Abstract
We present new planar dyonic black hole solutions of the \( \mathfrak{s}\mathfrak{u}(N) \) Einstein-Yang-Mills equations in asymptotically anti-de Sitter space-time, focussing on \( \mathfrak{s}\mathfrak{u}(2) \) and \( \mathfrak{s}\mathfrak{u}(3) \) gauge groups. The magnetic part of the gauge field forms a condensate close to the planar event horizon. We compare the free energy of a non-Abelian hairy black hole with that of an embedded Reissner-Nordström-anti-de Sitter (RN-AdS) black hole having the same Hawking temperature and electric charge. We find that the hairy black holes have lower free energy. We present evidence that there is a phase transition at a critical temperature, above which the only solutions are embedded RN-AdS black holes. At the critical temperature, an RN-AdS black hole can decay into a hairy black hole, and it is thermodynamically favourable to do so. Working in the probe limit, we compute the frequency-dependent conductivity, and find that enlarging the gauge group from \( \mathfrak{s}\mathfrak{u}(2) \) to \( \mathfrak{s}\mathfrak{u}(3) \) eliminates a divergence in the conductivity at nonzero frequency.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M.S. Volkov and D.V. Gal’tsov, Gravitating non-Abelian solitons and black holes with Yang-Mills fields, Phys. Rept. 319 (1999) 1 [hep-th/9810070] [INSPIRE].
E. Winstanley, Classical Yang-Mills black hole hair in anti-de Sitter space, Lect. Notes Phys. 769 (2009) 49 [arXiv:0801.0527] [INSPIRE].
E. Winstanley, A menagerie of hairy black holes, arXiv:1510.01669 [INSPIRE].
M.S. Volkov, Hairy black holes in the XX-th and XXI-st centuries, arXiv:1601.08230 [INSPIRE].
D.V. Galtsov and A.A. Ershov, Non-Abelian baldness of colored black holes, Phys. Lett. A 138 (1989) 160 [INSPIRE].
P. Bizon and O.T. Popp, No hair theorem for spherical monopoles and dyons in SU(2) Einstein-Yang-Mills theory, Class. Quant. Grav. 9 (1992) 193 [INSPIRE].
M.S. Volkov and D.V. Galtsov, Non-Abelian Einstein-Yang-Mills black holes, JETP Lett. 50 (1989) 346 [Pisma Zh. Eksp. Teor. Fiz. 50 (1989) 312] [INSPIRE].
M.S. Volkov and D.V. Galtsov, Black holes in Einstein-Yang-Mills theory, Sov. J. Nucl. Phys. 51 (1990) 747 [Yad. Fiz. 51 (1990) 1171] [INSPIRE].
P. Bizon, Colored black holes, Phys. Rev. Lett. 64 (1990) 2844 [INSPIRE].
H.P. Kuenzle and A.K.M. Masood-ul Alam, Spherically symmetric static SU(2) Einstein-Yang-Mills fields, J. Math. Phys. 31 (1990) 928 [INSPIRE].
D.V. Galtsov and M.S. Volkov, Charged non-Abelian SU(3) Einstein-Yang-Mills black holes, Phys. Lett. B 274 (1992) 173 [INSPIRE].
B. Kleihaus, J. Kunz and A. Sood, SU(3) Einstein-Yang-Mills sphalerons and black holes, Phys. Lett. B 354 (1995) 240 [hep-th/9504053] [INSPIRE].
B. Kleihaus, J. Kunz and A. Sood, Charged SU(N ) Einstein-Yang-Mills black holes, Phys. Lett. B 418 (1998) 284 [hep-th/9705179] [INSPIRE].
B. Kleihaus, J. Kunz, A. Sood and M. Wirschins, Sequences of globally regular and black hole solutions in SU(4) Einstein-Yang-Mills theory, Phys. Rev. D 58 (1998) 084006 [hep-th/9802143] [INSPIRE].
N.E. Mavromatos and E. Winstanley, Existence theorems for hairy black holes in SU(N ) Einstein-Yang-Mills theories, J. Math. Phys. 39 (1998) 4849 [gr-qc/9712049] [INSPIRE].
W.H. Ruan, Hairy black hole solutions to SU(3) Einstein-Yang-Mills equations, Commun. Math. Phys. 224 (2001) 373 [INSPIRE].
