Abstract
We consider anisotropic black holes in the context of holographic renormalization group (RG) flows. We construct an a-function that is stationary at the boundary and the horizon and prove that it is also monotonic in both the exterior and the interior of the black hole. In spite of the reduced symmetry, we find that the “radial” null energy condition is sufficient to ensure the existence of this monotonic a-function. After constructing the a-function, we explore a holographic anisotropic p-wave superfluid state as a concrete example and numerical testing grounds. In doing so, we find that the a-function exhibits nontrivial oscillations in the trans-IR regime while preserving monotonicity. We find evidence that such oscillations appear to drive the trans-IR flow into nontrivial fixed points. We conclude by briefly discussing how our work fits into both the broader program of holographic RG flow and quantum information approaches to probing the black hole interior.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
V. Balasubramanian and P. Kraus, Space-time and the holographic renormalization group, Phys. Rev. Lett. 83 (1999) 3605 [hep-th/9903190] [INSPIRE].
J. de Boer, E.P. Verlinde and H.L. Verlinde, On the holographic renormalization group, JHEP 08 (2000) 003 [hep-th/9912012] [INSPIRE].
J. de Boer, The Holographic renormalization group, Fortsch. Phys. 49 (2001) 339 [hep-th/0101026] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].
M. Fukuma, S. Matsuura and T. Sakai, Holographic renormalization group, Prog. Theor. Phys. 109 (2003) 489 [hep-th/0212314] [INSPIRE].
I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, IRMA Lect. Math. Theor. Phys. 8 (2005) 73 [hep-th/0404176] [INSPIRE].
I. Papadimitriou and K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes, JHEP 08 (2005) 004 [hep-th/0505190] [INSPIRE].
A. Frenkel, S.A. Hartnoll, J. Kruthoff and Z.D. Shi, Holographic flows from CFT to the Kasner universe, JHEP 08 (2020) 003 [arXiv:2004.01192] [INSPIRE].
E. Caceres, A. Kundu, A.K. Patra and S. Shashi, Trans-IR flows to black hole singularities, Phys. Rev. D 106 (2022) 046005 [arXiv:2201.06579] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
J.L. Cardy, Is There a c Theorem in Four-Dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Renormalization group flows from holography supersymmetry and a c theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [INSPIRE].
R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].
H. Casini and M. Huerta, A c-theorem for the entanglement entropy, J. Phys. A 40 (2007) 7031 [cond-mat/0610375] [INSPIRE].
H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].
B. Swingle, Entanglement does not generally decrease under renormalization, J. Stat. Mech. 1410 (2014) P10041 [arXiv:1307.8117] [INSPIRE].
D. Giataganas, U. Gürsoy and J.F. Pedraza, Strongly-coupled anisotropic gauge theories and holography, Phys. Rev. Lett. 121 (2018) 121601 [arXiv:1708.05691] [INSPIRE].
C.-S. Chu and D. Giataganas, c-Theorem for Anisotropic RG Flows from Holographic Entanglement Entropy, Phys. Rev. D 101 (2020) 046007 [arXiv:1906.09620] [INSPIRE].
M. Ghasemi and S. Parvizi, Constraints on anisotropic RG flows from holographic entanglement entropy, Phys. Rev. D 104 (2021) 086028 [arXiv:1907.01546] [INSPIRE].
S.S. Gubser and S.S. Pufu, The Gravity dual of a p-wave superconductor, JHEP 11 (2008) 033 [arXiv:0805.2960] [INSPIRE].
M. Ammon, J. Erdmenger, V. Grass, P. Kerner and A. O’Bannon, On Holographic p-wave Superfluids with Back-reaction, Phys. Lett. B 686 (2010) 192 [arXiv:0912.3515] [INSPIRE].
M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
J.L. Friedman, K. Schleich and D.M. Witt, Topological censorship, Phys. Rev. Lett. 71 (1993) 1486 [gr-qc/9305017] [INSPIRE].
R. Bousso, Z. Fisher, S. Leichenauer and A.C. Wall, Quantum focusing conjecture, Phys. Rev. D 93 (2016) 064044 [arXiv:1506.02669] [INSPIRE].
R. Bousso, Z. Fisher, J. Koeller, S. Leichenauer and A.C. Wall, Proof of the Quantum Null Energy Condition, Phys. Rev. D 93 (2016) 024017 [arXiv:1509.02542] [INSPIRE].
C. Hoyos, N. Jokela, J.M. Penín, A.V. Ramallo and J. Tarrío, Risking your NEC, JHEP 10 (2021) 112 [arXiv:2104.11749] [INSPIRE].
S.A. Hartnoll, G.T. Horowitz, J. Kruthoff and J.E. Santos, Diving into a holographic superconductor, SciPost Phys. 10 (2021) 009 [arXiv:2008.12786] [INSPIRE].
