, B. Deng, X.W. Li, arXiv:1908.06565
- 33.D. Carmi, R.C. Myers, P. Rath, JHEP
**1703**, 118 (2017)ADSCrossRefGoogle Scholar - 34.B. Czech, Phys. Rev. Lett.
**120**, 031601 (2018)ADSMathSciNetCrossRefGoogle Scholar - 35.P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi, K. Watanabe, JHEP
**1711**, 097 (2017)ADSCrossRefGoogle Scholar - 36.M. Alishahiha, Phys. Rev. D
**92**, 126009 (2015)ADSMathSciNetCrossRefGoogle Scholar - 37.P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi, K. Watanabe, Phys. Rev. Lett.
**119**, 071602 (2017)ADSMathSciNetCrossRefGoogle Scholar - 38.A.R. Brown, L. Susskind, Phys. Rev. D
**97**, 086015 (2018)ADSMathSciNetCrossRefGoogle Scholar - 39.C.A. Agon, M. Headrick, B. Swingle, arXiv:1804.01561
- 40.O. Ben-Ami, D. Carmi, JHEP
**1611**, 129 (2016)ADSCrossRefGoogle Scholar - 41.S. Chapman, M.P. Heller, H. Marrochio, F. Pastawski, Phys. Rev. Lett.
**120**, 121602 (2018)ADSMathSciNetCrossRefGoogle Scholar - 42.Y. Zhao, Phys. Rev. D
**97**, 126007 (2018)ADSMathSciNetCrossRefGoogle Scholar - 43.Z.Y. Fan, H.Z. Liang, arXiv:1908.09310
- 44.Z.Y. Fan, M. Guo, Phys. Rev. D
**100**, 026016 (2019)ADSCrossRefGoogle Scholar - 45.Z. Fu, A. Maloney, D. Marolf, H. Maxfield, Z. Wang, JHEP
**02**, 072 (2018)ADSCrossRefGoogle Scholar - 46.L. Hackl, R.C. Myers, arXiv:1803.10638
- 47.M. Alishahiha, A. Faraji Astaneh, M.R. Mohammadi Mozaffar, A. Mollabashi, arXiv:1802.06740
- 48.J. Couch, S. Eccles, W. Fischler, M.L. Xiao, JHEP
**1803**, 108 (2018)ADSCrossRefGoogle Scholar - 49.B. Swingle, Y. Wang, arXiv:1712.09826
- 50.M. Moosa, JHEP
**1803**, 031 (2018)ADSMathSciNetCrossRefGoogle Scholar - 51.B. Chen, W.M. Li, R.Q. Yang, C.Y. Zhang, S.J. Zhang, arXiv:1803.06680
- 52.S. Lloyd, Nature
**406**, 1047 (2000)ADSCrossRefGoogle Scholar - 53.P. Bueno, P.A. Cano, Phys. Rev. D
**94**, 104005 (2016)ADSMathSciNetCrossRefGoogle Scholar - 54.X.H. Feng, H. Huang, Z.F. Mai, H. Lu, Phys. Rev. D
**96**, 104034 (2017)ADSMathSciNetCrossRefGoogle Scholar

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