N. Straumann and Z.H. Zhou, Instability of a colored black hole solution, Phys. Lett. B 243 (1990) 33 [INSPIRE].
D.V. Galtsov and M.S. Volkov, Instability of Einstein-Yang-Mills black holes, Phys. Lett. A 162 (1992) 144 [INSPIRE].
M.S. Volkov and D.V. Galtsov, Odd parity negative modes of Einstein-Yang-Mills black holes and sphalerons, Phys. Lett. B 341 (1995) 279 [hep-th/9409041] [INSPIRE].
S. Hod, Lifetime of unstable hairy black holes, Phys. Lett. B 661 (2008) 175 [arXiv:0803.0608] [INSPIRE].
G.V. Lavrelashvili and D. Maison, A remark on the instability of the Bartnik-McKinnon solutions, Phys. Lett. B 343 (1995) 214 [hep-th/9409185] [INSPIRE].
M.S. Volkov, O. Brodbeck, G.V. Lavrelashvili and N. Straumann, The number of sphaleron instabilities of the Bartnik-McKinnon solitons and non-Abelian black holes, Phys. Lett. B 349 (1995) 438 [hep-th/9502045] [INSPIRE].
O. Brodbeck and N. Straumann, Instability proof for Einstein-Yang-Mills solitons and black holes with arbitrary gauge groups, J. Math. Phys. 37 (1996) 1414 [gr-qc/9411058] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
E. Winstanley, Existence of stable hairy black holes in SU(2) Einstein-Yang-Mills theory with a negative cosmological constant, Class. Quant. Grav. 16 (1999) 1963 [gr-qc/9812064] [INSPIRE].
J. Bjoraker and Y. Hosotani, Stable monopole and dyon solutions in the Einstein-Yang-Mills theory in asymptotically anti-de Sitter space, Phys. Rev. Lett. 84 (2000) 1853 [gr-qc/9906091] [INSPIRE].
J. Bjoraker and Y. Hosotani, Monopoles, dyons and black holes in the four-dimensional Einstein-Yang-Mills theory, Phys. Rev. D 62 (2000) 043513 [hep-th/0002098] [INSPIRE].
J.E. Baxter, M. Helbling and E. Winstanley, Abundant stable gauge field hair for black holes in anti-de Sitter space, Phys. Rev. Lett. 100 (2008) 011301 [arXiv:0708.2356] [INSPIRE].
J.E. Baxter, M. Helbling and E. Winstanley, Soliton and black hole solutions of SU(N ) Einstein-Yang-Mills theory in anti-de Sitter space, Phys. Rev. D 76 (2007) 104017 [arXiv:0708.2357] [INSPIRE].
J.E. Baxter and E. Winstanley, On the existence of soliton and hairy black hole solutions of SU(N ) Einstein-Yang-Mills theory with a negative cosmological constant, Class. Quant. Grav. 25 (2008) 245014 [arXiv:0808.2977] [INSPIRE].
J.E. Baxter and E. Winstanley, On the stability of soliton and hairy black hole solutions of \( \mathfrak{s}\mathfrak{u}(N) \) Einstein-Yang-Mills theory with a negative cosmological constant, J. Math. Phys. 57 (2016) 022506 [arXiv:1501.07541] [INSPIRE].
O. Sarbach and E. Winstanley, On the linear stability of solitons and hairy black holes with a negative cosmological constant: the odd parity sector, Class. Quant. Grav. 18 (2001) 2125 [gr-qc/0102033] [INSPIRE].
E. Winstanley and O. Sarbach, On the linear stability of solitons and hairy black holes with a negative cosmological constant: the even parity sector, Class. Quant. Grav. 19 (2002) 689 [gr-qc/0111039] [INSPIRE].
R.B. Mann, E. Radu and D.H. Tchrakian, Non-Abelian solutions in AdS 4 and D = 11 supergravity, Phys. Rev. D 74 (2006) 064015 [hep-th/0606004] [INSPIRE].
B.L. Shepherd and E. Winstanley, Characterizing asymptotically anti-de Sitter black holes with abundant stable gauge field hair, Class. Quant. Grav. 29 (2012) 155004 [arXiv:1202.1438] [INSPIRE].
Z.-Y. Fan and H. Lü, SU(2)-colored (A)dS black holes in conformal gravity, JHEP 02 (2015) 013 [arXiv:1411.5372] [INSPIRE].