Y.-S. An, L. Li, F.-G. Yang and R.-Q. Yang, Interior structure and complexity growth rate of holographic superconductor from M-theory, JHEP 08 (2022) 133 [arXiv:2205.02442] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
M. Kulaxizi and A. Parnachev, Energy Flux Positivity and Unitarity in CFTs, Phys. Rev. Lett. 106 (2011) 011601 [arXiv:1007.0553] [INSPIRE].
S.A. Hartnoll, Wheeler-DeWitt states of the AdS-Schwarzschild interior, arXiv:2208.04348 [INSPIRE].
P. Caputa, D. Das and S.R. Das, Path integral complexity and Kasner singularities, JHEP 01 (2022) 150 [arXiv:2111.04405] [INSPIRE].
E.M. Lifshitz and I.M. Khalatnikov, Investigations in relativistic cosmology, Adv. Phys. 12 (1963) 185 [INSPIRE].
V.A. Belinsky, I.M. Khalatnikov and E.M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology, Adv. Phys. 19 (1970) 525 [INSPIRE].
V.A. Belinsky, I.M. Khalatnikov and E.M. Lifshitz, A General Solution of the Einstein Equations with a Time Singularity, Adv. Phys. 31 (1982) 639 [INSPIRE].
V. Balasubramanian and S.F. Ross, Holographic particle detection, Phys. Rev. D 61 (2000) 044007 [hep-th/9906226] [INSPIRE].
L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, The Black hole singularity in AdS/CFT , JHEP 02 (2004) 014 [hep-th/0306170] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 24 [arXiv:1403.5695] [INSPIRE].
D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action, and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].
A. Belin, R.C. Myers, S.-M. Ruan, G. Sárosi and A.J. Speranza, Does Complexity Equal Anything?, Phys. Rev. Lett. 128 (2022) 081602 [arXiv:2111.02429] [INSPIRE].
N. Engelhardt and A.C. Wall, Extremal Surface Barriers, JHEP 03 (2014) 068 [arXiv:1312.3699] [INSPIRE].
D. Carmi, S. Chapman, H. Marrochio, R.C. Myers and S. Sugishita, On the Time Dependence of Holographic Complexity, JHEP 11 (2017) 188 [arXiv:1709.10184] [INSPIRE].
R. Auzzi, S. Bolognesi, E. Rabinovici, F.I. Schaposnik Massolo and G. Tallarita, On the time dependence of holographic complexity for charged AdS black holes with scalar hair, JHEP 08 (2022) 235 [arXiv:2205.03365] [INSPIRE].
S. Cremonini and X. Dong, Constraints on renormalization group flows from holographic entanglement entropy, Phys. Rev. D 89 (2014) 065041 [arXiv:1311.3307] [INSPIRE].
R.-G. Cai, C. Ge, L. Li and R.-Q. Yang, Inside anisotropic black hole with vector hair, JHEP 02 (2022) 139 [arXiv:2112.04206] [INSPIRE].
E. Caceres, S. Shashi, H.-Y. Sun, Imprints of geometric phase transitions behind black holes, work in progress (2023).
M. Baggioli and O. Pujolas, Electron-Phonon Interactions, Metal-Insulator Transitions, and Holographic Massive Gravity, Phys. Rev. Lett. 114 (2015) 251602 [arXiv:1411.1003] [INSPIRE].
M. Baggioli, K.-Y. Kim, L. Li and W.-J. Li, Holographic Axion Model: a simple gravitational tool for quantum matter, Sci. China Phys. Mech. Astron. 64 (2021) 270001 [arXiv:2101.01892] [INSPIRE].
M. Baggioli and G. Frangi, Holographic supersolids, JHEP 06 (2022) 152 [arXiv:2202.03745] [INSPIRE].
M. Baggioli and B. Goutéraux, Colloquium: Hydrodynamics and holography of charge density wave phases, arXiv:2203.03298 [INSPIRE].
Y. Liu and H.-D. Lyu, Interior of helical black holes, JHEP 09 (2022) 071 [arXiv:2205.14803] [INSPIRE].
A. Donos and J.P. Gauntlett, Holographic striped phases, JHEP 08 (2011) 140 [arXiv:1106.2004] [INSPIRE].
A. Donos and J.P. Gauntlett, Holographic charge density waves, Phys. Rev. D 87 (2013) 126008 [arXiv:1303.4398] [INSPIRE].
S. Kobayashi, D. Mateos, S. Matsuura, R.C. Myers and R.M. Thomson, Holographic phase transitions at finite baryon density, JHEP 02 (2007) 016 [hep-th/0611099] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2209.06818
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Caceres, E., Shashi, S. Anisotropic flows into black holes. J. High Energ. Phys. 2023, 7 (2023). https://doi.org/10.1007/JHEP01(2023)007
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2023)007