O. Kichakova, J. Kunz, E. Radu and Y. Shnir, Thermodynamic properties of asymptotically anti-de Sitter black holes in D = 4 Einstein-Yang-Mills theory, Phys. Lett. B 747 (2015) 205 [arXiv:1503.01268] [INSPIRE].
B.C. Nolan and E. Winstanley, On the existence of dyons and dyonic black holes in Einstein-Yang-Mills theory, Class. Quant. Grav. 29 (2012) 235024 [arXiv:1208.3589] [INSPIRE].
B.L. Shepherd and E. Winstanley, Dyons and dyonic black holes in \( \mathfrak{s}\mathfrak{u}(N) \) Einstein-Yang-Mills theory in anti-de Sitter spacetime, Phys. Rev. D 93 (2016) 064064 [arXiv:1512.03010] [INSPIRE].
B.C. Nolan and E. Winstanley, On the stability of dyons and dyonic black holes in Einstein-Yang-Mills theory, Class. Quant. Grav. 33 (2016) 045003 [arXiv:1507.08915] [INSPIRE].
J.E. Baxter, Existence of topological hairy dyons and dyonic black holes in anti-de Sitter \( \mathfrak{s}\mathfrak{u}(N) \) Einstein-Yang-Mills theory, J. Math. Phys. 57 (2016) 022505 [arXiv:1507.05314] [INSPIRE].
M.S. Volkov, Gravitating non-Abelian solitons and hairy black holes in higher dimensions, in Recent developments in theoretical and experimental general relativity, gravitation and relativistic field theories. Proceedings, 11th Marcel Grossmann Meeting, MG11, Berlin Germany July 23-29 2006, pg. 1379 [hep-th/0612219] [INSPIRE].
D. Birmingham, Topological black holes in anti-de Sitter space, Class. Quant. Grav. 16 (1999) 1197 [hep-th/9808032] [INSPIRE].
D.R. Brill, J. Louko and P. Peldan, Thermodynamics of (3 + 1)-dimensional black holes with toroidal or higher genus horizons, Phys. Rev. D 56 (1997) 3600 [gr-qc/9705012] [INSPIRE].
J.P.S. Lemos, Two-dimensional black holes and planar general relativity, Class. Quant. Grav. 12 (1995) 1081 [gr-qc/9407024] [INSPIRE].
J.P.S. Lemos, Cylindrical black hole in general relativity, Phys. Lett. B 353 (1995) 46 [gr-qc/9404041] [INSPIRE].
J.P.S. Lemos and V.T. Zanchin, Rotating charged black string and three-dimensional black holes, Phys. Rev. D 54 (1996) 3840 [hep-th/9511188] [INSPIRE].
L. Vanzo, Black holes with unusual topology, Phys. Rev. D 56 (1997) 6475 [gr-qc/9705004] [INSPIRE].
R.-G. Cai and Y.-Z. Zhang, Black plane solutions in four-dimensional space-times, Phys. Rev. D 54 (1996) 4891 [gr-qc/9609065] [INSPIRE].
R.B. Mann, Pair production of topological anti-de Sitter black holes, Class. Quant. Grav. 14 (1997) L109 [gr-qc/9607071] [INSPIRE].
W.L. Smith and R.B. Mann, Formation of topological black holes from gravitational collapse, Phys. Rev. D 56 (1997) 4942 [gr-qc/9703007] [INSPIRE].
R.B. Mann, Charged topological black hole pair creation, Nucl. Phys. B 516 (1998) 357 [hep-th/9705223] [INSPIRE].
J.J. Van der Bij and E. Radu, New hairy black holes with negative cosmological constant, Phys. Lett. B 536 (2002) 107 [gr-qc/0107065] [INSPIRE].
J.E. Baxter, On the existence of topological hairy black holes in \( \mathfrak{s}\mathfrak{u}(N) \) EYM theory with a negative cosmological constant, Gen. Rel. Grav. 47 (2015) 1829 [arXiv:1403.0171] [INSPIRE].
J.E. Baxter and E. Winstanley, Topological black holes in \( \mathfrak{s}\mathfrak{u}(N) \) Einstein-Yang-Mills theory with a negative cosmological constant, Phys. Lett. B 753 (2016) 268 [arXiv:1511.04955] [INSPIRE].
J.E. Baxter, Stable topological hairy black holes in \( \mathfrak{s}\mathfrak{u}(N) \) EYM theory with Λ < 0, arXiv:1507.03127 [INSPIRE].
S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].
C.P. Herzog, Lectures on holographic superfluidity and superconductivity, J. Phys. A 42 (2009) 343001 [arXiv:0904.1975] [INSPIRE].
G.T. Horowitz, Introduction to holographic superconductors, Lect. Notes Phys. 828 (2011) 313 [arXiv:1002.1722] [INSPIRE].
G.T. Horowitz, Surprising connections between general relativity and condensed matter, Class. Quant. Grav. 28 (2011) 114008 [arXiv:1010.2784] [INSPIRE].
M. Kaminski, Flavor superconductivity and superfluidity, Lect. Notes Phys. 828 (2011) 349 [arXiv:1002.4886] [INSPIRE].
S. Sachdev, What can gauge-gravity duality teach us about condensed matter physics?, Ann. Rev. Condensed Matter Phys. 3 (2012) 9 [arXiv:1108.1197] [INSPIRE].
F. Benini, Holography and condensed matter, Fortsch. Phys. 60 (2012) 810 [arXiv:1202.6008] [INSPIRE].
A. Salvio, Superconductivity, superfluidity and holography, J. Phys. Conf. Ser. 442 (2013) 012040 [arXiv:1301.0201] [INSPIRE].
D. Musso, Introductory notes on holographic superconductors, PoS(Modave 2013)004 [arXiv:1401.1504] [INSPIRE].
R.-G. Cai, L. Li, L.-F. Li and R.-Q. Yang, Introduction to holographic superconductor models, Sci. China Phys. Mech. Astron. 58 (2015) 060401 [arXiv:1502.00437] [INSPIRE].
S.S. Gubser, Colorful horizons with charge in anti-de Sitter space, Phys. Rev. Lett. 101 (2008) 191601 [arXiv:0803.3483] [INSPIRE].
S.S. Gubser and S.S. Pufu, The gravity dual of a p-wave superconductor, JHEP 11 (2008) 033 [arXiv:0805.2960] [INSPIRE].
R.E. Arias and I.S. Landea, Backreacting p-wave superconductors, JHEP 01 (2013) 157 [arXiv:1210.6823] [INSPIRE].
M.M. Roberts and S.A. Hartnoll, Pseudogap and time reversal breaking in a holographic superconductor, JHEP 08 (2008) 035 [arXiv:0805.3898] [INSPIRE].
C.P. Herzog, K.-W. Huang and R. Vaz, Linear resistivity from non-Abelian black holes, JHEP 11 (2014) 066 [arXiv:1405.3714] [INSPIRE].
S. Gangopadhyay and D. Roychowdhury, Analytic study of properties of holographic p-wave superconductors, JHEP 08 (2012) 104 [arXiv:1207.5605] [INSPIRE].
R.E. Arias and I.S. Landea, Hydrodynamic modes of a holographic p-wave superfluid, JHEP 11 (2014) 047 [arXiv:1409.6357] [INSPIRE].
H.-B. Zeng, Z.-Y. Fan and H.-S. Zong, Superconducting coherence length and magnetic penetration depth of a p-wave holographic superconductor, Phys. Rev. D 81 (2010) 106001 [arXiv:0912.4928] [INSPIRE].
H.-B. Zeng, W.-M. Sun and H.-S. Zong, Supercurrent in p-wave holographic superconductor, Phys. Rev. D 83 (2011) 046010 [arXiv:1010.5039] [INSPIRE].
P. Basu, J. He, A. Mukherjee and H.-H. Shieh, Hard-gapped holographic superconductors, Phys. Lett. B 689 (2010) 45 [arXiv:0911.4999] [INSPIRE].
S.S. Gubser, F.D. Rocha and A. Yarom, Fermion correlators in non-Abelian holographic superconductors, JHEP 11 (2010) 085 [arXiv:1002.4416] [INSPIRE].
G.L. Giordano, N.E. Grandi and A.R. Lugo, Fermionic spectral functions in backreacting p-wave superconductors at finite temperature, arXiv:1610.04268 [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].
C.P. Herzog and S.S. Pufu, The second sound of SU(2), JHEP 04 (2009) 126 [arXiv:0902.0409] [INSPIRE].
A. Akhavan and M. Alishahiha, p-wave holographic insulator/superconductor phase transition, Phys. Rev. D 83 (2011) 086003 [arXiv:1011.6158] [INSPIRE].
R.-G. Cai, Z.-Y. Nie and H.-Q. Zhang, Holographic phase transitions of p-wave superconductors in Gauss-Bonnet gravity with back-reaction, Phys. Rev. D 83 (2011) 066013 [arXiv:1012.5559] [INSPIRE].
J. Erdmenger, D. Fernandez and H. Zeller, New transport properties of anisotropic holographic superfluids, JHEP 04 (2013) 049 [arXiv:1212.4838] [INSPIRE].
R. Manvelyan, E. Radu and D.H. Tchrakian, New AdS non-Abelian black holes with superconducting horizons, Phys. Lett. B 677 (2009) 79 [arXiv:0812.3531] [INSPIRE].
P.T. Chrusciel and W. Kondracki, Some global charges in classical Yang-Mills theory, Phys. Rev. D 36 (1987) 1874 [INSPIRE].
R.A. Brandt and F. Neri, Magnetic monopoles in SU(N ) gauge theories, Nucl. Phys. B 186 (1981) 84 [INSPIRE].
P. Forgacs and N.S. Manton, Space-time symmetries in gauge theories, Commun. Math. Phys. 72 (1980) 15 [INSPIRE].
P.G. Bergmann and E.J. Flaherty, Symmetries in gauge theories, J. Math. Phys. 19 (1978) 212 [INSPIRE].
J.P. Harnad, L. Vinet and S. Shnider, Group actions on principal bundles and invariance conditions for gauge fields, J. Math. Phys. 21 (1980) 2719 [INSPIRE].
B.L. Shepherd, Einstein-Yang-Mills black holes in anti-de Sitter space, Ph.D. thesis, University of Sheffield, Sheffield U.K. (2012).
W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical recipes in FORTRAN: the art of scientific computing, Cambridge University Press, Cambridge U.K. (1992).
P. Alken et al., GNU scientific library reference manual, third ed., Network Theory Ltd., U.K. (2009).
H.-M. Chan and S.T. Tsou, On the characterization of monopoles in non-Abelian gauge theories, Phys. Lett. B 95 (1980) 395 [INSPIRE].
J.D.E. Creighton and R.B. Mann, Quasilocal thermodynamics of dilaton gravity coupled to gauge fields, Phys. Rev. D 52 (1995) 4569 [gr-qc/9505007] [INSPIRE].
P. Goddard, J. Nuyts and D.I. Olive, Gauge theories and magnetic charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].
B. Kleihaus, J. Kunz, A. Sood and M. Wirschins, Horizon properties of Einstein-Yang-Mills black holes, Phys. Rev. D 65 (2002) 061502 [gr-qc/0110084] [INSPIRE].
D. Sudarsky and R.M. Wald, Extrema of mass, stationarity and staticity and solutions to the Einstein-Yang-Mills equations, Phys. Rev. D 46 (1992) 1453 [INSPIRE].
J. Tafel and A. Trautman, Can poles change color?, J. Math. Phys. 24 (1983) 1087 [INSPIRE].
C.H. Oh, C.P. Soo and C.H. Lai, Global gauge transformations and conserved, gauge invariant electric and magnetic charges in Yang-Mills gauge theories, Phys. Rev. D 36 (1987) 2532 [INSPIRE].
A. Corichi and D. Sudarsky, Mass of colored black holes, Phys. Rev. D 61 (2000) 101501 [gr-qc/9912032] [INSPIRE].
B. Kleihaus, J. Kunz and F. Navarro-Lerida, Rotating Einstein-Yang-Mills black holes, Phys. Rev. D 66 (2002) 104001 [gr-qc/0207042] [INSPIRE].
V. Balasubramanian and P. Kraus, A stress tensor for anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1611.04162
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as 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.
About this article
Cite this article
Shepherd, B.L., Winstanley, E. Black holes with \( \mathfrak{s}\mathfrak{u}(N) \) gauge field hair and superconducting horizons. J. High Energ. Phys. 2017, 65 (2017). https://doi.org/10.1007/JHEP01(2017)065
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2017)065